Damanik, David (ed.); Gesztesy, Fritz (ed.); Yuditskii, Peter (ed.) Mini-workshop: Reflectionless operators: the Deift and Simon conjectures. Abstracts from the mini-workshop held October 22–28, 2017. (English) Zbl 1409.00082 Oberwolfach Rep. 14, No. 4, 2943-2985 (2017). Summary: Reflectionless operators in one dimension are particularly amenable to inverse scattering and are intimately related to integrable systems like KdV and Toda. Recent work has indicated a strong (but not equivalent) relationship between reflectionless operators and almost periodic potentials with absolutely continuous spectrum. This makes the realm of reflectionless operators a natural place to begin addressing Deift’s conjecture on integrable flows with almost periodic initial conditions and Simon’s conjecture on gems of spectral theory establishing correspondences between certain coefficient and spectral properties. MSC: 00B05 Collections of abstracts of lectures 00B25 Proceedings of conferences of miscellaneous specific interest 35Q53 KdV equations (Korteweg-de Vries equations) 35B15 Almost and pseudo-almost periodic solutions to PDEs 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations 47A55 Perturbation theory of linear operators 35-06 Proceedings, conferences, collections, etc. pertaining to partial differential equations 35P25 Scattering theory for PDEs 35R30 Inverse problems for PDEs 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Software:numericaluniversality; Genius; Proof General PDFBibTeX XMLCite \textit{D. Damanik} (ed.) et al., Oberwolfach Rep. 14, No. 4, 2943--2985 (2017; Zbl 1409.00082) Full Text: DOI References: [1] J. S. Christiansen, B. Simon, and M. Zinchenko, Asymptotics of Chebyshev Polynomials, I. Subsets of R, Invent. Math. 208 (2017), 217-245. · Zbl 1369.41031 [2] J. S. Christiansen, B. Simon, P. Yuditskii, and M. Zinchenko, Asymptotics of Chebyshev Polynomials, II. DCT subsets of R, Submitted preprint (2017). · Zbl 1426.41035 [3] V. Totik, Chebyshev constants and the inheritance problem, J. Approx. Theory 160 (2009), 187-201. · Zbl 1190.41002 [4] H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. in Math. 3 (1969), 127-232. Universality for numerical calculations with random data. Percy Deift (joint work with Govind Menon, Sheehan Olver, and Thomas Trogdon) In [5], the authors considered the computation of the eigenvalues of a random n×n matrix M using various standard algorithms. Let Σndenote the set of real n× n symmetric matrices. Associated with each algorithmA there is, in the discrete case, a map φ = φA: Σn→ Σnwith the properties • (isospectral) spec(φA(H)) = spec(H) • (convergence) the iterates Xk+1= φA(Xk), k≥ 0, X0= M , converge to a diagonal matrix X∞, Xk→ X∞as k→ ∞. and in the continuous case, there is a flow t→ X(t) ∈ Σnwith the properties • (isospectral) spec(X(t) = spec(X(0) 2950Oberwolfach Report 49/2017 • (convergence) the flow X(t), t ≥ 0, X(0) = M, converges to a diagonal matrix X∞, X(t)→ X∞as t→ ∞. In both cases, necessarily the diagonal entries of X∞are the eigenvalues of the given matrix M . The QR algorithm is a prime example of such a discrete algorithm, while the Toda algorithm is an example of the continuous case. In the discrete case, the authors in [5] recorded the number of steps (i.e. the stopping time) T (M ) = Tǫ,n,A,E(M ) for the algorithmA applied to a matrix M ∈ Σnchosen from an ensembleE, to compute the eigenvalues of M to an accuracy ǫ. They then plotted the histogram for the normalized stopping time τ (M ) = τǫ,nA,E(M ) = (T (M )− hT i)/σ wherehT i and σ2=h(T − hT i)2i denote the sample average and sample variance of the T (M )′s taken over a large number (approx.15,000) of matrices M chosen fromE. What they found was that, for ǫ and n in a suitable scaling range (ǫ small, n large), the histogram for τ was universal, independent of the ensembleE. The histogram does, however, depend on the algorithm. Similar phenomena were observed in the continuous case. Subsequently in [2] the authors raised the question of whether the universality results in [5] were limited to eigenvalue algorithms for real symmetric matrices, or whether they were present more generally in numerical computations. And indeed, the authors in [2] found similar universality results for a wide variety of numerical algorithms, including (a) other eigenvalue algorithms such as QR with shifts, the Jacobi eigenvalue algorithm, and also algorithms applied to complex Hermitian ensembles (b) the conjugate gradient and GMRES iterative algorithms to solve linear linear n× n systems Hx = b with H and b random (c) an iterative algorithm to solve the Dirichlet problem ∆u = 0 in a random star-shaped region Ω⊂ R2with random boundary data f on ∂Ω (d) a genetic algorithm to compute the equilibrium measure for orthogonal polynomials on the line. In [2] the authors also discussed similar universality results obtained by Bakhtin and Correll [1] in a series of experiments with live participants recording (e) decision making times for a specified task. All of the above results are numerical and experimental. In more recent work [5] [4] the authors have proved universality rigorously for a number of algorithms: They showed in particular that the limiting histograms for these algorithms can Reflectionless Operators: The Deift and Simon Conjectures2951 be expressed in terms of the distribution of the inverse of the difference of the two top (or bottom, depending on the algorithm) eigenvalues of the matrices. The proofs of these universality results rely on the very latest results in random matrix theory. Many problems remain open. In particular proving universality rigorously for more general algorithms, including those listed above (a) ...(e). References [6] Y.Bakhtin and J.Correll, A neural computation model for decision-making times. J.Math.Psychol., 56, 2012, 333-340. · Zbl 1282.91289 [7] P.Deift, G.Menon, S.Olver and T.Trogdon, Universality in numerical computations with random data. Proc. Natl. Acad. Sci. USA, 111(42), 2014, 14973-14978 · Zbl 1355.65019 [8] P.Deift and T.Trogdon, Universality for the Toda algorithm to compute the largest eigenvalue of a random matrix. ArXiv 1604.07384. To appear in Comm. Pure Appl. Math. · Zbl 1454.60012 [9] P.Deift and T.Trogdon, Universality for eigenvalue algorithms on sample covariance matrices. ArXiv 1701.01896. To appear in SIAM J. Numer. Anal. · Zbl 1378.65089 [10] C.Pfrang, P.Deift and G.Menon, How long does it take to compute the eigenvalues of a random symmetric matrix? Random Matrix Theory, Interact. Part. Syst., Integr. Syst. MSRI Publ. 65, 2014, 411-442. Asymptotics for the recurrence coefficients of polynomials orthogonal with respect to a logarithmic weight. Percy Deift (joint work with Thomas Conway) Let dµ(x) be a measure on the line with finite moments Z |x|mdµ(x) <∞, m ≥ 0, R and let pn(x) = γnxn+ ... , γn> 0, n≥ 0, be the associated orthonormal polynomials Z pn(x)pm(x)dµ(x) = δn,m, n, m≥ 0. R The polynomials automatically satisfy a three term recurrence relation bnpn+1(x) + (an− x)pn(x) + bn−1pn−1(x) = 0, n≥ 0 with recurrence coefficients bn> 0, an∈ R and b−1≡ 0. Given dµ(x), it is of basic interest to determine the asymptotic behavior of the b′ns and a′ns as n→ ∞. In [1] the authors considered logarithmic weights dµ(x) = log(2k/(1− x)) on [−1, 1], where k > 1. Such weights arise in various problems in physics and in mathematics. The main result in [1] is the following: As n→ ∞ (1) an= (2C)/(n log n)2+ O(1/n2log n3) (2) bn= 1/2 + 1/(16n2) + C/(n log n)2+ O(1/n2log n3) 2952Oberwolfach Report 49/2017 where C =−3/32. This result, and more, was conjectured by A.Magnus [2], up to the precise value for the constant C. The authors in [1] use Riemann-Hilbert/steepest-descent methods to prove (1)(2), but not in the standard way, as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at x = 1. The authors overcame this difficulty by using the operator theory that underlies Riemann-Hilbert problems, together with a new formula for the difference of the solutions of two Riemann-Hilbert problems on the same contour. Many open problems remain. Firstly, to prove the analog of (1)(2) for the case k = 1 (note that for k = 1, we no longer have log(2/(1− x)) ≥ c > 0 for x∈ [−1, 1]). Secondly, it is of great interest to compute the asymptotic behavior of the polynomials pnthemselves as n→ ∞, particularly in a neighborhood of the logarithmic singularity at x = 1. References [11] T.O.Conway and P.Deift, Asymptotics of polynomials orthogonal with respect to a logarithmic weight. ArXiv 1711.01590 · Zbl 1391.33027 [12] A.Magnus,Gaussianintegrationformulasforlogarithmicweightsandapplicationto2-dimensionalsolidstatelattices.VersionfromAugust20,2016. https://perso.uclouvain.be/alphonse.magnus/grapheneR2.pdf The Toda shock and rarefaction waves Iryna Egorova (joint work with Johanna Michor, Gerald Teschl) We are concerned with the long-time asymptotics of the Cauchy problem for the Toda equation ˙b(n, t) = 2(a(n, t)2− a(n − 1, t)2), (1)(n, t)∈ Z × R+, ˙a(n, t) = a(n, t)(b(n + 1, t)− b(n, t)), with steplike initial data a(n, 0)→ a > 0, b(n, 0) → b ∈ R, as n → −∞, (2)1 2,b(n, 0)→ 0,as n→ +∞. The initial value problem (1)–(2) is uniquely solvable for initial data which approach their limiting constants with a polynomial rate. Moreover, for each t6= 0 the solution tends as n→ ±∞ to the same constants, and with the same rate as the initial data (cf. [7]). We study the asymptotic behaviour of the solution in the regime when n→ ∞, t → +∞, but the ratio ξ =ntslowly varying. Qualitatively (up to a phase shift), the long-time asymptotics are determined by the mutual Reflectionless Operators: The Deift and Simon Conjectures2953 location of the intervals [b− 2a, b + 2a] and [−1, 1], and by the discrete spectrum λ1, ..., λNof the underlying Jacobi operator (3)H(t)y(n) := a(n− 1, t)y(n − 1) + b(n, t)y(n) + a(n, t)y(n + 1), n ∈ Z. Our goal is to rigorously justify these asymptotics by means of the nonlinear steepest descent (NSD) method developed by Deift and Zhou [6]. In fact, our investigation requires an extension of the original NSD analysis based on a suitably chosen g-function as first introduced in Deift, Venakides, and Zhou [5]. Here we restrict our considerations to the case of disjoint background spectra. For b + 2a <−1 we deal with the Toda shock problem, the case 1 < b − 2a is known as the Toda rarefaction problem. The long-time asymptotics of the rarefaction problem were studied rigorously by Deift, Kamvissis, Kriecherbauer and Zhou [4] in the transitional region where ξ :=nt≈ 0 as t → +∞. To this end the authors applied the NSD approach for a vectorRiemann-Hilbert problem. Using a similar approach, in [8] we show that there are four principal sectors with the following asymptotic behavior: • In the region n > t, the solution {a(n, t), b(n, t)} is asymptotically close to the constant right background solution{21, 0} plus a sum of solitons corresponding to the eigenvalues λj<−1. • In the region 0 < n < t, as t → ∞ we have n11− n 2t+ Ot,b(n, t) = 1 +2t+ O1t. • In the region −2at < n < 0, as t → ∞ we have n + 11n +3 2t+ Ot,b(n, t) = b− 2a −t2+ O1t. • In the region n < −2at, the solution of (1)–(2) is asymptotically close to the left background solution{a, b} plus a sum of solitons corresponding to the eigenvalues λj> b + 2a. Note that the main terms of the asymptotics (4) and (5) are solutions of the Toda lattice equation. In turn, the error terms O(t−1) are uniformly bounded with respect to n for εt≤ n ≤ (1 − ε)t in (4), and for (−2a + ε)t ≤ n ≤ −εt in (5), where ε > 0 is an arbitrary small value. In the two middle regions we also derive a precise formula for these error terms. The first investigation of shock waves in the Toda lattice was done by Venakides, Deift, and Oba [10] employing the Lax-Levermore method. As their main result they showed (for a =12) that in a sector|nt| < ξ′crthe solution can be asymptotically described by a period two solution, while in a sector|nt| > ξcrthe particles are close to the unperturbed lattice. For the remaining region ξcr′<|nt| < ξcr the solution was conjectured to be asymptotically close to a modulated singlephase quasi-periodic solution but this case was not solved there. Investigation of the Toda shock problem by the NSD analysis was singled out by Deift [3] as an important open problem in the theory of nonlinear systems. 2954Oberwolfach Report 49/2017 We perform this analysis for arbitrary a > 0, assuming that the discrete spectrum of the Jacobi operator (3) consists of the single point λ0∈ (2a + b, −1). A short qualitative description of the asymptotics for the Toda shock waves derived in [9] is the following. There are five principal regions on the half plane (n, t) divided by rays n/t = ˜ξ, with ˜ξ = ξcr,1, ξcr,1′, ξcr,0, ξ′cr, ξcrwhere ξcr,1< ξcr,1′< ξcr,0< ξcr′< ξcr. In the domain ξ > ξcr, the solution is asymptotically close to the constant right background solution{12, 0}, and in the domain ξ < ξcr,1it is close to the left background{a, b}. In the domain ξ′cr< ξ < ξcr, there appears a monotonous smooth function γ(ξ)∈ R such that γ(ξcr′) = b + 2a, γ(ξcr) = b− 2a. When the parameter ξ starts to decay from the point ξcr, the point γ(ξ) “opens” a band [b− 2a, γ(ξ)] (the Whitham zone, cf. [2]). This interval and [−1, 1] can be treated as the bands of a (slowly modulated) two band solution of the Toda lattice, which turns out to give the leading asymptotical term of our solution with respect to large t. This two band solution is defined uniquely by its initial divisor. We compute this divisor precisely via the values of the right transmission coefficient on the interval [b− 2a, γ(ξ)] . Thus, in a vicinity of any raynt= ξ the solution of (1)–(2) is asymptotically finite-gap. This asymptotical term also can be treated as a function of n, t, andntin the whole domain t(ξcr′+ ε) < n < t(ξcr− ε). Next, in the domains ξcr,0< ξ < ξcr′and ξcr,1′< ξ < ξcr,0, the asymptotic of the solution of (1)–(2) is described by two finite-gap solutions. They are connected with one and the same intervals [b− 2a, b + 2a] and [−1, 1] and the initial divisors (or shifts of the phase) do not depend on the slow variable ξ, but differ due to the presence of the point of the discrete spectrum λ0which generates a soliton. In particular, for a =12our asymptotics in these domains agree with the asymptotics obtained in [10]. The situation in the domain ξcr,1< ξ < ξ′cr,1is similar to the Whitham zone described above. There appears a monotone smooth function γ1(ξ)∈ R such that γ1(ξcr,1) = 1, γ1(ξcr,1′) =−1. The finite-gap asymptotic here is again local along the ray, and is defined by the intervals [b− 2a, b + 2a] and [γ1(ξ), 1]. An interesting open problem is to understand asymptotics in transitional regions. In particular, for the Toda shock problem in vicinities of the raysnt= ξcr andnt= ξcr,1one can expect the appearance of asymptotic solitons (see [1]). Another interesting problem is to describe long-time asymptotics of a steplike solution for (1)–(2) in the case of intersecting background spectra. Acknowledgment:I.E. was supported in part by the direction of MFO. References [13] A. Boutet de Monvel, I. Egorova, and E. Khruslov, Soliton asymptotics of the Cauchy problem solution for the Toda lattice, Inverse Problems 13-2 (1997), 223-237. · Zbl 0872.35107 [14] A. M. Bloch and Y. Kodama, Dispersive regularization of the Whitham equation for the Toda lattice, SIAM J. Appl. Math. 52 (1992), 909-928. · Zbl 0757.34014 [15] P. Deift, Some open problems in random matrix theory and the theory of integrable systems. II, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 016, 23 pp. Reflectionless Operators: The Deift and Simon Conjectures2955 · Zbl 1375.37160 [16] P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou, The Toda rarefaction problem, Comm. Pure Appl. Math. 49, No. 1 (1996), 35-83. · Zbl 0857.34025 [17] P. Deift, S. Venakides, and X. Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Applied Math. 47 (1994), 199-206. · Zbl 0797.35143 [18] P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Ann. of Math. (2) 137 (1993), 295-368. · Zbl 0771.35042 [19] I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds, J. Math. Physics 50 (2009), 103522. · Zbl 1283.37067 [20] I. Egorova, J. Michor, and G. Teschl, Rarefaction waves for the Toda equation via nonlinear steepest descent, Discrete Contin. Dyn. Syst. (to appear) · Zbl 1394.37106 [21] I. Egorova, J. Michor, and G. Teschl, Long-time asymptotics for the Toda shock problem: non-overlapping spectra, arXiv:1406.0720. · Zbl 1394.37106 [22] S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl. Math. 44, No.8-9 (1991), 1171-1242. Abelian coverings, discrete Schr¨odinger operators, and KdV-equation Benjamin Eichinger (joint work with Tom VandenBoom and Peter Yuditskii) Let E⊂ R be compact and D/Γ be a uniformization of C E. That is, there exists a local homeomorphism λ : D→ C E and Γ is the Fuchsian group of its deck transformations. If E has positive logarithmic capacity, then Γ is of convergent type, and hence the Blaschke product b(z; Γ) = zY|γ(0)|γ(0)− z γ(0)1− γ(0)z γ∈Γ converges. Γ (or equivalently E) is called of Widom type, if in addition b′(z; Γ) is of bounded characteristic in D. Let Γ∗be the group of characters of Γ, i.e., the set of homomorphism from Γ to R/Z. A function f is called character automorphic with character α∈ Γ∗, if f◦ γ = e2πiα(γ)f,for all γ∈ Γ. Note that b(z; Γ) is character automorphic with some character µ∈ Γ∗. By H2(α; Γ) we denote the Hardy space of character automorphic functions with character α on D. That is, a function f∈ H2(α; Γ) is a multi-valued function on the Riemann surface D/Γ. Widom [5] showed that if Γ is of Widom type, then H2(α; Γ) is non-empty for all α∈ Γ∗. Let J(E) denote the isospectral torus of reflectionless Jacobi matrices whose spectrum is the set E. If E is of Widom type such that the so-called Direct Cauchy Theorem holds, then Sodin and Yuditskii [3] showed that J(E) is homeomorphic to Γ∗. The homeomorphism is called generalized Abel map and associates to each Jacobi matrix J∈ J(E) a Hardy space H2(α; Γ). As a corollary of this construction it follows that all elements in J(E) are almost periodic. Hence, this model is based on the fact that there exist sufficiently many admissible analytic functions on the Riemann surface D/Γ. 2956Oberwolfach Report 49/2017 Assume now that E consists of three points, say E ={0, 1, ∞}. This is the smallest number of punctures in the Riemann sphere C such that the universal covering is the unit disk D. Lyons and McKean [2] proved that already in this case the commutator subgroup Γ′of Γ is of convergent type. In other words, the boundary of the Riemann surface D/Γ′has positive capacity. Moreover, note that if E has positive analytic capacity, then the Ahlfors function yields a single valued function on D/Γ′. Hence, we conclude: By passing from the Riemann surface D/Γ to D/Γ′the “quality of the boundary”, in the sense of number of admissible analytic functions, increases essentially. D/Γ′is called universal Abelian covering for D/Γ. Therefore, in this talk we shall discuss the Riemann surface D/Γ′in detail. We will describe it by means of the covering map λ and by means of b(z; Γ), which becomes a single valued function on D/Γ′. Moreover, we will discuss function theory on the Riemann surface D/Γ0, where Γ0= ker µ. In particular, we give a criterion by means of reproducing kernels of the Hardy space H2(α; Γ0) such that the Jacobi matrix J(α) is in fact a discrete Schr¨odinger operator. Vinnikov and Yuditskii [4] gave an interpretation of the Toda flow an Jacobi matrices by means of the fact that multiplication by functions on D/Γ′trivially commute. We conjecture that the same would also be possible for the KdV hierarchy. To be more precise. Conjecture.Let ∞ E = [0,∞) \(\)(aj, bj) j=1 such that E is of Widom type and the DCT holds in CE. Assume in addition that for some k∈ N we have ∞ (1)bk+2j− ak+2j<∞ j=1 Then there exists a function θk, on C+/Γ′corresponding to the uniformization (C+/Γ, λ) such that the kth element of the KdV hierarchy is generated by the functions θkand λ. Let uVkdenote the solution of the kth element of the KdV hierarchy with initial potential V . Deift asked the following question: Question.Does almost periodicity of the initial data V imply almost periodicity of the solution uV1in t-direction. As a consequence, we would get an affirmative answer to this question for initial data V , which are almost periodic and satisfy σ(LV) = σac(LV) = E, where LV=−∂x2+ V (x) is the corresponding Schr¨odinger operator. This improves a result of Binder, Damanik, Goldstein and Lukic [1] in two directions. First, we weaken the assumption on the spectral set E and secondly under the regularity condition (1) we Reflectionless Operators: The Deift and Simon Conjectures2957 obtain the corresponding result for all elements of the KdV hierarchy, where the aforementioned authors only discuss the case k = 1. References [23] I. Binder, D. Damanik, and Lukic M. Goldstein, M. and, Almost periodicity in time of solutions of the KdV equation, arXiv:1509.07373 (2015). · Zbl 1406.35325 [24] T. J. Lyons and H. P. McKean, Winding of the plane Brownian motion, Adv. in Math. 51 (1984), no. 3, 212-225. · Zbl 0541.60075 [25] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), no. 3, 387-435. · Zbl 1041.47502 [26] V. Vinnikov and P. Yuditskii, Functional models for almost periodic Jacobi matrices and the Toda hierarchy, Mat. Fiz. Anal. Geom. 9 (2002), no. 2, 206-219. · Zbl 1102.47020 [27] H. Widom, Hpsections of vector bundles over Riemann surfaces, Ann. of Math. 94 (1971), 304-324. Limit-Periodic Schr¨odinger Operators with Zero-Measure Spectrum Jake Fillman (joint work with David Damanik, Milivoje Lukic) We consider continuum Schr¨odinger operators acting in L2(R) via LVu =−u′′+ V u, where the potential V : R→ R is limit-periodic, that is, V is a uniform limit of continuous periodic functions. We denote the class of limit-periodic potentials by LP and equip it with the topology inherited from the L∞metric. For an archetypal example, consider ∞ V (x) =X2−jcos 2πx. j! j=1 These operators are interesting for spectral theory as they provide a large tractable class of aperiodic almost-periodic Schr¨odinger operators which may exhibit rich and subtle spectral properties. These operators and their discrete analogs may exhibit absolutely continuous spectrum [3, 8, 13, 15, 16], singularly continuous spectrum [2, 5, 6], or even pure point spectrum [7, 10, 17]. As the rate of approximation grows worse, the character of the associated quantum dynamics transitions from ballistic motion (free propagation of wave packets) [1, 11, 14] to localization [7, 11, 17]. Moreover, limit-periodic potentials are uniformly almost-periodic and hence also provide an interesting class of initial data for the Cauchy problem for the KdV equation: ∂tu− 6u∂xu + ∂x3u = 0,u(x, 0)≡ V0(x). Recent work of Binder-Damanik-Goldstein-Lukic achieved success in solving Deift’s conjecture (cf. [12]) for reflectionless almost-periodic initial data as long as σ(LV0) is sufficiently thick [4]. It is then natural to ask how badly the hypotheses of the general results may fail, and how often such failures may occur within 2958Oberwolfach Report 49/2017 the class of limit-periodic potentials. Broadly speaking, within the class of limitperiodic potentials, the answers to these questions are “quite badly” and “rather often.” To wit: Theorem 1(Damanik-F.–Lukic, (2015) [5]). There is a dense Gδsubset Z⊆ LP with the property that σ(LλV) is a perfect set of zero Lebesgue measure for every V∈ Z and every λ > 0. There is a dense set H ⊆ LP with the property that σ(LλV) is a perfect set of zero Hausdorff dimension for all V∈ H and every λ > 0. From the KdV point of view, the examples in this theorem are quite bad, as it shows that the (topologically) generic behavior of limit-periodic potentials lies well outside the tractable regime. References [28] J. Asch, A. Knauf, Motion in periodic potentials, Nonlinearity 11 (1998) 175-200. · Zbl 0896.34027 [29] A. Avila, On the spectrum and Lyapunov exponent of limit-periodic Schr¨odinger operators, Commun. Math. Phys. 288 (2009), 907-918. · Zbl 1188.47023 [30] J. Avron, B. Simon, Almost periodic Schr¨odinger operators. I. Limit periodic potentials, Commun. Math. Phys. 82 (1981), 101-120. · Zbl 0484.35069 [31] I. Binder, D. Damanik, M. Goldstein, M. Lukic, Almost periodicity in time of solutions of the KdV equation, preprint. Arxiv:1509.07373. · Zbl 1406.35325 [32] D. Damanik, J. Fillman, M. Lukic, Limit-periodic continuum Schr¨odinger operators with zero-measure Cantor spectrum, J. Spectral Th., in press. arXiv:1508.04696. · Zbl 1432.34113 [33] D. Damanik and Z. Gan, Limit-periodic Schr¨odinger operators in the regime of positive Lyapunov exponents, Journal of Functional Analysis 258 (2010), 4010-4025. · Zbl 1191.47058 [34] D. Damanik and Z. Gan, Limit-periodic Schr¨odinger operators with uniformly localized eigenfunctions, J. Anal. Math. 115 (2011), 33-49. · Zbl 1314.47053 [35] D. Damanik and Z. Gan, Spectral properties of limit-periodic Schr¨odinger operators, Commun. Pure Appl. Anal. 10 (2011), 859-871. · Zbl 1242.47023 [36] D. Damanik, M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data, J. Amer. Math. Soc. 29 (2016), 825-856. · Zbl 1342.35300 [37] D. Damanik, A. Gorodetski, An extension of the Kunz-Souillard approach to localization in one dimension and applications to almost-periodic Schr¨odinger operators, Adv. Math. 297 (2016), 149-173. · Zbl 1344.47021 [38] D. Damanik, M. Lukic, W. Yessen, Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many body problems, Commun. Math. Phys. 337 (2015), 1535-1561. · Zbl 1332.82027 [39] P. Deift, Some open problems in random matrix theory and the theory of integrable systems, Integrable Systems and Random Matrices, 419-430, Contemp. Math. 458, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1147.15305 [40] I. E. Egorova, Spectral analysis of Jacobi limit-periodic matrices, Dokl. Akad. Nauk Ukrain. SSR Ser. A 3 (1987), 7-9. (in Russian) · Zbl 0649.35069 [41] J. Fillman, Ballistic transport for limit-periodic Jacobi matrices with applications to quantum many-body problems, Commun. Math. Phys. 350 (2017), 1275-1297. · Zbl 1359.81179 [42] L. Pastur, V. A. Tkachenko, On the spectral theory of the one-dimensional Schr¨odinger operator with limit-periodic potential, Dokl. Akad. Nauk SSSR 279 (1984) 1050-1053. (in Russian) [43] L. Pastur, V. Tkachenko, Spectral theory of a class of one-dimensional Schr¨odinger operators with limit-periodic potentials, Trudy Moskov. Mat. Obshch. 51 (1988), 114-168. [44] J. P¨oschel, Examples of discrete Schr¨odinger operators with pure point spectrum, Commun. Math. Phys. 88 (1983), 447-463. Reflectionless Operators: The Deift and Simon Conjectures2959 On the Spectrum of Multi-Frequency Quasiperiodic Schr¨odinger Operators with Large Coupling Michael Goldstein (joint work with Wilhelm Schlag, Mircea Voda) In this talk I will give a review of recent joint work M.Goldstein,W.Schlag,M.Voda on the spectrum of multi-frequency quasi-periodic operators at large coupling constant. In the last 40 years after the groundbreaking paper [9] the theory of quasiperiodic Schr¨odinger operators has been developed extensively, see the monograph [5] for an overview and [14] for a survey of the more recent results. For shifts on a onedimensional torus T most of the results have been established non-perturbatively, i.e., either in the regime of almost reducibility or in the regime of positive Lyapunov exponent, and Avila’s global theory, see [3], gives a qualitative spectral picture, covering both regimes, for generic potentials. One of the main results of the one-dimensional theory is the fact that the spectrum is a Cantor set. For the case of the almost Mathieu operator (corresponding to a cosine potential), this result has been proved for any non-zero coupling and any irrational shift, see [19] and [1, 2]. For general analytic potentials in the regime of positive Lyapunov exponent with generic shift the Cantor structure of the spectrum has been obtained in [12]. On the other hand, shifts on a multidimensional torus Tdturned out to be harder to analyze and the theory is less developed, even in the perturbative setting. In particular, not much is known about the geometry of the spectrum for multidimensional shifts. In their pioneering paper [7], Chulaevsky and Sinai conjectured that in contrast to the shift on the one-dimensional torus, for the two-dimensional shift the spectrum can be an interval for generic large smooth potentials. In this paper we prove this conjecture for large analytic potentials. Heuristically, gaps in the spectrum of the one-frequency operators are created by horizontal “forbidden zones” appearing at the points of intersection of the graphs of shifted finite scale eigenvalues parametrized by phase, see [20, 12]. In contrast to this, the heuristic principle underlying [7] is that for multiple frequencies, the intersection curves of the graphs of shifted finite scale eigenvalues may not be too flat, thus preventing the appearance of the horizontal “forbidden zones” and stopping the formation of gaps. It is clear that some genericity assumption on the potential function is needed for this to be true, since potentials like V (x, y) = v(x) lead to flat intersection curves and have Cantor spectrum. Furthermore, the largeness of the potential is also needed. Indeed, it is known that for small potentials with atypical frequency vector the spectrum has gaps, see [4]. Implementing such an argument, appears to be very challenging for a number of reasons. First, the analytical techniques available in finite volume are less favorable (mainly the large deviation theorems and everything that depends on them) as compared to the case of one frequency. In particular, it is difficult to implement an approach based on finite scale localization as in [12]. This is due 2960Oberwolfach Report 49/2017 to the fact that it is hard to handle long chains of resonances and to control the intersections of the resonant curves with the level sets of the eigenvalues. Second, it is inevitable that the intersection curves of the graphs of shifted finite scale eigenvalues flatten near the absolute extrema and handling this situation seems to be a delicate matter. In [13] we addressed some of the issues regarding the analytical techniques, including establishing finite scale localization. We will use most of the basic tools from [13]. However, for the purpose of this paper one would need a refined version of finite scale localization, beyond what is achieved in that paper. We analyze the spectrum of the operator HN(x), x∈ Td, on a finite interval [1, N ] subject to Dirichlet boundary conditions. To keep this spectrum under control requires resolving the following problem. Given E letRN(E) be the set of all phases x such that E is in the spectrum of the operator HN(x). One has to identify phases x∈ RN(E) for which x + nω is not too close toRN(E) as n runs in the interval N≪ n < NA, A≫ 1. This issue, commonly referred to as double resonances, is well-known. Similar strategies, leading to the formation of intervals in the spectrum, have been implemented for the skew-shift in [15] and for continuous two-dimensional Schr¨odinger operators in [16]. The main new device that we develop in this work, consists of an elimination of double resonances for all shifts x + h, and not just the “arithmetic ones” x + nω. Of course the shift h cannot be too small. Although this problem looks less accessible, it turns out to provide more control on the resonant setRN(E) of the previous scale. The level sets V (x) = E of the potential in question must satisfy the requirements of this more general elimination in order to launch the multi-scale analysis. Furthermore, in order to show that the spectrum is actually an interval, we develop a Cartan type estimate that controls the intersections of the level sets of an analytic function near a non-degenerate extremum with their shifts. The core of our approach is non-perturbative and works in the regime of positive Lyapunov exponent. More precisely, we develop two non-perturbative inductive schemes, one leading to the formation of intervals in the bulk of the spectrum and the other leading to intervals at the edges of the spectrum. We will only use the largeness of the potential to check that the initial inductive conditions are satisfied. We introduce some notation and definitions that we need to state our main result. We work with operators (1)[Hλ(x)ψ](n) =−ψ(n + 1) − ψ(n − 1) + λV (x + nω)ψ(n), with λ > 0 being a real parameter, and with the potential V a real analytic function on the torus Td, T = R/Z, d≥ 2. We assume that the frequency vector ω∈ Tdobeys the standard Diophantine condition. We introduce also the class of “generic trigonometric polynomials of a given degree”. The formal definition is pretty lengthy. Obviously it has to be a Morse function. On top of that the level sets shifts should be transversal to themselves unless the shift is too small. We denote the st of such trigonometric polynomials by G. Reflectionless Operators: The Deift and Simon Conjectures2961 Theorem 1.There exists λ0= λ0(V, a, b, d) such that the following statements hold for λ≥ λ0. (a) Assume that V attains its global minimum at exactly one non-degenerate critical point x. Then there exists E∈ R, |λ−1E− V (x)| < λ−1/4, such that [E, E + λ exp(−(log λ)1/2)]⊂ Sλand(−∞, E) ∩ Sλ=∅. An analogous statement holds relative to the global maximum of V (using the notation x, E). (b) Assume that V∈ G and let E, E be as in (a). Then Sλ= [E, E]. Remark 1. (a) The constant λ0(V, a, b, d) can be expressed explicitly, see the proof of Theorem 1. (b) The genericity of the assumptions on V will be addressed in [21]. More precisely, the following result will be established. Consider real trigonometric polynomials of the form X V (x) =cme2πim·x,x∈ Rd m∈Zd:|m|≤n of a given cumulative degree n≥ 1, |m| :=P1≤j≤d|mj|. Then for almost all vectors (cm)|m|≤none has V∈ G. (c) For the completeness of our paper we include a particular example of potential V∈ G that can be obtained by the methods from [21]. Namely, we show that V (x, y) = cos(2πx) + s cos(2πy) belongs to G for all s∈ R {−1, 0, 1}. We note that as s approaches {−1, 0, 1} our explicit value for λ0diverges to∞ and the geometry of the spectrum cannot be decided by continuity. Of course, for s = 0 the spectrum is a Cantor set. However, for s =±1, part (a) of Theorem 1 still applies and guarantees the existence of intervals at the edges of the spectrum. As mentioned above, the derivation of Theorem 1 is based on two non-perturbative statements in the regime of positive Lyapunov exponent. References [45] Artur Avila and Svetlana Jitomirskaya. The Ten Martini Problem. Ann. of Math. (2), 170(1):303-342, 2009. · Zbl 1166.47031 [46] Artur Avila and Svetlana Jitomirskaya. Almost localization and almost reducibility. J. Eur. Math. Soc. (JEMS), 12(1):93-131, 2010. · Zbl 1185.47028 [47] Artur Avila. Global theory of one-frequency Schr¨odinger operators. Acta Math., 215(1):1-54, 2015. · Zbl 1360.37072 [48] J. Bourgain. On the spectrum of lattice Schr¨odinger operators with deterministic potential. J. Anal. Math., 87:37-75, 2002. Dedicated to the memory of Thomas H. Wolff. · Zbl 1022.47024 [49] J. Bourgain. Green’s function estimates for lattice Schr¨odinger operators and applications, volume 158 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2005. · Zbl 1137.35001 [50] J. Bourgain. Positivity and continuity of the Lyapounov exponent for shifts on Tdwith arbitrary frequency vector and real analytic potential. J. Anal. Math., 96:313-355, 2005. 2962Oberwolfach Report 49/2017 · Zbl 1089.81021 [51] V. A. Chulaevsky and Ya. G. Sina˘ı. Anderson localization for the 1-D discrete Schr¨odinger operator with two-frequency potential. Comm. Math. Phys., 125(1):91-112, 1989. [52] P. Duarte and S. Klein. Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles. ArXiv e-prints, March 2016. · Zbl 1427.37042 [53] E. I. Dinaburg and Ja. G. Sina˘ı. The one-dimensional Schr¨odinger equation with quasiperiodic potential. Funkcional. Anal. i Priloˇzen., 9(4):8-21, 1975. [54] Michael Goldstein and Wilhelm Schlag. H¨older continuity of the integrated density of states for quasi-periodic Schr¨odinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2), 154(1):155-203, 2001. · Zbl 0990.39014 [55] Michael Goldstein and Wilhelm Schlag. Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal., 18(3):755-869, 2008. · Zbl 1171.82011 [56] Michael Goldstein and Wilhelm Schlag. On resonances and the formation of gaps in the spectrum of quasi-periodic Schr¨odinger equations. Ann. of Math. (2), 173(1):337-475, 2011. · Zbl 1268.82013 [57] M. Goldstein, W. Schlag, and M. Voda. On localization and spectrum of multi-frequency quasi-periodic operators. ArXiv e-prints, 2016. [58] S. Jitomirskaya and C. A. Marx. Dynamics and spectral theory of quasi-periodic Schr¨odingertype operators. Ergodic Theory and Dynamical Systems, pages 1-41, Jul 2016. [59] Helge Kr¨uger. The spectrum of skew-shift Schr¨odinger operators contains intervals. J. Funct. Anal., 262(3):773-810, 2012. · Zbl 1252.35211 [60] Y. Karpeshina and R. Shterenberg. Extended States for the Schr¨odinger Operator with Quasi-periodic Potential in Dimension Two. ArXiv e-prints, August 2014. · Zbl 1442.35003 [61] Serge Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002. [62] Morris Marden. Geometry of polynomials. Second edition. Mathematical Surveys, No. 3. American Mathematical Society, Providence, R.I., 1966. · Zbl 0162.37101 [63] Joaquim Puig. Cantor spectrum for the almost Mathieu operator. Comm. Math. Phys., 244(2):297-309, 2004. · Zbl 1075.39021 [64] Ya. G. Sina˘ı. Anderson localization for one-dimensional difference Schr¨odinger operator with quasiperiodic potential. J. Statist. Phys., 46(5-6):861-909, 1987. · Zbl 0682.34023 [65] Mircea Voda. Cartan type estimates for shifts along level sets of trigonometric polynomials. In preparation, 2017. Exponential estimates on the size of spectral gaps for quasi-periodic Schr¨odinger operators Martin Leguil (joint work with Jiangong You, Zhiyan Zhao, and Qi Zhou) In the following, we consider one-dimensional discrete Schr¨odinger operators on ℓ2(Z): (HV,α,θu)n= un+1+ un−1+ V (θ + nα)un,∀ n ∈ Z, for some phase θ∈ Td:= (R/Z)d, some analytic potential V : Td→ R, and where the (multi-)frequency α = (α1, . . . , αd)∈ Tdis chosen in such a way that (1, α1, . . . , αd) is rationally independent. In this case, the spectrum of HV,α,θis a compact subset of R, independent of θ, denoted by ΣV,α. By the Gap-Labelling Theorem, it is of the form ΣV,α= [E, E]∪k∈Zd{0 [66] then proved global existence and uniqueness for a Diophantine frequency and small quasi-periodic analytic initial datum. Recently, Binder-Damanik-GoldsteinLukic [7] showed that in the same setting, the solution is in fact almost periodic in time, thus proving Deift’s conjecture in this case. In our work, we consider the discrete version of Deift’s conjecture, namely that for almost periodic initial data, the Toda flow is almost periodic in the time variable. Recall the Toda lattice equation a′n(t) =an(t) (bn+1(t)− bn(t)) , (1)n∈ Z. b′n(t) =2(a2n(t)− a2n−1(t)), In view of Theorem 12.6 in [23], given an initial condition (a(0), b(0))∈ ℓ∞(Z)× ℓ∞(Z), there is a unique solution (a, b)∈ C∞(R, ℓ∞(Z)× ℓ∞(Z)) to (1). We can identify (a(t), b(t)) with a doubly infinite Jacobi matrix J(t): (2)(J(t)u)n:= an−1(t) un−1+ bn(t) un+ an(t) un+1. As a consequence of homogeneity (Theorem 4) and purely absolutely continuous spectrum of subcritical Schr¨odinger operators [2], we prove a discrete version of Deift’s conjecture for almost periodic initial data, building on an previous result of Vinnikov-Yuditskii [25]. We show the following generalization of the result of Binder-Damanik-Goldstein-Lukic [7] to Avila’s subcritical regime (see also the recent paper [8] for related advance on this problem). Theorem 5(L.-You-Zhao-Zhou [17]). Let α∈ R\Q with β(α) = 0. Let V : T → Rbe a subcritical analytic potential, i.e., such that (α, SVE) is subcritical for all E∈ ΣV,α. We consider the Toda flow (1) with initial condition (an, bn)(0) = (1, V (θ + nα)), n∈ Z. Then (1) For any θ∈ T, (1) admits a unique solution (a(t), b(t)) defined for all t∈ R. (2) For every t, the Jacobi matrix J(t) given by (2) is almost periodic and has constant spectrum ΣV,α. (3) The solution (a(t), b(t)) is almost periodic in t in the following sense: there is a continuous mapM: TZ→ ℓ∞(Z)× ℓ∞(Z), a point ϕ∈ TZand a direction ̟∈ RZ, such that (a(t), b(t)) =M(ϕ + ̟t). 2966Oberwolfach Report 49/2017 In particular, the above conclusion holds for V = 2λ cos 2π(·) with 0 < λ < 1. Some ideas of the proofs.Our approach is from the perspective of dynamical systems, and is based on quantitative (strong) almost reducibility. To obtain bounds on the size of spectral gaps, we analyze the behavior of Schr¨odinger cocycles close to the boundary of some spectral gap. At the edge points, the cocycles are reducible to constant parabolic cocycles. The key points in our proof are the exponential decay of the off-diagonal coefficient of the parabolic matrix, and the subexponential growth of the conjugacy (in restriction to T) with respect to the label k. We first consider the case of small analytic potentials, and we distinguish between two cases in the proof. If the frequency is Diophantine, we develop a new KAM scheme to show almost reducibility with nice estimates (this result works for multifrequencies, and for both continuous and discrete cocycles). Moreover, in order to get a sharp decay on the size of spectral gaps (Theorem 2), we prove almost reducibility of the cocycle in a fixed band, arbitrarily close to the initial band. On the other hand, for a one-dimensional frequency α satisfying β(α) = 0, we use the almost localization argument (via Aubry duality) given by Avila [1] (initially developed by Avila-Jitomirskaya [5]); one key ingredient in the proof is the Corona Theorem. The generalization to the global subcritical regime is based on Avila’s global theory of analytic SL(2, R)−cocycles [3], especially his proof of the Almost Reducibility Conjecture [2, 3]. Homogeneity of the spectrum in the subcritical regime is derived from the upper bounds on the size of spectral gaps, together with H¨older continuity of the IDS. Thanks to Avila’s global theory of one-frequency Schr¨odinger operators [3], one can then prove Theorem 3 by combining our results in the subcritical case with previous work of Damanik-Goldstein-Schlag-Voda [12] in the supercritical regime. References [67] Avila, A.; The absolutely continuous spectrum of the almost Mathieu operator, preprint. [68] Avila, A.; KAM, Lyapunov exponents and the spectral dichotomy for one-frequency Schr¨odinger operators, preprint. [69] Avila, A.; Global theory of one-frequency Schr¨odinger operators, Acta Math., 215, 1-54 (2015). · Zbl 1360.37072 [70] Avila, A., Jitomirskaya, S.; The Ten Martini Problem, Ann. of Math., 170, 303-342 (2009). · Zbl 1166.47031 [71] Avila, A., Jitomirskaya, S.; Almost localization and almost reducibility, J. Eur. Math. Soc., 12, 93-131 (2010). · Zbl 1185.47028 [72] Avila, A., You, J., Zhou, Q.; Dry ten Martini problem in the non-critical case, preprint. [73] Binder, I., Damanik, D., Goldstein, M., Lukic, M.; Almost periodicity in time of solutions of the KdV equation, arXiv:1509.07373. · Zbl 1406.35325 [74] Binder, I., Damanik, D., Lukic, M., VandenBoom, T.; Almost periodicity in time of solutions of the Toda Lattice, arXiv:1603.04905. · Zbl 1396.37067 [75] Damanik, D., Goldstein, M.; On the inverse spectral problem for the quasi-periodic Schr¨odinger equation, Publ. Math. Inst. Hautes ´Etudes Sci., 119, 217-401 (2014). · Zbl 1296.35168 [76] Damanik, D., Goldstein, M.; On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data, J. Ame. Math. Soc. 29 (3), 825-856 (2016). · Zbl 1342.35300 [77] Damanik, D., Goldstein, M., Lukic, M.; The spectrum of a Schr¨odinger operator with small quasi-periodic potential is homogeneous, J. Spec. Theory, 6, 415-427 (2016). Reflectionless Operators: The Deift and Simon Conjectures2967 · Zbl 1350.34067 [78] Damanik, D., Goldstein, M., Schlag, W., Voda, M.; Homogeneity of the spectrum for quasiperiodic Schr¨odinger operators, to appear in J. Eur. Math. Soc.. · Zbl 1478.47023 [79] Deift, P.; Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices, 419-430, Contemp. Math. 458, Amer. Math. Soc., Providence, RI (2008). · Zbl 1147.15305 [80] Deift, P.; Some open problems in random matrix theory and the theory of integrable systems. II Symmetry, Integrability and Geometry: Methods and Applications, 13 (016), 23 pages. (2017). · Zbl 1375.37160 [81] Gesztesy, F., Yuditskii, P.; Spectral properties of a class of reflectionless Schr¨odinger operators, J. Func. Anal., 241, 486-527 (2006). · Zbl 1387.34120 [82] Hadj Amor, S.; H¨older continuity of the rotation number for quasi-periodic cocycles in SL(2, R), Commun. Math. Phys., 287 (2), 565-588 (2009). · Zbl 1201.37066 [83] Leguil M., You J., Zhao Z., Zhou Q.; Asymptotics of spectral gaps of quasi-periodic Schr¨odinger operators, in preparation. [84] Moser, J., P¨oschel, J.; An extension of a result by Dinaburg and Sinai on quasi-periodic potentials, Commun. Math. Helv., 59 (1), 39-85 (1984). · Zbl 0533.34023 [85] Puig, J.; Cantor spectrum for the almost Mathieu operator, Commun. Math. Phys. 244, 297-309 (2004). · Zbl 1075.39021 [86] Simon, B.; Almost periodic Schr¨odinger operators: A review, Adv. Appl. Math. 3, 463-490 (1982). · Zbl 0545.34023 [87] Sodin, M., Yuditskii, P.; Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum, Comment. Math. Helv., 70 (4), 639-658 (1995). · Zbl 0846.34024 [88] Sodin, M., Yuditskii, P.; Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal., 7 (3), 387-435 (1997). · Zbl 1041.47502 [89] Teschl, G.; Jacobi operators and completely integrable nonlinear lattices, volume 72 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. · Zbl 1056.39029 [90] Tsugawa, T., Local well-posedness of KdV equations with quasi-periodic initial data, SIAM Journal of Mathematical Analysis, 44, 3412-3428 (2012). · Zbl 1257.35168 [91] Vinnikov, V., Yuditskii, P.; Functional models for almost periodic Jacobi matrices and the Toda hierarchy, Matematicheskaya fizika, analiz, geometriya, 9 (2), 206-219 (2002). KdV Equation with Quasi-Periodic Initial Data Milivoje Lukic (joint work with Ilia Binder, David Damanik, Michael Goldstein) The KdV equation [1] (1)∂tu− 6u∂xu + ∂x3u = 0 was introduced in the 19th century as a model for the propagation of shallow water waves in one dimension; in the 1960s it was found to have infinitely many conserved quantities [2] and a Lax pair representation [3], making KdV the first of many integrable partial differential equations to be discovered. Integrability of the KdV equation was heavily used for the study of the Cauchy problem with rapidly decaying and for periodic initial data (2)u(0) = V. In particular, for initial data V∈ Hn(T) for nonnegative integer n, the solution u(t) is an Hn(T)-almost periodic function of t, which motivated the conjecture of 2968Oberwolfach Report 49/2017 Deift [4] whether, for almost periodic initial data V , the solution of the Cauchy problem is almost periodic as a function of t. The analysis of the KdV Cauchy problem with almost periodic initial data presents significant new obstacles, and even short time existence of solutions is not known in general. Our work has focused on quasiperiodic initial data V of the form (3)V (x) =Xc(n)e2πinωx n∈Zν where ω∈ Rν. In particular, we define for ε > 0 and κ∈ (0, 1] the space P(ω, ε, κ) of functions of the form (3) such that|c(n)| ≤ ε exp(−κ|n|) for n ∈ Zν. Our work proves Deift’s conjecture for the case of small quasiperiodic analytic initial data with Diophantine frequency. Theorem 1. [92] Let ω∈ Rνobey the Diophantine condition |nω| ≥ a0|n|−b0,n∈ Zν{0} for some 0 < a0< 1, ν < b0<∞. There exists ε0(a0, b0, κ) > 0 such that, if ε < ε0and V∈ P(ω, ε, κ), then there exists a global solution u of (1), (2) with the following properties: √ (1) for every t√∈ R, u(·, t) is quasiperiodic in x and u(·, t) ∈ P(ω,4ε, κ/4) (2) u isP(ω,4ε, κ/4)-almost periodic in t, i.e., there is a compact (finite or infinite dimensional) torus Td, a continuous map √ M : Td→ P(ω,4ε, κ/4), a base point α∈ Td, and a direction vector ζ∈ Rdsuch that u(t) = M(α + ζt) (3) the solution is unique, in the following sense: if ˜u is a solution of (1), (2) on R× [−T, T ] for some T > 0, and (4)˜u, ∂xxxu˜∈ L∞(R× [−T, T ]), then ˜u = u. In the conclusion (2) above,P(ω, ε, κ) can be taken as a metric space with the metric induced by the L∞(R)-norm, or the Wk,∞(R)-norm for any k∈ N; all such metrics are mutually equivalent onP(ω, ε, κ). The theorem therefore implies that besides u, derivatives of u are also almost periodic in t, and so is each Fourier coefficient c(n, t) of u(x, t). In fact, Theorem 1 is a corollary of our more general result, which proves existence, uniqueness, and almost periodicity in t whenever V is almost periodic and the spectrum of the associated Schr¨odinger operator−∂x2+ V is absolutely continuous and not too thin, in a sense quantified by Craig-type conditions. The spectrum S = σ(HV) is closed and bounded from below but not from above, so it can be written in the form [ S = [E,∞) \(Ej−, Ej+), j∈J Reflectionless Operators: The Deift and Simon Conjectures2969 where E = inf S and (Ej−, Ej+) are the maximal open intervals in R\ S, called gaps. We denote γj= Ej+− Ej− for j∈ J and ηj,l= dist((Ej−, Ej+), (El−, El+)),ηj,0= dist((Ej−, Ej+), E) for j, l∈ J (notationally, we assume here that our abstract index set J does not contain 0 as an element). We also denote Cj= (ηj,0+ γj)1/2Y1 +γl1/2. ηj,l l∈J l6=j We assume that S satisfies a set of moment conditions and Craig-type conditions: X (5)(1 + η2j,0)γj<∞ j∈J γj1/2<∞, supγj1/21 + ηj,0C (6)j∈Jj∈Jηj,0j<∞ γj1/2γl1/2!a (7)sup(1 + ηj,0)(Cj+ 1) <∞ for a ∈12, 1 . j∈Jl∈Jηj,l l6=j Theorem 2. [93] Let the initial data V : R→ R be uniformly almost periodic. Denote S = σ(HV) and assume that S = σac(HV) and that S obeys the Craig-type conditions (5), (6), (7). Then there exists a global solution u of (1), (2) with the following properties: (1) for every t∈ R, the function u(·, t) is uniformly almost periodic with frequency module equal to the frequency module of V ; (2) u is almost periodic in t, in the following sense: there is a continuous map M : TJ→ W4,∞(R), a base point α∈ TJ, and a direction vector ζ∈ RJsuch that u(·, t) = M(α + ζt); (3) the solution is unique, in the following sense: if ˜u is a solution of (1), (2) on R× [−T, T ] for some T > 0, which obeys (4), then ˜u = u. The existence and almost periodicity of a solution of the KdV equation under Craig-type conditions were previously studied by Egorova [6] (with the analog for the nonlinear Schr¨odinger equation studied by Boutet de Monvel-Egorova [94] ). Another paper of Egorova [8] used a different approach to construct almost periodic solutions for limit periodic initial data with superexponential periodic approximants. 2970Oberwolfach Report 49/2017 Acknowledgments.I. B. was supported in part by an NSERC Discovery grant. D. D. was supported in part by NSF grant DMS-1361625. M. G. was supported in part by an NSERC Discovery grant. M. L. was supported in part by NSF grant DMS-1301582. References [95] D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 539 (1895), 422-443. · JFM 26.0881.02 [96] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, A method of solving the Korteweg– de Vries equation, Phys. Rev. Lett. 19 (1967), 1095-1097. · Zbl 1061.35520 [97] P. Lax, Integrals of non-linear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. · Zbl 0162.41103 [98] P. Deift, Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices, 419-430, Contemp. Math. 458, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1147.15305 [99] I. Binder, D. Damanik, M. Goldstein, M. Lukic, Almost Periodicity in Time of Solutions of the KdV Equation, preprint (arXiv:1509.07373). · Zbl 1406.35325 [100] I. E. Egorova, Almost periodicity of some solutions of the KdV equation with Cantor spectrum, Dopov./Dokl. Akad. Nauk Ukraini, (1993), 26-29. · Zbl 0900.35327 [101] A. Boutet de Monvel, I. Egorova, On solutions of nonlinear Schr¨odinger equations with Cantor-type spectrum, J. Anal. Math. 72 (1997), 1-20. · Zbl 0898.35095 [102] I. E. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, Adv. Soviet Math., 19 (1994), 181-208. Long-time asymptotics for KdV with steplike initial data Gerald Teschl (joint work with Kyrylo Andreiev, Iryna Egorova, Till Luc Lange) In this work we were concerned with the Cauchy problem for the Korteweg-de Vries (KdV) equation (1)qt(x, t) = 6q(x, t)qx(x, t)− qxxx(x, t),(x, t)∈ R × R+, with steplike initial data q(x, 0) = q0(x) satisfying (2)q0(x)→ 0,as x→ +∞, q0(x)→ c2,as x→ −∞. This case is known as rarefaction problem. The corresponding long-time asymptotics of q(x, t) as t→ ∞ are well understood on a physical level of rigor ([18, 14, 16]) and can be split into three main regions: • In the region x < −6c2t the solution is asymptotically close to the background c2. • In the region −6c2t < x < 0 the solution can asymptotically be described by−6tx. • In the region 0 < x the solution is asymptotically given by a sum of solitons. Reflectionless Operators: The Deift and Simon Conjectures2971 For the corresponding shock problem we refer to [1, 6, 11, 12, 13, 15, 17]. Our aim was to rigorously justify these results. In addition, we were also able to compute the second terms in the asymptotic expansion, which was, to the best of our knowledge, not obtained before. Our approach is based on the nonlinear steepest descent method for oscillatory Riemann-Hilbert (RH) problems developed by Deift and Zhou [5] based on earlier work by Its and Manakov (see [10] for an easy introduction in the case c = 0). In turn, this approach rests on the inverse scattering transform for steplike initial data originally developed by Buslaev and Fomin [2] with later contributions by Cohen and Kappeler [3]. For recent developments and explicit conditions on the initial data q0ensuring unique solvability of the Cauchy problem we refer to [9, 8, 7]. As is known, the solution of the initial value problem (1), (2) can be computed by the inverse scattering transform from the right scattering data of the initial profile. Here the right scattering data are given by the reflection coefficient R(k), k∈ R, a finite number of eigenvalues −κ21, . . . ,−κ2N, and positive norming constants γ1, . . . , γN. The difference with the decaying case c = 0 consists of the fact, that the modulus of the reflection coefficient is equal to 1 on the interval [−c, c]. This implies that the deformation of the initial Riemann-Hilbert problem requires a new phase function, the so-called g function, as first outlined in [4]. At the point k = 0 the reflection coefficient takes the values±1 (cf. [3]). The case R(0) = −1 is known as the nonresonant case (which is generic), whereas the case R(0) = 1 is called the resonant case. Note, that the right transmission coefficient T (k) can be reconstructed uniquely from these data (cf. [2]). Our main results is the following: Let the initial data q0(x)∈ C8(R) of the Cauchy problem (1)–(2) satisfy Z+∞ (3)eκx(|q0(x)| + |q0(−x) − c2|)dx < ∞, 0 for some small κ > 0. Let q(x, t) be the solution of this problem. Then for arbitrary small ǫj> 0, j = 1, 2, 3, and for ξ =12tx, the following asymptotics are valid as t→ ∞ uniformly with respect to ξ: A. In the domain (−6c2+ ǫ1)t < x <−ǫ1t: x + Q(ξ) 6t(1 + O(t−1/3)),as t→ +∞, where √ 2Z−2ξdN π−√−2ξdslog R(s)− 4iXs2κ+ κj2ds∓√1, j=1jps2+ 2ξ2−2ξ with± corresponding to the resonant/nonresonant case, respectively. B. In the domain x < (−6c2− ǫ2)t in the nonresonant case: r 4ντ (6)q(x, t) = c2+sin(16tτ3− ν log(192tτ3) + δ)(1 + o(1)), 3t 2972Oberwolfach Report 49/2017 q where τ = τ (ξ) =c22− ξ, ν = ν(ξ) = −2π1log 1− |R(τ)|2 and 3π δ(ξ) =−+ arg(R(τ )− 2T (τ) + Γ(iν)) 4 1Z1− |R(s)|2s ds πR \(\)−τ,τ ]log1− |R(τ)|2s2− c2− (c2+ ξ)1/2(c2. 2− s2)1/2 Here Γ is the Gamma function. C. In the domain x > ǫ3t: N q(x, t) =−X2κ2j−ǫ3t/2). j=1cosh2κjx− 4κ3jt−12log2κγj−PNlogκi−κj + O(e ji=j+1κi+κj References [103] R.F. Bikbaev, Structure of a shock wave in the theory of the Korteweg-de Vries equation, Phys. Lett. A 141 (1989), 289-293. [104] V.S. Buslaev and V.N. Fomin, An inverse scattering problem for the one-dimensional Schr¨odinger equation on the entire axis, Vestnik Leningrad. Univ. 17 (1962), 56-64. (Russian) [105] A. Cohen and T. Kappeler, Scattering and inverse scattering for steplike potentials in the Schr¨odinger equation, Indiana Univ. Math. J. 34 (1985), 127-180. · Zbl 0553.34015 [106] P. Deift, S. Venakides, and X. Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Applied Math. 47 (1994), 199-206. · Zbl 0797.35143 [107] P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Ann. of Math. (2) 137 (1993), 295-368. · Zbl 0771.35042 [108] I. Egorova, Z. Gladka, V. Kotlyarov, and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation with steplike initial data, Nonlinearity 26 (2013), 1839-1864 . · Zbl 1320.35308 [109] I. Egorova, Z. Gladka, T. L. Lange, and G. Teschl, Inverse scattering theory for Schr¨odinger operators with steplike potentials, Zh. Mat. Fiz. Anal. Geom. 11 (2015), 123-158. · Zbl 1333.34127 [110] I. Egorova, K. Grunert, and G. Teschl, On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data. I. Schwartz-type perturbations, Nonlinearity, 22 (2009), 1431-1457. · Zbl 1171.35103 [111] I. Egorova and G. Teschl, On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data II. Perturbations with finite moments, J. d’Analyse Math. 115 (2011), 71-101. · Zbl 1314.35136 [112] K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom. 12 (2009), 287-324. · Zbl 1179.37098 [113] A.V. Gurevich and L.P. Pitaevskii, Decay of initial discontinuity in the Korteweg-de Vries equation, JETP Letters 17:5 (1973), 193-195. [114] A. V. Gurevich, L.P. Pitaevskii, Nonstationary structure of a collisionless shock wave, Soviet Phys. JETP 38 (1974), 291-297. [115] E.Ya. Khruslov, V.P. Kotlyarov, Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations, in “Spectral operator theory and related topics”, Adv. Soviet Math. 19, 129-180, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0819.58015 [116] J. A. Leach and D. J. Needham, The large-time development of the solution to an initialvalue problem for the Korteweg-de Vries equation: I. Initial data has a discontinuous expansive step, Nonlinearity 21 (2008), 2391-2408. · Zbl 1155.35437 [117] J. A. Leach and D. J. Needham, The large-time development of the solution to an initialvalue problem for the Korteweg-de Vries equation: II. Initial data has a discontinuous compressive step, Mathematika 60 (2014), 391-414. Reflectionless Operators: The Deift and Simon Conjectures2973 · Zbl 1306.37086 [118] V. Yu. Novokshenov, Time asymptotics for soliton equations in problems with step initial conditions, J. Math. Sci. 125 (2005), 717-749. · Zbl 1075.35071 [119] S. Venakides, Long time asymptotics of the Korteweg-de Vries equation, Trans. Amer. Math. Soc. 293 (1986), 411-419. · Zbl 0619.35084 [120] V.E. Zaharov, S.V. Manakov, S.P. Novikov, and L.P. Pitaevskii, Theory of solitons. The method of the inverse problem (Russian), Nauka, Moscow, 1980. Reflectionless discrete Schr¨odinger operators are spectrally atypical Tom VandenBoom A discrete Schr¨odinger operator (DSO) is a self-adjoint linear operator HVon ℓ2(Z) which acts entrywise via (1)(HVu)n= un+1+ un−1+ V (n)un, u∈ ℓ2(Z), where V is a bounded potential funtion V : Z→ R. The Schr¨odinger operator and DSO tend to share many spectral characteristics, and as such the question of identifying the spectral characteristics of a DSO with a fixed almost-periodic potential V is thoroughly studied and reasonably well-understood. On the other hand, one can likewise ask which bounded self-adjoint operators on ℓ2(Z) demonstrate particular spectral characteristics. In this context, the Jacobi operator (2)(Ju)n= anun−1+ bnun+ an+1un+1, u∈ ℓ2(Z) is natural to consider. When a whole-line Jacobi operator is reflectionless – that is, when the diagonal entries of its resolvent matrix tend to purely imaginary values almost everywhere on the spectrum – one can reconstruct the sequences a and b from spectral data. Examples of reflectionless Jacobi operators abound, and are subtly but intimately related to the presence of absolutely continuous spectrum [3, 5, 7, 12]. Fix a positive-measure compact E⊂ R, and define the isospectral torus for E as J (E) := {J : σ(J) ⊂ E and J reflectionless on E}. When E has uniformly positive Lebesgue density,J (E) is homeomorphic to a torus with dimension the number of spectral gaps in E [9]. For such compacts E, certain potential-theoretic properties are directly related to spectral properties of elements ofJ (E). For example, the logarithmic capacity of a finite-gap compact E can be determined as the limit of the geometric means of the off-diagonal sequences of Jacobi operators in the isospectral torus [8, 9]: (3)cap(E) = lim(a1a2· · · an)1/n, J(a, b)∈ J (E). n→∞ Note that, by (3), if E = σ(HV) for some DSO HV, then cap(E) = 1. Consequently, the following question is natural: for a compact E with cap(E) = 1, does there exist a DSO HV∈ J (E)? The titular result provides a negative answer: 2974Oberwolfach Report 49/2017 Theorem 1.For a full-measure dense Gδof finite-gap compacts E having cap(E) = 1,J (E) contains no DSO. This result is related to a result of Hur [4] regarding the sparsity of DSO mfunctions among those for Jacobi operators; however, our result makes stronger claims about this atypicality. In fact, this theorem arises as a straightforward corollary of a dynamical statement. The actionS of conjugation by the left-shift S : δn7→ δn+1preserves both the spectrum and the reflectionless condition, and thus (J (E), S) is a discrete-time dynamical system. This action is minimal if theS orbit of every point J ∈ J (E) is dense. We prove that Theorem 2.Fix a positive-measure compact E⊂ R, and suppose there exists a DSO HV∈ J (E). Then either V is a constant potential V = C, or the dynamical system (J (E), S) is not minimal. At first glance, this seems paradoxical, because it is not hard to find examples of reflectionless, finite-gap DSOs; in particular, any DSO having p-periodic potential function V has at most p− 1 spectral gaps. Another result suggests that further examples may not exist! Theorem 3.Suppose E has cap(E) = 1 and 0, 1, or 2 spectral gaps. If there exists a DSO HV∈ J (E), then V (and the dynamical system (J (E), S)) is periodic. That this result holds for 0-gap spectrum is an old and well-known result [1, 8]; similarly, this result has probably been observed for 1-gap spectra by way of trace formulas. That the theorem holds for 2-gap spectra is quite surprising, but follows via a certain novel application of the Toda flow, which we now describe. Consider a bounded linear operator A on ℓ2(Z). Denote by A±the restrictions of A to ℓ2(Z±) ֒→ ℓ2(Z), where the inclusion map is given by assigning zeros to the left- or right-half line. Fix a polynomial P of degree n + 1≥ 1. The nthToda flow (for P ) is the integral curve J(t) of Jacobi operators satisfying the Lax pair (4)∂tJ = [P (J)+− P (J)−, J]. There exist unique solutions to (4) for any bounded Jacobi initial condition J0[10, Theorem 12.6]. When there exists a monic polynomial P so that (5)[P (J)+− P (J)−, J] = 0. we say J is stationary for P . This definition can even be extended to bounded functions [11]. Stationary solutions to the Toda hierarchy are closely related to reflectionlessness; in a sense, isospectral tori are the level sets of commutators like in (5) [2, 10, 11] For particular choices of polynomial P , the Toda flow induces a system of differential equations on the parametrizing sequences a, b∈ ℓ∞. The critical facts about the Toda flow that we employ are summarized in Proposition 1.For any non-constant polynomial P : Reflectionless Operators: The Deift and Simon Conjectures2975 (1) [6, Corollary 1.3] Suppose J(t) is the unique solution to (4) with J(0) = J0∈ J (E), where E = σ(J0). Then J(t)∈ J (E) for all t ∈ R. (2) [10, Theorem 12.8] The stationary solutions ∂tJ = 0 of (4) are finite-gap reflectionless Jacobi operators. (3) [10, Corollary 12.10] min{deg(P ) : J stationary for P } = # {spectral gaps in σ(J)} + 1, where we interpret min{∅} = ∞. We prove our theorems by leveraging these powerful results against the relative simplicity of the induced differential system under the assumption an= 1 for all n. References [121] G¨oran Borg. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math., 78:1-96, 1946. · Zbl 0063.00523 [122] Fritz Gesztesy, Helge Holden, Johanna Michor, and Gerald Teschl. Soliton equations and their algebro-geometric solutions. Vol. II, volume 114 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008. (1 + 1)-dimensional discrete models. · Zbl 1151.37056 [123] Fritz Gesztesy, Konstantin A. Makarov, and Maxim Zinchenko. Essential closures and AC spectra for reflectionless CMV, Jacobi, and Schr¨odinger operators revisited. Acta Appl. Math., 103(3):315-339, 2008. · Zbl 1165.34050 [124] Injo Hur. The m-functions of discrete Schr¨odinger operators are sparse compared to those for Jacobi operators. arXiv preprint arXiv:1703.03494, 2017. · Zbl 06792228 [125] Shinichi Kotani. Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schr¨odinger operators. In Stochastic analysis (Katata/Kyoto, 1982), volume 32 of North-Holland Math. Library, pages 225-247. North-Holland, Amsterdam, 1984. [126] Alexei Poltoratski and Christian Remling. Approximation results for reflectionless Jacobi matrices. Int. Math. Res. Not., (16):3575-3617, 2011. · Zbl 1225.47033 [127] Christian Remling. The absolutely continuous spectrum of Jacobi matrices. Ann. of Math. (2), 174(1):125-171, 2011. · Zbl 1235.47032 [128] Barry Simon. Szeg˝o’s theorem and its descendants. M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 2011. Spectral theory for L2perturbations of orthogonal polynomials. [129] Mikhail Sodin and Peter Yuditskii. Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal., 7(3):387-435, 1997. · Zbl 1041.47502 [130] Gerald Teschl. Jacobi operators and completely integrable nonlinear lattices, volume 72 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. · Zbl 1056.39029 [131] Victor Vinnikov and Peter Yuditskii. Functional models for almost periodic Jacobi matrices and the Toda hierarchy. Mat. Fiz. Anal. Geom., 9(2):206-219, 2002. · Zbl 1102.47020 [132] Alexander Volberg and Peter Yuditskii. Kotani-Last problem and Hardy spaces on surfaces of Widom type. Invent. Math., 197(3):683-740, 2014. 2976Oberwolfach Report 49/2017 Solutions to the KdV hierarchy via finite-gap approximation Tom VandenBoom (joint work with Benjamin Eichinger and Peter Yuditskii) Consider the one-dimensional Schr¨odinger operator Lq=−∆ + q where ∆ = d2/dx2is the Laplacian and q : R→ R is a potential function. For a potential function q of sufficient regularity (namely, q∈ Cm+3(R, R)), one can iteratively construct a sequence of differential polynomials depending on L = Lq by (1)f0(q) = 1 (2)4∂x3fm−1+ q∂xfm−1+12fm−1∂xq The kthKdV hierarchy, whose initial member was proposed by Korteweg and de Vries in the late 19th century [4], is defined by KdVk(q) :=−2∂xfk+1(q). We study the Cauchy problem for the KdV hierarchy; that is, we study solutions q∈ C2k+1,1(R× R, R) to the partial differential equation (3)∂tkq = KdVk(q), (4)q(·, 0) = q0 for initial conditions q0satisfying certain regularity properties. Consider a closed set E⊂ R which is bounded from below such that the domain Ω = CE is of Widom type. By translation, it is no assumption to let inf E = 0. We thus can write E as the right half-line with an at-most countable set of maximal gaps removed; that is, E can be written as ∞ (5)E = [0,∞) \(\)(aj, bj). j=1 Denote byQ(E) the set of potentials q for which Lqis reflectionless on its spectrum E. Joint with B. Eichinger and P. Yuditskii, we proved the following result: Theorem 1.Suppose E⊂ R is closed and bounded below of the form (5) such that ∞ X (6)bk+2j− ak+2j<∞, j=1 and suppose q0∈ Q(E). Then there exists a classical solution q = q(x, tk)∈ C2(k+1),1(R× R, R) to the Cauchy problem (3), (4). Reflectionless Operators: The Deift and Simon Conjectures2977 Furthermore, this solution is almost periodic in both the x and tkcoordinates, in the sense that there exists a continuous mapM : T∞→ Q(E) and vectors α0∈ T∞and δx, δtk∈ R∞so that q(x, tk) =M(α0+ xδx+ tkδtk). Results of this kind are contributed to a conjecture of Deift [2] and have been explored to a certain extent in a variety of previous results [1, 3]. The condition (6) is optimal for the existence of classical solutions to the Cauchy problem (3), (4). As an example, consider the case k = 0: if one only assumes finite total gap length, one can only conclude continuity of the associated potentials, which cannot in general be classical solutions to the associated KdV equation ∂tq = KdV0(q) =−∂xq. Explicit examples which are not differentiable can be constructed. Our methods of proving Theorem 1 can be viewed to a certain extent as a refinement of those methods developed in [1] for KdV1; specifically, in their paper they prove an analogous result (with the additional conclusion of uniqueness) to Theorem 1 via the following approximate scheme: (1) Find the KdV flow on the Dirichlet data of an initial condition q∈ Q(E) (2) Assume Craig-type conditions to achieve existence and uniqueness for the associated flowon Dirichlet data. (3) Use previous results for finite-gap spectra to conclude almost-periodicity for finite-gap approximants. (4) Pass to the infinite-gap limit under Craig-type conditions and uniform convergence. Our methods replace items (1) and (2) above by the character-automorphism techniques of Sodin and Yuditskii [5]. Specifically, we find the flow associated to the KdV hierarchy on the characters of the finite-gap approximants, and pass to the limit using the condition (6). This allows for some simplification of the assumptions, but at the moment only allows for the weaker conclusion of existence. We believe that by using the theory of Abelian coverings, one can find the precise conditions for existence and uniqueness in an approach similar to that developed for the Toda flow by Vinnikov and Yuditskii [7]. References [133] Ilia Binder, David Damanik, Michael Goldstein, Milivoje Lukic. Almost Periodicity in Time of Solutions of the KdV Equation. Preprint (arXiv:1509.07373). · Zbl 1406.35325 [134] Percy Deift. Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices, 419-430, Contemp. Math. 458, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1147.15305 [135] Iryna Egorova. Almost periodicity of some solutions of the KdV equation with Cantor spectrum. Dopov./Dokl. Akad. Nauk Ukraini, 26-29, 1993. · Zbl 0900.35327 [136] Diederick Korteweg and Gustav de Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag., 539:422-443, 1895. · JFM 26.0881.02 [137] Mikhail Sodin and Peter Yuditskii. Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum. Comment. Math. Helvetici, 70:639-658, 1995. 2978Oberwolfach Report 49/2017 · Zbl 0846.34024 [138] Mikhail Sodin and Peter Yuditskii. Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal., 7(3):387-435, 1997. · Zbl 1041.47502 [139] Victor Vinnikov and Peter Yuditskii. Functional models for almost periodic Jacobi matrices and the Toda hierarchy. Mat. Fiz. Anal. Geom., 9(2):206-219, 2002. Killip-Simon problem and Jacobi flow on GMP matrices Peter Yuditskii One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuous (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schr¨odinger operator and B is an arbitrary Jacobi matrix (of Hilbert-Schmidt class) the a.c. spectrum remains perfectly the same (Deift-Killip [3]), that is, the interval [−2, 2]. Moreover, Killip and Simon [5] described explicitly the spectral properties for such A + B. Jointly with Damanik [2] they generalized this result to the case of perturbations of periodic Jacobi matrices in the non-degenerated case. Recall that the spectrum of a periodic Jacobi matrix is a system of intervals of a very specific nature. Christiansen, Simon and Zinchenko [1] posed in a review dedicated to F. Gesztesy the following question: “is there an extension of the Damanik-Killip-Simon theorem to the general finite system of intervals case?” In [7] this problem was solved completely. Our method deals with the Jacobi flow on GMP matrices. GMP matrices are probably a new object in the spectral theory. They form a certain Generalization of matrices related to the strong Moment Problem [4], the latter ones are a very close relative of Jacobi and CMV matrices. The Jacobi flow on them is also a probably new member of the rich family of integrable systems. Finally, related to Jacobi matrices of Killip-Simon class, analytic vector bundles and their curvature play a certain role in our construction and, at least on the level of ideology, this role is quite essential. In this talk we concentrate on the functional model for periodic GMP matrices and prove the so-called “magic formula” for them as an evident consequence of this model. For a finite gap set E = [b0, a0]\∪gj=1(aj, bj), let D/Γ≃ C\E be a uniformization of the given domain with the covering map function z : D→ C E and the Fuchsian group Γ, z◦ γ = z, ∀γ ∈ Γ. Let Γ∗be the corresponding group of unitary characters. For α∈ Γ∗we define the Hardy space of character automorphic functions as H2(α) ={f ∈ H2: f◦ γ = e2πiα(γ)f, γ∈ Γ}, where H2denotes the standard Hardy class in D. We define two special functions: the so-called Green function b = b∞of the group Γ, which is the Blaschke product with zeros at z−1(∞) = {γ(ζ0)}γ∈Γ, and Reflectionless Operators: The Deift and Simon Conjectures2979 the reproducing kernel kα= kα∞of the space H2(α), i.e., hf, kαi = f(ζ0)∀f ∈ H2(α). Using these special functions, one can give a parametric description of the class of reflectionless Jacobi matrices J(E) with the spectrum E (in a much more general form the following theorem was proved in [6]). Theorem 1.The system of functions eαn(ζ) = bn(ζ)kα−nµ(ζ) pkα−nµ(0) forms an orthonormal basis in H2(α) for n∈ N. The multiplication operator by z with respect to the basis{eαn(ζ)}n∈Zis the Jacobi matrix J(α) with the coefficients {a(n; α), b(n; α)}n∈Z: zeαn= a(n; α)eαn−1+ b(n; α)eαn+ a(n + 1; α)eαn+1. Moreover, J(E) ={J(α) : α ∈ Γ∗}. Recall that the set E is the spectrum of a periodic Jacobi matrix if and only if there exists a polynomial Tn(z) such that E = Tn−1[−2, 2]. In this case the isospectral set J(E) can be described as a collection of Jacobi matrices J, which satisfies the following (“magic”) formula Tn(J) = Sn+ S−n, where S is the standard shift operator. One of our main ideas in solving the Killip-Simon problem deals with the fact that for an arbitrary finite gap set E there exists an essentially unique rational function ∆(z) of the form g ∆(z) = λ0z + c0+Xλj, λ cj− zj> 0, cj∈ (aj, bj), j=1 such that E = ∆−1[−2, 2]. Its Zhukovskii transform Ψ(z), ∆(z) =Ψ(z)1+ Ψ(z), is a single valued function in the domain ¯CE. Moreover, this is a product of the complex Green functions Ψ(z(ζ)) = b(ζ)Qgj=1bcj(ζ). We substitute the orthonormal system{eαn(ζ)} by the system (1)fαn= fαn(ζ; c1, . . . , cg) = Ψmfαj,n = (g + 1)m + j, j∈ [0, . . . , g] where (with a suitable constants φj∈ R/Z) fα0=q, fα1=c1kζα−µ2c1, ..., fαQg kζα(ζ1)qkα−µc1(ζg=j=1bcjkα+µ. 1ζ22)pkα+µ(0) 2980Oberwolfach Report 49/2017 Theorem 2.In the above notations the multiplication operator by z with respect to the basis{fαn}n∈Zis a periodic GMP matrix A(α). Moreover, the isospectral set A(E) of periodic GMP matrices has the form A(E) ={A(α) : α ∈ Γ∗}, and can be described as the collection of GMP matrices A, which satisfies the following (“magic”) formula ∆(A) = Sg+1+ S−(g+1). References [140] J. S. Christiansen, B. Simon, and M. Zinchenko, Finite gap Jacobi matrices: A review, Proc. Sympos. in Pure Math. 87 (2013), 87-103. · Zbl 1319.47028 [141] D. Damanik, R. Killip, and B. Simon, Perturbations of orthogonal polynomials with periodic recursion coefficients, Annals of Math. 171 (2010), no. 3., 1931-2010. · Zbl 1194.47031 [142] P. A. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schr¨odinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), 341-347. · Zbl 0934.34075 [143] B. Eichinger, F. Puchhammer, and P. Yuditskii, Jacobi Flow on SMP Matrices and KillipSimon Problem on Two Disjoint Intervals, Comput. Methods and Funct., 16 (2016), no. 1, 3-41. · Zbl 1372.47041 [144] R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Annals of Math. 158 (2003), no. 2, 253-321. · Zbl 1050.47025 [145] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), 387-435. · Zbl 1041.47502 [146] P. Yuditskii, Killip-Simon problem and Jacobi flow on GMP matrices, Advances in Mathematics (to appear). Lieb-Thirring inequalities for finite and infinite gap Jacobi matrices Maxim Zinchenko (joint work with Jacob Christiansen) In 1976, Lieb and Thirring in their work on stability of matter [9, 10] obtained an upper bound on the moments of discrete eigenvalues of a Schr¨odinger operator. For one-dimensional Schr¨odinger operators the bound takes the form Z (1)|λ|p−1/2≤ Lp|V (x)|pdx,p≥ 1, λ∈σd(A)R where Lpis a constant independent of V . The bound is false for p < 1. In the original work the inequality was derived for p > 1 and the endpoint result for p = 1 was proven only 20 years later by Weidl [11]. Lieb-Thirring inequalities have found applications in the studies of quantum mechanics, differential equations, and dynamical systems, see e.g., [7] for a history of the subject. Reflectionless Operators: The Deift and Simon Conjectures2981 An analogous Lieb-Thirring inequality for discrete Schr¨odinger operators and more generally for Jacobi operators on ℓ2(Z), represented by the tridiagonal Jacobi matrices . .. ... ... a0b1a1 (2)J =a1b2a2, a2b3a3 . .. ... ... was obtained in [8] for perturbations of the discrete Laplacian and in [4] for perturbations of periodic Jacobi matrices. In either of these settings denoting the unperturbed Jacobi matrix by J0, its spectrum by E, and letting the Jacobi matrix J = J0+ δJ be a compact self-adjoint perturbation of J0leads to the following Lieb-Thirring bound for all p≥ 1, ∞ (3)dist λ, Ep−12≤ Lp, EX|δan|p+|δbn|p, λ∈σd(J)n=−∞ where the constant Lp, Eis independent of both J0and δJ and may depend only on p and E. In [5] the case p = 1 of the above Lieb-Thirring bound was proven for perturbations of Jacobi matrices from arbitrary finite gap isospectral toriTE, hence extending the bound to perturbations of certain quasi-periodic Jacobi matrices. In the joint work with J. S. Christiansen [2], we investigate possible extensions of the Lieb-Thirring bound to more general classes of Jacobi matrices, in particular, to perturbations of certain almost periodic Jacobi matrices with infinite gap spectrum. In [2] we obtain the following abstract version of the Lieb-Thirring bound. Suppose J0and J = J0+ δJ are two-sided Jacobi matrices and δJ is a compact self-adjoint perturbation. Let E be the spectrum of J0and denote the gaps of E by (αk, βk), k≥ 0, so that E =β0, α0Sj≥1αj, βj. In addition, let dρnbe the spectral measures of (J0, δn), n∈ Z, and suppose that supn(t)≤Ck,x∈ (α (4)n∈ZE|t − x|dist(x, E)1/2k, βk),k≥ 0, for some summable sequence{Ck}k≥0. Then σess(J) = E and the discrete eigenvalues of J satisfy the Lieb-Thirring bound (3) with p > 1 and the constant Lp,E that depends only on p and the sequence{Ck}k≥0. We then show that the abstract result applies (i.e., assumption (4) is satisfied with the constants{Ck}k≥0depending only on E) in particular situations of almost periodic J0from isospectral toriTE for sets E of three types: (i) arbitrary finite gap sets; (ii) Cantor sets E =T∞k=0Ek, where E0= [β0, α0] and Ekis obtained from Ek−1by removing the middle ǫk portion from each of the 2k−1bands in Ek−1, with parameters{ǫk}k≥1⊂ (0, 1) satisfying lim supk→∞(ǫk)1/k< 1/4; (iii) infinite band sets E =T∞k=0Ek, where E0= [β0, α0] and Ekis obtained from Ek−1by removing the middle ǫkportion from the first of the k bands in Ek−1, with parameters{ǫk}k≥1⊂ (0, 1) satisfying 2982Oberwolfach Report 49/2017 P∞√ k=1ǫklog(1/ǫk) <∞. The endpoint case p = 1 of the Lieb-Thirring inequality (3) for perturbations of Jacobi matrices from infinite gap isospectral tori is an open problem. Another open problem is to find a characterization of infinite gap sets for which there is a Lieb-Thirring bound. In recent years, Lieb-Thirring-type bounds have been also obtained for Schr¨odinger operators with complex potentials. In the setting of tridiagonal Jacobi matrices with complex coefficients . .. ... ... a0b1c1 (5)J =a1b2c2, a2b3c3 . .. ... ... a Lieb-Thirring-type bound was derived in [1, 6] for compact non-self-adjoint perturbations J = J0+ δJ of the discrete Laplacian J0. The bound asserts that for every 0 < ǫ < 1, dist z, [−2, 2]p+ǫ∞ (6)|z2− 4|12≤ Lp,ǫn=−∞X|δan|p+|δbn|p+|δcn|p,p≥ 1, z∈σd(J) where the eigenvalues are repeated according to their algebraic multiplicity and the constant Lp,ǫis independent of δJ. It is currently not known whether or not the inequality continues to hold for ǫ = 0. In the joint work with J. S. Christiansen [3], we improve the non-self-adjoint Lieb-Thirring bound (6) and extend it to perturbations of periodic Jacobi matrices and more generally to perturbations of quasi-periodic Jacobi matrices from finite gap isospectral toriTE. Give an arbitrary finite gap set E⊂ R, let J0∈ TEand suppose that J = J0+ δJ is a compact non-self-adjoint perturbation of J0. In [3] we show that for every 0 < ǫ < 1, dist z, Ep+ǫ(1 +|z|)12−2ǫ∞ (7)dist(z, ∂E)21≤ Lǫ,p,En=−∞X|δan|p+|δbn|p+|δcn|p, z∈σd(J) where the eigenvalues are repeated according to their algebraic multiplicity and the constant Lǫ,p,Eis independent of both J0and δJ and may depend only on ǫ, p, and E. We note that (7) is new even when J0is the discrete Laplacian since, unlike (6), it is nearly optimal not only for small but also for large perturbations. As with (6), it is an open problem whether (7) remains true for ǫ = 0. Another 1 open problem is to determine whether it is possible to replace dist(z, ∂E)2in the 1 denominator on the left hand-side by dist(z, E)2. Reflectionless Operators: The Deift and Simon Conjectures2983 References [147] A. Borichev, L. Golinskii, and S. Kupin, A Blaschke-type condition and its application to complex Jacobi matrices, Bull. Lond. Math. Soc. 41 (2009), no. 1, 117-123. · Zbl 1175.30007 [148] J. S. Christiansen and M. Zinchenko, Lieb-Thirring inequalities for finite and infinite gap Jacobi matrices, Ann. Henri Poincar´e 18 (2017), 1949-1976. · Zbl 1368.81073 [149] J. S. Christiansen and M. Zinchenko, Lieb-Thirring inequalities for complex finite gap Jacobi matrices, Lett. Math. Phys. 107 (2017), 1769-1780. · Zbl 06793826 [150] D. Damanik, R. Killip, and B. Simon, Perturbations of orthogonal polynomials with periodic recursion coefficients, Ann. of Math. 171 (2010), 1931-2010. · Zbl 1194.47031 [151] R. L. Frank and B. Simon, Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices, Duke Math. J. 157 (2011), no. 3, 461-493. · Zbl 1229.35157 [152] M. Hansmann and G. Katriel, Inequalities for the eigenvalues of non-selfadjoint Jacobi operators, Complex Anal. Oper. Theory 5 (2011), no. 1, 197-218. · Zbl 1222.47039 [153] D. Hundertmark, Some bound state problems in quantum mechanics. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math. 76, Part 1, Amer. Math. Soc., Providence, 2007, 463-496. · Zbl 1126.81025 [154] D. Hundertmark and B. Simon, Lieb-Thirring inequalities for Jacobi matrices, J. Approx. Theory 118 (2002), 106-130. · Zbl 1019.39013 [155] E. H. Lieb and W. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett. 35 (1975) 687-689; Phys. Rev. Lett. 35 (1975) 1116, Erratum. [156] E. H. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schr¨odinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann, E. H. Lieb, B. Simon, and A. S. Wightman (eds.), Princeton University Press, Princeton, NJ, 1976, pp. 269-303. [157] T. Weidl, On the Lieb-Thirring constants Lγ,1for γ ≥ 1/2, Comm. Math. Phys. 178 (1996), 135-146. · Zbl 0858.34075 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.