## Rank-one perturbations and Anderson-type Hamiltonians.(English)Zbl 1419.81013

Summary: Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $H_\omega=H+V_\omega$ on a separable Hilbert space $$\mathcal{H}$$, where the perturbation is given by $V_\omega=\sum_n\omega_n(\cdot,\phi_n)\phi_n$ with a sequence $$\{\phi_n\}\subset \mathcal{H}$$ and independent identically distributed random variables $$\omega_n$$. We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 47B80 Random linear operators 81Q15 Perturbation theories for operators and differential equations in quantum theory
Full Text:

### References:

 [1] E. Abakumov, C. Liaw, and A. Poltoratskiĭ, Cyclicity in rank-$$1$$ perturbation problems, J. Lond. Math. Soc. (2) 88 (2013), no. 2, 523-537. · Zbl 1283.47011 [2] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), no. 5, 1492-1505. [3] N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equations, Amer. J. Math. 79 (1957), no. 3, 597-610. · Zbl 0079.10802 [4] J. Bellissard, P. Hislop, A. Klein, and G. Stolz, Random Schrödinger Operators: Universal Localization, Correlations, and Interactions, conference lecture, April 2009, preprint, available at https://www.birs.ca/workshops//2009/09w5116/report09w5116.pdf. [5] R. V. Bessonov, Truncated Toeplitz operators of finite rank, Proc. Amer. Math. Soc. 142 (2014) no. 4, 1301-1313. · Zbl 1314.47040 [6] M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-adjoint Operators in Hilbert Space, Math. Appl. (Soviet Ser.) 5, Reidel, Dordrecht, 1987. [7] A. Blandignères, E. Fricain, F. Gaunard, A. Hartmann, and W. Ross, Reverse Carleson embeddings for model spaces, J. Lond. Math. Soc. (2) 88 (2013) no. 2, 437-464. · Zbl 1308.30061 [8] R. W. Carey and J. D. Pincus, Unitary equivalence modulo the trace class for self-adjoint operators, Amer. J. Math. 98 (1976), no. 2, 481-514. · Zbl 0362.47007 [9] J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy Transform, Math. Surveys Monogr. 125, Amer. Math. Soc., Providence, 2006. · Zbl 1096.30046 [10] H. Cycon, R. Froese, W. Kirsh, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer, Berlin, 1987. [11] W. F. Donoghue, On the perturbation of spectra, Comm. Pure Appl. Math. 18 (1965), 559-579. · Zbl 0143.16403 [12] V. Jakšić and Y. Last, Spectral structure of Anderson type Hamiltonians, Invent. Math. 141 (2000), no. 3, 561-577. · Zbl 0962.60056 [13] V. Jakšić and Y. Last, A new proof of Poltoratskii’s theorem, J. Funct. Anal. 215 (2004), no. 1, 103-110. · Zbl 1070.47012 [14] V. Jakšić and Y. Last, Simplicity of singular spectrum in Anderson-type Hamiltonians, Duke Math. J. 133 (2006), no. 1, 185-204. · Zbl 1107.47027 [15] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995. · Zbl 0836.47009 [16] W. King, R. C. Kirby, and C. Liaw, Delocalization for the $$3$$-D discrete random Schrödinger operators at weak disorder, J. Phys. A 47 (2014), no. 30, Art. ID 305202. [17] C. Liaw, Approach to the extended states conjecture, J. Stat. Phys. 153 (2013), no. 6, 1022-1038. · Zbl 1302.82011 [18] C. Liaw and S. Treil, Rank one perturbations and singular integral operators, J. Funct. Anal. 257 (2009), no. 6, 1947-1975. · Zbl 1206.42012 [19] C. Liaw and S. Treil, Clark model in the general situation, J. Anal. Math. 130 (2016), no. 1, 287-328. · Zbl 1491.47012 [20] C. Liaw and S. Treil, “Singular integrals, rank one perturbations and Clark model in general situation” in Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory, Vol. 2, Assoc. Women Math. Ser. 5, Springer, Cham, 2017, 85-132. · Zbl 1485.47016 [21] M. Martin and M. Putinar, Lectures on Hyponormal Operators, Oper. Theory Adv. Appl. 39, Birkhäuser, Basel, 1989. [22] A. G. Poltoratskiĭ, The boundary behavior of pseudocontinuable functions (in Russian), Algebra i Analiz 5 (1993), no. 2, 189-210; English translation in St. Petersburg Math. J. 5 (1994), no. 2, 389-406. [23] A. G. Poltoratskiĭ, Kreĭn’s spectral shift and perturbations of spectra of rank one (in Russian), Algebra i Analiz 10 (1998), no. 5, 143-183; English translation in St. Petersbg. Math. J. 10 (1999), no. 5, 833-859. [24] A. G. Poltoratskiĭ, Equivalence up to a rank one perturbation, Pacific J. Math. 194 (2000), no. 1, 175-188. [25] A. G. Poltoratskiĭ and D. Sarason, “Aleksandrov-Clark measures” in Recent Advances in Operator-related Function Theory (Dublin, 2004), Contemp. Math. 393, Amer. Math. Soc., Providence, 2006, 1-14. [26] B. Simon, Trace Ideals and Their Applications, 2nd ed., Math. Surveys Monogr. 120, Amer. Math. Soc., Providence, 2005. · Zbl 1074.47001 [27] B. Simon and T. Wolff, Localization in the general one-dimensional random systems, I: Jacobi matrices, Comm. Math. Phys. 102 (1985), no. 2, 327-336. · Zbl 0604.60062 [28] B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986) 75-90. · Zbl 0609.47001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.