Recent zbMATH articles in MSC 34Lhttps://zbmath.org/atom/cc/34L2021-06-15T18:09:00+00:00WerkzeugOn the deformation of linear Hamiltonian systems.https://zbmath.org/1460.350732021-06-15T18:09:00+00:00"Schmid, Harald"https://zbmath.org/authors/?q=ai:schmid.haraldSummary: For linear Hamiltonian \(2 n \times 2 n\) systems \(J y^\prime(x) = (\lambda W(x) + H(x)) y(x)\) we investigate the problem how the eigenvalues \(\lambda\) depend on the entries of the coefficient matrix \(H\). This question turns into a deformation equation for \(H\) and a partial differential equation for the eigenvalues \(\lambda \). We apply our results to various examples, including generalizations of the confluent Heun equation and the Chandrasekhar-Page angular equation. We are mainly concerned with the \(2 \times 2\) case, and in order to reduce the degrees of freedom in \(H\) as much as possible, we will first convert such systems into a complementary triangular form, which is a canonical form with a minimum number of free parameters. Furthermore, we discuss relations to monodromy preserving deformations and to matrix Lax pairs.
Reviewer: Reviewer (Berlin)Cauchy problem of the non-self-adjoint Gauss-Laguerre semigroups and uniform bounds for generalized Laguerre polynomials.https://zbmath.org/1460.340732021-06-15T18:09:00+00:00"Patie, Pierre"https://zbmath.org/authors/?q=ai:patie.pierre"Savov, Mladen"https://zbmath.org/authors/?q=ai:savov.mladen-svetoslavovSummary: We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be natural non-selfadjoint and non-local generalizations of the Laguerre differential operator. Our methods rely on intertwining relations that we establish between these semigroups and the classical Laguerre semigroup and combine with techniques based on non-harmonic analysis. As a by-product we also provide regularity properties for the semigroups as well as for their heat kernels. The biorthogonal sequences that appear in their eigenvalue expansion can be expressed in terms of sequences of polynomials, and they generalize the Laguerre polynomials. By means of a delicate saddle point method, we derive uniform asymptotic bounds that allow us to get an upper bound for their norms in weighted Hilbert spaces. We believe that this work opens a way to construct spectral expansions for more general non-self-adjoint Markov semigroups.
Reviewer: Reviewer (Berlin)A periodic boundary value problem for a fourth order differential operator with a summable potential.https://zbmath.org/1460.340282021-06-15T18:09:00+00:00"Mitrokhin, Sergeĭ Ivanovich"https://zbmath.org/authors/?q=ai:mitrokhin.sergei-ivanovichSummary: The paper is devoted to the study of a fourth-order differential operator with a summable potential and periodic boundary conditions. The method of studying of operators with a summable potential is an extension of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise when studying the oscillations of beams and bridges composed from materials of different density. The solution of the differential equation is reduced to the solution of the Volterra integral equation. The integral equation is solved by Picard's method of successive approximations. The aim of the, investigation of the integral equation is to obtain asymptotic formulas and estimates for the solutions of the differential equation that defines the differential operator. Questions of geophysics, quantum mechanics, kinetics, gas dynamics and the theory of oscillations of rods, beams and membranes require the development of asymptotic methods for the case of differential equations with nonsmooth coefficients. Asymptotic methods continue to evolve, despite the rapid progress in numerical methods associated with the advent of supercomputers; at present asymptotic and numerical methods complement each other. In the paper, for large values of the spectral parameter, the asymptotics of the solutions of the differential equation that defines the differential operator is obtained. Asymptotic estimates for solutions are established similarly to the asymptotic estimates of solutions of a second-order differential operator with smooth coefficients. The study of periodic boundary conditions leads to the study of the roots of a function represented in the form of a fourth order determinant. To obtain the roots of this function, an indicator diagram has been examined. The roots are in four sectors of an infinitesimal angle, determined by the indicator diagram. The behavior of the roots of this equation in each of the sectors of the indicator diagram is investigated. The asymptotics of eigenvalues of the differential operator under consideration is found. The formulas obtained for the asymptotics of the eigenvalues make it possible to study the spectral properties of the eigenfunctions. If the potential of the operator is not a summable function, but only piecewise smooth, then the obtained formulas for the asymptotics of the eigenvalues are sufficient to derive the formula for the first regularized trace of the differential operator under study.
Reviewer: Reviewer (Berlin)Floquet theory of the analytical solution of a periodically driven two-level system.https://zbmath.org/1460.811262021-06-15T18:09:00+00:00"Schmidt, Heinz-Jürgen"https://zbmath.org/authors/?q=ai:schmidt.heinz-jurgen"Schnack, Jürgen"https://zbmath.org/authors/?q=ai:schnack.jurgen"Holthaus, Martin"https://zbmath.org/authors/?q=ai:holthaus.martinSummary: We investigate the analytical solution of a two-level system subject to a monochromatical, linearly polarized external field that was first published in 2007. In particular, we derive an explicit expression for the quasienergy. Moreover, we calculate the time evolution of a typical two-level system over a full period by evaluating series solutions of the confluent Heun equation. This is possible without invoking the connection problem of this equation since the complete time evolution of the system under consideration can be reduced to that of the first quarter-period. As a physical application we consider the work performed on a two-level system.
Reviewer: Reviewer (Berlin)Completeness theorem for the system of eigenfunctions of the complex Schrödinger operator \(\mathcal{L}_c = - d^2 / d x^2 + c x^{2 / 3} \).https://zbmath.org/1460.341042021-06-15T18:09:00+00:00"Tumanov, Sergey"https://zbmath.org/authors/?q=ai:tumanov.sergeySummary: The completeness of the system of eigenfunctions of the complex Schrödinger operator \(\mathcal{L}_c = -d^2 / dx^2 + cx^{2/3}\) on the semi-axis in \(L_2 (\mathbb{R}_+)\) with Dirichlet boundary conditions is proved for all \(c: |\arg c| < \pi / 2 + \theta_0\), where \(\pi / 10 < \theta_0 < \pi / 2\) is determined as the only solution of a certain transcendental equation.
Reviewer: Reviewer (Berlin)Eigenvectors from eigenvalues: the case of one-dimensional Schrödinger operators.https://zbmath.org/1460.340332021-06-15T18:09:00+00:00"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritz"Zinchenko, Maxim"https://zbmath.org/authors/?q=ai:zinchenko.maximFollowing the study [\textit{P. B. Denton} et al., ``Eigenvectors from eigenvalues: a survey of a basic identity in linear algebra'', Preprint, \url{arXiv:1908.03795}], the authors extend the eigenvector-eigenvalue identity (this name was given by the authors of [loc. cit.]) for \(n\times n\) matrices to one dimensional self-adjoint Dirichlet Schrödinger operators on compact intervals.
Reviewer: Erdogan Sen (Tekirdağ)Canonical forms of self-adjoint boundary conditions for regular differential operators of order three.https://zbmath.org/1460.340342021-06-15T18:09:00+00:00"Niu, Tian"https://zbmath.org/authors/?q=ai:niu.tian"Hao, Xiaoling"https://zbmath.org/authors/?q=ai:hao.xiaoling"Sun, Jiong"https://zbmath.org/authors/?q=ai:sun.jiong"Li, Kun"https://zbmath.org/authors/?q=ai:li.kun.2Summary: In this paper, we find all canonical forms for third order self-adjoint boundary conditions. These canonical forms play an important role in the study of the dependence of the eigenvalues on the problem and for their numerical calculation. In order to obtain those canonical forms, we give a classification of self-adjoint boundary conditions. Those self-adjoint boundary conditions can be categorized into three mutually exclusive classes: coupled, strictly separated and mixed. Unlike the even order case, for the third order case, the strictly separated self-adjoint boundary conditions can not be realized. For coupled and mixed cases, there are some different types for the canonical forms: 2 for coupled and 4 for mixed boundary conditions.
Reviewer: Reviewer (Berlin)A constructive approach to topological invariants for one-dimensional strictly local operators.https://zbmath.org/1460.810292021-06-15T18:09:00+00:00"Tanaka, Yohei"https://zbmath.org/authors/?q=ai:tanaka.yoheiSummary: In this paper we shall focus on one-dimensional strictly local operators, the notion of which naturally arises in the context of discrete-time quantum walks on the one-dimensional integer lattice \(\mathbb{Z} \). In particular, we give an elementary constructive approach to the following two topological invariants associated with such operators: Fredholm index and essential spectrum. As a direct application, we shall explicitly compute and fully classify these topological invariants for a well-known quantum walk model.
Reviewer: Reviewer (Berlin)Distribution and number of focal points for linear Hamiltonian systems.https://zbmath.org/1460.370822021-06-15T18:09:00+00:00"Šepitka, Peter"https://zbmath.org/authors/?q=ai:sepitka.peter"Šimon Hilscher, Roman"https://zbmath.org/authors/?q=ai:simon-hilscher.romanSummary: In this paper we consider the question of distribution and number of left and right focal points for conjoined bases of linear Hamiltonian differential systems. We do not assume any complete controllability (identical normality) condition. Recently we obtained the Sturmian separation theorem for this case which provides optimal lower and upper bounds for the numbers of left and right focal points of every conjoined basis in terms of the principal solutions at the endpoints of the interval. In this paper we show that for any two given integers within these bounds there exists a conjoined basis with these prescribed numbers of left and right focal points. We determine such conjoined bases by their initial conditions. Our approach is to transfer the problem through the comparative index into matrix analysis. The main results are new even for completely controllable linear Hamiltonian systems. As an application we extend a classical result for controllable systems by \textit{W. T. Reid} [Ordinary differential equations. Applied Mathematics Series. New York, NY: John Wiley \& Sons (1971; Zbl 0212.10901)] about the existence of conjoined bases with an invertible first component.
Reviewer: Reviewer (Berlin)A representation for Jost solutions and an efficient method for solving the spectral problem on the half line.https://zbmath.org/1460.341052021-06-15T18:09:00+00:00"Delgado, Briceyda B."https://zbmath.org/authors/?q=ai:delgado.briceyda-b"Khmelnytskaya, Kira V."https://zbmath.org/authors/?q=ai:khmelnytskaya.kira-v"Kravchenko, Vladislav V."https://zbmath.org/authors/?q=ai:kravchenko.vladislav-vThe paper presents a method how to numerically solve the spectral problem for a half-line Schrödinger operator. The Schrödinger operator acting as \(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+q(x)\) on the half-line \(x\in (0,\infty)\) with a short-range potential \(q\) satisfying \((1+x)q(x) \in L_1(0,\infty)\) is considered. For the spectral problem (and for the scattering theory as well), the Jost solution \(e(\rho, x)\) asymptotically behaving for \(x\to \infty\) as \(\mathrm{e}^{i\rho x}\), \(\mathrm{Im\,}\rho \geq 0\) is useful. The authors review the Levin's representation of the Jost solution
\[
e(\rho,x) = \mathrm{e}^{i\rho x} + \int_x^\infty A(x,t) \,\mathrm{e}^{i\rho t}\,\mathrm{d}t\,,
\]
with the kernel \(A(x,x) = \frac{1}{2}\int_x^\infty q(t)\,\mathrm{d}t\), \(A(x,\cdot) \in L_2(x,\infty)\) (see, e.g. [\textit{K. Chadan} and \textit{P. C. Sabatier}, Inverse problems in quantum scattering theory. 2nd ed., revised and expanded. New York etc.: Springer-Verlag (1989; Zbl 0681.35088)]) and Fourier-Laguerre expansion of the above kernel (see [\textit{V.V. Kravchenko}, Math. Methods Appl. Sci. 42, No. 4, 1321--1327 (2019; Zbl 1414.34068)]).
The main contribution of the current paper is a method of constructing the coefficients of the Fourier-Laguerre expansion. This is used for numerically solving the spectral problem. The problem of computing eigenvalues is reduced to finding the roots of a polynomial. Moreover, the authors give formulæ to find the derivative of the spectral density. The procedures are illustrated in several simple examples. The authors claim that their method is very effective and that the computation of these examples took only several seconds.
Reviewer: Jiři Lipovský (Hradec Králové)On the quotient quantum graph with respect to the regular representation.https://zbmath.org/1460.580192021-06-15T18:09:00+00:00"Mutlu, Gökhan"https://zbmath.org/authors/?q=ai:mutlu.gokhanSummary: Given a quantum graph \(\Gamma\), a finite symmetry group \(G\) acting on it and a representation \(R\) of \(G\), the quotient quantum graph \(\Gamma/R\) is described and constructed in the literature [1,2,18]. In particular, it was shown that the quotient graph \(\Gamma/\mathbb{C}G\) is isospectral to \(\Gamma\) by using representation theory where \(\mathbb{C}G\) denotes the regular representation of \(G\) [18]. Further, it was conjectured that \(\Gamma\) can be obtained as a quotient \(\Gamma/\mathbb{C}G\) [18]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph \(\Gamma\) and a finite symmetry group \(G\) acting on it, the quotient quantum graph \(\Gamma/\mathbb{C}G\) is not only isospectral but rather identical to \(\Gamma\) for a particular choice of a basis for \(\mathbb{C}G\). Furthermore, we prove that, this result holds for an arbitrary permutation representation of \(G\) with degree \(|G|\), whereas it doesn't hold for a permutation representation of \(G\) with degree greater than \(|G|\).
Reviewer: Reviewer (Berlin)Direct and inverse scattering for the matrix Schrödinger equation.https://zbmath.org/1460.340042021-06-15T18:09:00+00:00"Aktosun, Tuncay"https://zbmath.org/authors/?q=ai:aktosun.tuncay"Weder, Ricardo"https://zbmath.org/authors/?q=ai:weder.ricardo-aThe monograph describes the (direct and inverse) scattering problem for the matrix Schrödinger operator on a half-line with a general self-adjoint boundary condition. The problem has applications to quantum star graphs and operators on the full line.
The authors define the input data set (the potential and the boundary condition) and the scattering data set (the scattering matrix and the bound state information). The direct scattering problem can then be viewed as a map from the Fadeev class of input data sets to the Marchenko class of scattering data sets. In other words, in this problem one finds the scattering matrix and bound states from the potential and the boundary condition. The inverse scattering problem is then viewed as the inverse map from the Marchenko class to the Fadeev class. Characterization of these classes using certain conditions is given in the book. Existence, uniqueness, construction, and characterisation of the problem are discussed.
Unlike other publications, the authors do not solve the problem separately for the Dirichlet and non-Dirichlet boundary conditions. They describe all possible self-adjoint conditions in a similar way how they are usually described in quantum graphs. The main result of the book is establishing one-to-one correspondence between Fadeev class and Marchenko class.
The monograph consists of six chapters and an appendix. The first chapter is introductory, the authors define the main aims and present an overview of the problem. In the second chapter, the direct and inverse problem are introduced mathematically, including Fadeev and Marchenko classes. Chapters 3 and 4 deal with the direct scattering problem; Chapter 5 with the inverse scattering problem. Examples are given in the sixth chapter. The necessary preliminaries are summarized in the appendix. The book has 624 pages.
Reviewer: Jiři Lipovský (Hradec Králové)Resolvent operator and spectrum of new type boundary value problems.https://zbmath.org/1460.340722021-06-15T18:09:00+00:00"Mukhtarov, Oktay Sh."https://zbmath.org/authors/?q=ai:mukhtarov.oktay-sh"Olǧar, Hayati"https://zbmath.org/authors/?q=ai:olgar.hayati"Aydemir, Kadriye"https://zbmath.org/authors/?q=ai:aydemir.kadriyeSummary: The aim of this study is to investigate a new type boundary value problems which consist of the equation \(-y''(x) + (\mathcal By)(x) = \lambda y(x)\) on two disjoint intervals \((-1,0)\) and \((0,1)\) together with transmission conditions at the point of interaction \(x = 0\) and with eigenparameter dependent boundary conditions, where \(\mathcal B\) is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces \(L_2(-1,0)\oplus (L_2(0,1)\). By suggesting an own approach, we introduce a modified Hilbert space and linear operator in such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further, we examine asymptotic behaviour of the eigenvalues.
Reviewer: Reviewer (Berlin)Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations.https://zbmath.org/1460.650872021-06-15T18:09:00+00:00"Blanes, Sergio"https://zbmath.org/authors/?q=ai:blanes.sergio"Casas, Fernando"https://zbmath.org/authors/?q=ai:casas.fernando"González, Cesáreo"https://zbmath.org/authors/?q=ai:gonzalez.cesareo"Thalhammer, Mechthild"https://zbmath.org/authors/?q=ai:thalhammer.mechthildSummary: This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge-Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker-Campbell-Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.
Reviewer: Reviewer (Berlin)Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set.https://zbmath.org/1460.810262021-06-15T18:09:00+00:00"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jake"Gorodetski, Anton"https://zbmath.org/authors/?q=ai:gorodetski.antonSummary: We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe-Sommerfeld criterion for sums of Cantor sets which may be of independent interest.
Reviewer: Reviewer (Berlin)