C. Dunning, K. E. Hibberd, J. Links
Spectral equivalences of the quasi-exactly solvable sectors of two classes of Schrodinger operators are established, using Gaudin-type Bethe ansatz equations. In some instances the results can be extended leading to full isospectrality. In this manner we obtain equivalences between PT-symmetric problems and Hermitian problems. We also find equivalences between some classes of Hermitian operators.
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On the Equivalence Between the Schrodinger Equation in Quantum Mechanics and the Euler-Bernoulli Equation in Elasticity Theory
本文证明了薛定谔方程与欧拉-伯努利方程在数学上的等价性，并探讨了其在量子计算、双缝实验及p-adic方程中的潜在应用。
We consider the classical integrable structure of two-dimensional non-linear sigma models with target space three-dimensional Schrodinger spacetimes. There are the two descriptions to describe the classical dynamics: 1) the left description based on SL(2,R)_L and 2) the right description based on U(1)_R. We have shown the sl(2,R)_L Yangian and q-deformed Poincare algebras associated with them. We proceed to argue an infinite-dimensional extension of the q-deformed Poincare algebra. The corresponding charges are constructed by using a non-local map from the flat conserved currents related to th…
A derivation of the spectral determinant of the Schrödinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant we first derive the spectral zeta function of the Schrödinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture.
In a previous work [Andrade \textit{et al.}, Phys. Rep. \textbf{647}, 1 (2016)], it was shown that the exact Green's function (GF) for an arbitrarily large (although finite) quantum graph is given as a sum over scattering paths, where local quantum effects are taken into account through the reflection and transmission scattering amplitudes. To deal with general graphs, two simplifying procedures were developed: regrouping of paths into families of paths and the separation of a large graph into subgraphs. However, for less symmetrical graphs with complicated topologies as, for instance, random …
We investigate the equivalence between dynamical localization and localization properties of eigenfunctions of Schrödinger Hamiltonians. We introduce three classes of equivalent properties and study the relationships between them. These relationships are shown to be optimal thanks to counter examples.
The purpose of the present work is to establish decorrelation estimates for some random discrete Schrodinger operator in dimension one. We prove that the Minami estimates are consequences of the Wegner estimates and Localization. We also prove decorrelation estimates at distinct energies for the random hopping model and Schrodinger operators with alloy-type potentials. These results are used to give a description of the spectral statistics.
The purpose of the present work is to establish decorrelation estimates at distinct energies for some random Schrödinger operator in dimension one. In particular, we establish the result for some random operators on the continuum with alloy-type potential. These results are used to give a description of the spectral statistics.
In this paper, we consider one dimensional adiabatic quasi-periodic Schrödinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion.
In this paper we constructively determine a family of the spectral invariants of the multidimensional Schrodinger operator with a periodic potential by the given band functions.
The spectral series of the Schrödinger operator with a delta-potential on a three-dimensional compact spherically symmetric manifold in the semiclassical limit as $h\to0$ are described.
In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.
We apply the methods of value distribution theory to the spectral asymptotics of Schrodinger operators with L^2-sparse potentials.
We study the inverse spectral problem for periodic Schrödinger opera\-tors of kind $- \frac{1}{2} \hbar^2 Δ_x + V(x)$ on the flat torus $\Bbb T^n := (\Bbb R / 2 π\Bbb Z)^n$ with potentials $V \in C^{\infty} (\Bbb T^n)$. We show that if two operators are isospectral for any $0 < \hbar \le 1$ then they have the same effective Hamiltonian given by the periodic homogenization of Hamilton-Jacobi equation. This result provides a necessary condition for the isospectrality of these Schrödinger operators. We also provide a link between our result and the spectral limit of quantum integrable systems.
This paper is devoted to the description of our recent results on the spectral behavior of one-dimensional adiabatic quasi-periodic Schrodinger operators. The specific operator we study is a slow periodic perturbation of an incommensurate periodic Schrodinger operator, and we are interested in energies where the perturbation creates a strong interaction between two consecutive bands of the background periodic operator. We describe the location of the spectrum and its nature and discuss the various new resonance phenomena due to the interaction of the spectral bands of the unperturbed periodic …
We consider $N$-body Schrödinger operators with $N\geq3$ particles in dimension $d\geq 3$ in the critical case when the lowest eigenvalue coincides with the bottom of the essential spectrum of the operator. We give the asymptotic behaviour of the corresponding eigenfunction in dependence of the dimension and the number of particles of the system.
This article deals with the numerical calculation of eigenvalues of perturbed periodic Schrödinger operators located in spectral gaps. Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. In this article, we give a precise description of the corresponding spurious states. We then prove that the supercell model does not produce spectral pollution. Lastly, we extend results by Lewin and Séré on some no-pollution criteria. In …
We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint operators, nor to Hilbert space or 1D.
In this paper the localization properties of the spectral expansions of distributions related to the self adjoint extension of the Schrodinger operator are investigated. Spectral decompositions of the distributions and some classes of distributions are defined. Estimations for Riesz means of the spectral decompositions of the distributions in the norm of the Sobolev classes with negative order are obtained.
We show that spectral Hausdorff dimensional properties of discrete Schröodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
In this paper, we investigate the three-dimensional Schrodinger operator with a periodic, relative to a lattice Ω of R3, potential q. A special class V of the periodic potentials is constructed, which is easily and constructively determined from the spectral invariants. First, we give an algorithm for the unique determination of the potential q in V of the three-dimensional Schrodinger operator from the spectral invariants that were determined constructively from the given Bloch eigenvalues. Then we consider the stability of the algorithm with respect to the spectral invariants and Bloch eigen…
It is shown that there exits a lower bound $α>0$ to the Hausdorff dimension of the spectral measures of the one-dimensional period doubling substitution Schrödinger operator, and, generically in the hull of such sequence, $α$ is also a lower bound to the upper packing dimension of spectral measures.
Generalizing previous results obtained for the spectrum of the Dirichlet and Neumann realizations in a bounded domain of a Schrödinger operator with a purely imaginary potential $h^2Δ+iV$ in the semiclassical limit $h\to 0$ we address the same problem in exterior domains. In particular we obtain the left margin of the spectrum, and the emptiness of the essential part of the spectrum under some additional assumptions.
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
We prove some new pointwise-in-energy bounds on the expectations of various spectral shift functions associated with random Schrödinger operators in the continuum having Anderson-type random potentials in both finite-volume and infinite-volume. These estimates are a consequence of our new Wegner estimate for finite-volume random Schrödinger operators. For lattice models, we also obtain a representation of the infinite-volume density of states in terms of a spectral shift function. For continuum models, the corresponding measure is absolutely continuous with respect to the density of states and…
On the spectral gap of one-dimensional Schrödinger operators on large intervals
该研究聚焦大区间极限下非负势对一维Schrödinger算子谱间隙的影响，推导了不同类别势下谱间隙的上下界以刻画其渐近行为。
In this article we study the semiclassical spectral measures associated with Schrödinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give an alternative proof of Colin de Verdiere's result on recovering one dimensional potential wells from semiclassical spectral data. We also study the relation of semiclassical spectra measures to Birkhoff normal forms and describe a generalization of the asymptotic expansion above to the Schrödinger operator in the presence of a magnetic field.
Exact Hausdorff dimensions are computed for singular continuous components of the spectral measures of a class of Schrödinger operators in bounded intervals.
Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schrödinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific contour on the spectral curve of the differential which is constructed by the wave function and its double. The formula is parametrized by the point on the spectral curve and for almost every parameter value it gives the Green's function with known growth at infinity.
We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time.
In this paper we obtain sharp Lieb-Thirring inequalities for a Schrödinger operator on semi-axis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and spectral inequalities for Schrödinger operators on half-spaces with Robin boundary conditions.
We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct examples of reflectionless Schrödinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain, which exhibit a singular component.
In this paper we continue the study of the spectral gap of Schrödinger operators on large intervals and subject to Neumann boundary conditions. The main goal is to derive a lower bound on the spectral gap which is polynomial in the interval length. This bound is derived for a class of bounded potentials of compact support which are weak enough in a suitable sense.
We prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschke-type condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E.
In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schrödinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.
We consider the Dirichlet realization of the operator $-h^2Δ+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic expansion, in powers of $h$, of each eigenvalue. In two dimensions we obtain the left margin of the spectrum, under some additional conditions.
Inverse spectral results for the magnetic Schrödinger operator in low dimensions
本文研究低维磁薛定谔算子的逆谱问题，证明一维偶势和二维径向磁场可由谱唯一确定，推广了Guillemin-Wang的结果并解答其猜想。
We study location of eigenvalues of one-dimensional discrete Schrödinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two different spectral bounds are obtained. The method relies on the Birman-Schwinger principle and various techniques for estimations of the norm of the Birman-Schwinger operator.
We consider a semi-classical Schrödinger operator, -h^2Δ+ V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the low-lying eigenvalues of the operator determine the Taylor series of the potential at the minimum.
We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local minimum. The main result, which computes the contribution of this equilibrium, is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point.
We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.
We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes $Λ\subset R^d$. We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes $Λ\subset R^d$, for $d \geq 1$. The result hol…
The scattering matrix of the Schrodinger operator with smooth short-range electric and magnetic potentials is considered. The asymptotic density of the eigenvalues of this scattering matrix in the high energy regime is determined. An explicit formula for this density is given. This formula involves only the magnetic vector-potential.
Some results on semiclassical spectral analysis of magnetic Schrödinger operators
本文综述了作者近期关于有界几何流形上 Bochner-Schrödinger 算子半经典谱问题的结果，并将其置于欧氏空间磁 Schrödinger 算子背景下，同时阐述证明思路。
The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.
We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.
We prove the complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrodinger operator $H=-Δ+b$ acting in $R^d$ when the potential $b$ is real and either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.
In general, Morita equivalence of spectral triples need not be a symmetric relation. In this paper, we show that Morita equivalence of spectral triples is an equivalence relation for equivariant torus spectral triples.
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.
In this paper we give a complete description of the spectral analysis of the Schrodinger operator L(V) with the optical potentil. First we consider the Bolch eigenvalues and spectrum of L(V). Then using it we investigate spectral singularities and essential spectral singularities (ESS). We prove that the operator L(V) has no ESS and has ESS respectively if and only if V is not a critical point and V is a critical point. Using it we classify the spectral expansion in term of the critical points. Finally we discuss the critical points, formulate some conjectures and describe the changes of the s…
We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we c…
Extrema of spectral band functions of two dimensional discrete periodic Schrödinger operators
本文利用贝祖定理和BKK定理分析二维离散周期薛定谔算子谱带函数极值的水平集，改进了先前结果。
We study spectral properties of convolution operators $\mathcal L$ and their perturbations $H=\mathcal L+v(x)$ by compactly supported potentials. Results are applied to determine the front propagation of a population density governed by operator $H$ with a compactly supported initial density provided that $H$ has positive eigenvalues. If there is no positive spectrum, then the stabilization of the population density is proved.
We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a give…
We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and $2\times 2$ matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on $(a,\infty)$ with a potential strongly singular at the end point $a$.…
Spectral Properties of Schrödinger Operators on Perturbed Lattices
本文研究了扰动格上Schrödinger算子的谱性质，证明了嵌入特征值的不存在性或离散性及预解式的极限吸收原理，构造了谱表示并定义了S矩阵，理论适用于多种格结构。
Spectral stability of multiple periodic waves for the Schrodinger system with cubic nonlinearity
本文研究具有三次非线性的薛定谔系统中 dnoidal 与 cnoidal 型多重周期波的存在性及谱稳定性，利用比较定理、Floquet 理论和 Krein 符号给出稳定性判据。
Equivalence of Sobolev norms involving generalized Hardy operators
本文研究带哈代位势的分数阶薛定谔算子所生成的齐次索伯列夫空间与通常齐次索伯列夫空间的等价性，得到广义及反向哈代不等式，并推广了非分数阶情形下的近期结果。
We consider Schrödinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field $h\to 0$.
Starting from the spectrum of Schrödinger operators on $\mathbb{R}^n$, we propose a method to detect critical points of the potential. We argue semi-classically on the basis of a mathematically rigorous version of Gutzwiller's trace formula which expresses spectral statistics in term of classical orbits. A critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum. Via sharp spectral estimates, this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential…
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