Steve Clark, Fritz Gesztesy
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general matrix-valued Schrödinger operators on a half-line.
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We consider discrete Schroedinger operator J with Wigner-von Neumann potential not belonging to l^2. We find asymptotics of orthonormal polynomials associated to J. We prove the Weyl-Titchmarsh type formula, which relates the spectral density of J to a coefficient in asymptotics of orthonormal polynomials.
Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from the upper half-plane and on the absolutely continuous spectrum is established. Weyl-Titchmarsh type formula for this operator is proven.
We show that the Herglotz functions that arise as Weyl-Titchmarsh $m$ functions of one-dimensional Schrödinger operators are dense in the space of all Herglotz functions with respect to uniform convergence on compact subsets of the upper half plane. This result is obtained as an application of de Branges theory of canonical systems.
Weil-Titchmarsh theory as applied to the singular non-sectorial Schrödinger operator. Conditions for discreteness of the spectrum and compactness of the resolvent
针对取值区域超出半平面的复值势导致奇异 Schrödinger 算子非扇形且数值域覆盖整个复平面的问题，本文利用 Weil-Titchmarsh 理论给出谱离散及预解式紧的充分条件。
In this note it is proved that the complete spectral data of selfadjoint Schrödinger operators on unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogues of classical facts from si…
An analogue of Rellich's theorem is proved for discrete Laplacian on square lattice, and applied to show unique continuation property on certain domains as well as non-existence of embedded eigenvalues for discrete Schr{ö}dinger operators.
We explore the sparsity of Weyl-Titchmarsh $m$-functions of discrete Schrödinger operators. Due to this, the set of their $m$-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral theory for discrete Schrödinger operators via their spectral measures should be difficult. To obtain the result, de Branges theory of canonical systems is applied to work on them, instead of Weyl-Titchmarsh $m$-functions.
Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schroedinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for "free" Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained.
We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schroedinger operators (also known as Bessel operators) under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.
We study to what extent Lieb--Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrödinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [Complex Anal. Oper. Theory 5, No. 1 (2011), 197-218] and answer another open question raised therein. The results are obtained by means of asymptotic analysis of eigenvalues of discrete Schrödinger operators with rectangular barrier potential and complex coupling. Applying the ideas in the continuous setting, we also solve a similar open problem for one-dimensional Schrödinger …
Asymptotics of Titchmarsh--Weyl functions near the real axis andan application to the KdV hierarchy
本文推导了一维Schrödinger与Dirac算子的Titchmarsh–Weyl函数在边界趋近谱的多项式速率区域内的高能渐近展开，验证了Kotani构造KdV流的高能假设，证明了对应势可生成KdV层级指定成员的全局经典解。
We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a genera…
We compute an explicit formula for the integrated density of states of the periodic Airy-Schr{ö}dinger operator on the real line. For this purpose, we study precisely the spectrum of the restriction of the periodic Airy-Schr{ö}dinger operator to a finite number of periods. We prove that all the eigenvalues of this restriction are in the spectral bands of the periodic Airy-Schr{ö}dinger operator and none of them are in its spectral gaps. We count exactly the number of eigenvalues in each of the spectral bands, this number being equal to the number of periods to which the periodic operator is re…
On a complete weighted Riemannian manifold $(M^n,g,μ)$ satisfying the doubling condition and the Poincar{é} inequalities, we characterize the class of function $V$ such that the Schr{ö}dinger operator $Δ-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector.
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
On the canonical $2$-sphere and for Schr{ö}dinger eigenfunctions, we obtain a simple geometric criterion on the potential under which we can improve, near a given point and for every $p\neq 6$, Sogge's estimates by a power of the eigenvalue. This criterion can be formulated in terms of the critical points of the Radon transform of the potential and it is independent of the choice of eigenfunctions.
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{ö}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to relax conditions on the derivatives of the potential that were required up to now.
We develop Weyl-Titchmarsh theory for self-adjoint Schrödinger operators $H_α$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $τ=-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $α= α^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(α)u'(a)+\cos(α)u(a)=0, $$ with $u$ lying in the domain of the underl…
The boundary control approach to the Titchmarsh-Weyl $m-$function
本文将边界控制理论与Titchmarsh-Weyl理论相结合，为Simon的$A-$幅度提供自然解释，并提出一种计算Schrödinger算子在Dirichlet边界条件下Titchmarsh-Weyl $m-$函数的高效新方法。
Uniform observation of semiclassical Schr{ö}dinger eigenfunctions on an interval
本文研究区间上单阱势与 Dirichlet 边界条件下的半经典 Schrödinger 特征函数，利用 Agmon 估计、Gronwall 论证及半经典测度，建立在半经典和高能极限下均一致且最优的 L^2 密度上下界。
Spectral flow and Levinson's theorem for Schrö dinger operators
本文利用谱流为具有光滑紧支撑势的 Schrödinger 算子提供了 Levinson 定理的新证明，该证明适用于所有维度及存在共振的情况，并基于谱位移函数与高能修正时间延迟表述结论。
The original Weyl-Titchmarsh functions and sectorial Shrödinger L-systems
本文研究非负极小对称薛定谔算子下原始Weyl-Titchmarsh函数生成的L-系统实现，推导其阻抗函数属于Stieltjes扇形类的充要条件，给出L-系统几何结构及算子扇形角异同的参数化描述，并附示例验证。
We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with $s$ or p-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centres. When the operator is of regular type we prove that the wave operators are bo…
We give necessary and sufficient conditions for the controllability of a Schr\''odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\''odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\''odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main…
We consider discrete Schr{ö}dinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long-range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugat…
Long-time dynamics for the magnetic Schr{ö}dinger equation on the torus
本文研究二维环面上磁薛定谔流的长时间半经典测度，在几何非消失条件下证明量子唯一遍历性的动力学形式，并给出构型与动量边际的收敛阈值。
How and why may an interacting system of many particles be described assuming that all particles are independent and identically distributed ? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular the experimental creation of Bose-Einstein condensates directly asks the following variant: why and how can a large assembly of very cold interacting bosons (quantum particles deprived of the Pauli exclusion principle) all populate the same quantum state ? In this text I review the various mathemati…
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