E B Davies
We prove that the spectrum of certain non-self-adjoint Schrodinger operators is unstable in the semi-classical limit. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum.
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We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner--Wienholtz--Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.
Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators
本文将广义边界三元组及Weyl函数推广至算子伴随对，给出诱导非空预解集闭算子的边界参数准则，并应用于具紧致边界的Lipschitz域上带复Lp势的Schrödinger算子。
We examine semiclassical magnetic Schrödinger operators with complex electric potentials. Under suitable conditions on the magnetic and electric potentials, we prove a resolvent estimate for spectral parameters in an unbounded parabolic neighborhood of the imaginary axis.
We describe the general qualitative behaviour of the resolvent norm for a very wide class of non-self-adjoint Schroedinger operators in the semi-classical regime, as the spectral parameter varies over the complex plane.
We study eigenvalues of non-self-adjoint Schrödinger operators on non-trapping asymptotically conic manifolds of dimension $n\ge 3$. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the $L^p$-norm of the potential, while the second one is a Lieb-Thirring type bound controlling sums of powers of eigenvalues in terms of the $L^p$-norm of the potential. We extend the results of Frank (2011), Frank-Sabin (2017), and Frank-Simon (2017) on th…
In this note, we study existence of the outgoing/incoming resolvents of repulsive Schrödinger operators which may not be essentially self-adjoint on the Schwartz space. Moreover, we recover the classical result: The repulsive Schroödinger operators with large repulsive constant is not essentially self-adjoint on the Schwartz space.
In this paper, we give Lieb-Thirring type inequalities for isolated eigenvalues of $d$-dimensional non-selfadjoint Schrödinger operators with complex-valued and dilation analytic potentials. In order to derive them, we prove that isolated eigenvalues and their multiplicities are invariant under complex dilation.
For non-self-adjoint almost-periodic Schrödinger operators, a criterion is given to guarantee that they have both the same spectrum and same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the Moser-Pöschel argument for opening gaps may not be valid for non-self-adjoint operators.
Using the concept of intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.
We give sufficient conditions for essential self-adjointness of magnetic Schrödinger operators on locally finite graphs. Two of the main theorems of the present paper generalize recent results of Torki-Hamza.
In this paper we investigate the one-dimensional Schrodinger operator L(q) with complex-valued periodic potential q when q\inL_{1}[0,1] and q_{n}=0 for n=0,-1,-2,..., where q_{n} are the Fourier coefficients of q with respect to the system {e^{i2πnx}}. We prove that the Bloch eigenvalues are (2πn+t)^{2} for n\inZ, t\inC and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.
On two properties of positively perturbed discrete Schrödinger operators
本文研究离散加权图上非负势扰动对 Schrödinger 算子自伴性的影响，证明型和与 Friedrichs 扩张一致，并给出本质自伴性的稳定性条件。
For general non-symmetric operators $A$, we prove that the moment of order $γ\ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $γ$ of negative eigenvalues of its symmetric part $H = {1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schrödinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schrödinger self-adjoint operators.
The purpose of this paper is to study spectral properties of non-self-adjoint Schrödinger operators $-Δ-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type inequalities which measure the radius of a disc containing the discrete spectrum, in terms of the $L^{p,\infty}$ norm of $V$. Similar inequalities also hold if the inverse-square potential is replaced by a large class of subcritical potentials with critical singularities at the origin. The main new ingredient in the proof is the uniform Sobolev inequality of Kenig-R…
We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.
We define the magnetic Schrödinger on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges . We discuss essential self-adjointness of this operator for graphs of bounded degree. The main result is a discrete version of a result of two authors of the present paper.
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph. Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the weighted graph Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us …
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 investigate the multidimensional Schrodinger operator L(q) with complex-valued periodic, with respect to a lattice, potential q when the Fourier coefficients of q with respect to the orthogonal system {exp(i(a,x))}, where a changes in the dual lattice, vanish if a belong to a half-space We prove that the Bloch eigenvalues of L(q) and of the free operator L(0) are the same and find explicit formulas for the Bloch functions. It implies that the Fermi surfaces of L(q) and L(0) are the same. The considered set of operators includes a large class of PT symmetric operators used in the PT symmetri…
The compression of the resolvent of a non-self-adjoint Schrödinger operator $-Δ+V$ onto a subdomain $Ω\subset\mathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $Ω$, the Dirichlet-to-Neumann maps, and certain solution operators of closely related boundary value problems on $Ω$ and $\mathbb R^n\setminus\overlineΩ$ are being used. In a more abstract operator theory framework this topic is closely connected and very much inspired by the so-called coupling method that has been developed for the self-adjoint case by Henk de Snoo and his coauthors.
We study positivity, localization of binding and essential self-adjointness properties of a class of Schroedinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials.
We prove upper and lower bounds for sums of eigenvalues of Lieb-Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases.
We revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n \times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schrödinger operators on a compact interval.
Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in $L^2(\mathbb{R})$ associated with Dirichlet eigenvalues was proved in [9]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential $Ke^{2ix}$, where $K\in\mathbb{C}$.
In the context of a weighted graph with vertex set $V$ and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator $Δ_σ+W$, where $Δ_σ$ is the magnetic Laplacian and $W\colon V\to\mathbb{R}$ is a function satisfying $W(x)\geq -q(x)$ for all $x\in V$, with $q\colon V\to [1,\infty)$. The condition is expressed in terms of completeness of a metric that depends on $q$ and the weights of the graph. The main result is a discrete analogue of the results of I. Oleinik and M. A. Shubin in the setting of non-compact Riemannian manifolds.
In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schrödinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.
Under the assumption that $V \in L^2([0,π]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators $-d^2/dx^2+V$ in $L^2([0,π]; dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $-d^2/dx^2+V$ in $L^p([0,π]; dx)$, $p \in (1,\infty)$.
Essential self-adjointness of non-semibounded Schrödinger operators on infinite graphs
本文研究无穷加权图上未必下半有界的离散薛定谔算子的本质自伴性，给出了相关充分条件，并推导得到势函数满足特定二次下界时算子本质自伴的推论。
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 …
Eigenvalue bounds for non-self-adjoint Schrödinger operators and pseudodifferential generalizations
本文综述了确定性和随机Schrödinger算子在R^d和紧致流形上特征值界的理论，同时提出新定理将紧致流形上的结果推广到分数Laplacian，使用Cuenin和Sogge的方法，以L^p范数描述势函数。
Non-self-adjoint Schrodinger operators A which correspond to non-symmetric zero-range potentials are investigated. For a given A, the description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of A as a self-adjoint operator in a Krein space is studied, the problem of similarity of A to a self-adjoint operator in a Hilbert space is solved.
Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast se…
We study two- and three-dimensional matrix Schrödinger operators with $m\in \mathbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by the other authors in this field. For instance, we parametrize all self-adjoint extensions of the initial minimal symmetric Schrödinger operator by abstract boundary conditions and characterize their spectra. Particularly, we find a sufficient condition in terms of distances and intensities for the self-adjoint extension $H_{α,X}^{(3)}$ to have $m'$ negative eig…
We study the eigenvalues of the non-self adjoint problem $-y^{\prime\prime}+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\geq 3$. We obtain a Bohr-Sommerfeld-like asymptotic formula for $E_n$ that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential $V$ and boundary condition from the first $(m+2)$ terms of the asymptotic formula.
Let $Ω$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-Δ+V(x)$ be a Schrödinger operator on $Ω$ with potential V locally bounded. We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^\infty(Ω)$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\geq \frac{3}{4d(x)^2}$, where $d(x)=dist(x,\partialΩ)$. The constant 1 in front of each logarithmic term in Theorem 2 is optimal. The proof…
We consider Dirac, Pauli and Schrödinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. In particular, we establish point spectrum analogous of Bögli results [Bög17] obtained for non-magnetic Laplacians, and hence showing that classical Lieb-Thirring inequalities cannot hold for our magnetic models. On the …
Non-self-adjoint Schrodinger operators which correspond to non-symmetric zero-range potentials are investigated. We show that various properties of these operators (eigenvalues, exceptional points, spectral singularities and the property of similarity to a self-adjoint operator) are completely determined by poles of the corresponding S-matrix.
For integers $m\geq 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime\prime}(x)+(x^m+P(x))u(x)=E u(x)$, $0\leq x<+\infty$, with the boundary conditions $u(+\infty)=0$ and $αu(0)+βu^{\prime}(0)=0$ for some $α, β\in\C$ with $|α|+|β|\not=0$, where $P(x)=a_1 x^{m-1}+a_2 x^{m-2}+...+a_{m-1} x$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $E_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.
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 $a$.…
We consider non-self-adjoint Schrödinger operators $Δ+V$ where $Δ$ is the Laplace-Beltrami operator on a Zoll manifold $X$ and $V\in C^\infty(X,\mathbb C)$. We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.
Schrödinger operators on the half-line with integrable complex potentials
本文研究了半线上带有可积复势的薛定谔算子的谱对，推导了显式公式并分析了Jost解的存在性。
We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator can completely change the spectrum of the operator.
New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of $L_{p}$-norms of the potentials. The results extend and improve those obtained previously. In particular, diverse versions of an assertion conjectured by Laptev and Safronov are discussed. Schrödinger operators with slowly decaying potentials are also considered.
Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $τ= - (d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line (assuming the limit-point property of $τ$ at $\pm \infty$). Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in \mathcal{B}(\mathcal{H})$ such that $V(\cdot)$ is weakly measurable, the operator norm $\|V(\cdot)\|_{\mathcal{B}(\mathcal{H})}$ is locally integrable, and $V(\cdot) = V(\cdot)^*$ a.e. In a nutshell, a Donoghue-type $m$-function $M_{A,\mathcal{N}_i}^{Do}(\cdot)$ assoc…
A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh-Weyl $m$-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.
The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrödinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schrödinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite un…
In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\mathbb{1}_{(-\infty,λ^2]}(P)$ has a full asymptotic expansion in powers of $λ$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supp…
In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenv…
We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on the finite interval $(0,R)$ converge in the infinite volume limit $R\to\infty$ to the half-line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at $x=0$ and $x=R$.
We provide a systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrodinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context. Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrodinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrodinger operators corre…
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph . Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us to write any S…
In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrödinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted $L^2$ spaces on the torus, and an oscillatory integral theorem.
Bounds on complex eigenvalues and resonances
本文给出了复势非自伴薛定谔算子复特征值与自伴薛定谔算子复共振的界，并与数值结果比较，证明其有效性。
We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schrödinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum.
We consider the one-dimensional Schrödinger equation $-f''+q_αf = Ef$ on the positive half-axis with the potential $q_α(r)=(α-1/4)r^{-2}$. It is known that the value $α=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r + q_α(r)$ for the Hamiltonian have infinitely many eigenvalues for $α<0$ and at most one eigenvalue for $α\geq 0$. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in $α$ and, in particular, is not singular at $α= 0$. Employing suitable singular Ti…
We investigate the connections between Weyl-Titchmarsh-Kodaira theory for one-dimensional Schrödinger operators and the theory of $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be $n$-entire in terms of square integrability of derivatives (w.r.t. the spectral parameter) of the Weyl solution. We also show that this is equivalent to the Weyl function being in a generalized Herglotz-Nevanlinna class. As an application we show that perturbed Bessel operators are $n$-entire, improving the previously known condition…
Harmonic Approximation and Resolvent Estimates for Semiclassical Non-Self-Adjoint Operators
研究一类具有临界点的非自伴非椭圆h-伪微分算子的预解估计和低端谱界，通过主象征实部沿虚部哈密顿流的平均动力学条件，在半经典伪谱边界附近建立精确预解估计。
The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-N…
We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.
We obtain slow dynamics for self-adjoint semigroups and unitary evolution groups. For semigroups, the slow dynamics is for orbits, and for the average return probability in the case of unitary evolution groups. We present an application to the quantum dynamics of purely absolutely continuous systems
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