We investigate the spectrum of a typical non-self-adjoint differential operator $AD=-d^2/dx^2\otimes A$ acting on $\Lp(0,1)\otimes \mathbb{C}^2$, where $A$ is a $2\times 2$ constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate respectively at both ends of $[0,1]\subset\mathbb{R}$. For $A\in \mathbb{R}^{2\times 2}$ we explore in detail the connection between the entries of $A$ and the spectrum of $AD$, we find necessary conditions to ensure similarity to a self-adjoint operator and give numerical evidence that suggests a non-trivial spectral evolution.
In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than $-\varepsilon$ $(\varepsilon >0)$ of a self-adjoint operator $L$ which is defined by the following differential expression $$\ell(y)=-(p(x)y'(x))'-Q(x)y(x)$$ with the boundary condition $y(0) = 0$ in the space in the space $L_{2}(0,\infty ;H)$.
It is well known that the standard projection methods allow one to recover the whole spectrum of a bounded self-adjoint operator but they often lead to spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the essential spectrum. Methods using second order relative spectra are free from this problem, but they have not been proven to approximate the whole spectrum. L. Boulton (2006, 2007) has shown that second order relative spectra approximate all isolated eigenvalues of finite multiplicity. The main result of the present paper is that second order relative spectra do not in ge…
The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.
In this paper we construct the spectral expansion for the non-self-adjoint differential operators generated in the space of vektor functions by the ordinary differential expression of arbitrary order with the periodic matrix coefficients by using the essential spectral singularities, singular quasimomenta and the series with parenthesis.
For a given self-adjoint operator $A$ with discrete spectrum, we completely characterize possible eigenvalues of its rank-one perturbations~$B$ and discuss the inverse problem of reconstructing $B$ from its spectrum.
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.
We investigate the behaviour of the eigenvalues of two-dimensional Pauli operators with nonconstant magnetic fields perturbed by a sign-indefinite decaying electric potential V. We prove new eigenvalues asymptotics.
In this paper we construct the spectral expansion for the differential operator generated in all real line by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.
We study the spectral theory of operators, generated as direct sums of self-adjoint extensions of quasi-differential minimal operators on a multi-interval set (self-adjoint vector-operators), acting in a Hilbert space. Spectral theorems for such operators are discussed, the structure of the ordered spectral representation is investigated for the case of differential coordinate operators. One of the main results is the construction of spectral resolutions. Finally, we study the matters connected with analytical decompositions of generalized eigenfunctions of such vector-operators and build a ma…
In this paper we investigate the non-self-adjoint operator H generated in all real line by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.
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 obtain uniform, with respect to t asymptotic formulas for the eigenvalues of the operators generated in (0,1) by the Mathieu-Hill equation with a complex-valued potential and by the t-periodic boundary conditions. Then using it we investigate the non-self-adjoint Mathieu-Hill operator H generated in the allreal line by the same equation and establish the necessary and sufficient conditions for the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator.
We study the spectrum of unbounded J-self-adjoint block operator matrices. In particular, we prove enclosures for the spectrum, provide a sufficient condition for the spectrum being real and derive variational principles for certain real eigenvalues even in the presence of non-real spectrum. The latter lead to lower and upper bounds and asymptotic estimates for eigenvalues.
The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is derived as a corollary of the main result.
The paper pursues three objectives. Firstly, we provide an expanded version of spectral analysis of self-adjoint Toeplitz operators, initially built by M. Rosenblum in the 1960's. We offer some improvements to Rosenblum's approach: for instance, our proof of the absolute continuity, relying on a weak version of the limiting absorption principle, is more direct. Secondly, we study in detail Toeplitz operators with finite spectral multiplicity. In particular, we introduce generalized eigenfunctions and investigate their properties. Thirdly, we develop a more detailed spectral analysis for piecew…
In the first part of this manuscript a relationship between the spectrum of self-adjoint operator matrices and the spectra of their diagonal entries is found. This leads to enclosures for spectral points and in particular, enclosures for eigenvalues. We also consider graph invariant subspaces, and their corresponding angular operators. The existence of a bounded angular operator leads to basis properties of the first component of eigenvectors of operator matrices for which the corresponding eigenvalues lie in a half line. The results are applied to an example from magnetohydrodynamics.
We obtain explicit upper and lower bounds on the norms of the spectral projections of the non-self-adjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials.
The matrix Sturm-Liouville operator on a finite interval with the boundary conditions in the general self-adjoint form and with the singular potential from the class $W_2^{-1}$ is studied. This operator generalizes Sturm-Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.
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.
We develop the spectral and scattering theory for self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We construct wave operators relating simple "model" (that is, explicitly diagonalizable) Hankel operators for each jump and the given Hankel operator $H$. We show that the set of all these wave operators is asymptotically complete. This determines the absolutely continuous part of $H$. We also prove that the singular continuous spectrum of $H$ is empty and that its eigenval…
A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.
We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.
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…
This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $ \left[ b,π\right] $ with $b\in \left( d,π\right] $ or $b=d$, then $h,$ $β,$ $γ\ $and $q$ on $\left[ 0,π\right] \ $can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the cas…
Let $u=\int_{-\infty}^{+\infty}λdE_λ$ be a self-adjoint operator in a Hilbert space $H$. Our purpose is to provide a non-standard description of the spectral family $(E_λ)$ and the generalized Gelfand eigenvectors.
The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator $B$. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for $A_{[B]}$ to have a non-empty resolvent set are provided in terms of the parameter $B$ and the Weyl function. Special attention…
We obtain the uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L_{t}(q) with a potential q\inL_{1}[0,1] and with t-periodic boundary conditions, t\in(-π,π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L_{2}(-\infty,\infty) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies the sufficient conditions.
Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $φ(H)-φ(H_0)$ for piecewise continuous functions $φ$. This description involves the scattering matrix for the pair $H$, $H_0$.
The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric operators.
This text is a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations.
Let $H_0$, $H$ be a pair of self-adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true. We give an explicit description of the absolutely continuous spectrum of the operator $\mathcal{D}_θ=θ(H)-θ(H_0)$ for piecewise continuous functions $θ$. This description involves the scattering matrix for the pair $H_0$, $H$, evaluated at the discontinuities of $θ$. We also prove that the singular continuous spectrum of $\mathcal{D}_θ$ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the "thresholds"…
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 study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe their eigenvalues by the method of matched asymptotic expansions. These eigenvalues acquire a strange behavi…
In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,λ)$. The purpose of the paper is to study the spectral properties of the difference $D(λ)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(λ)$ in terms of the eigenvalues of the scattering matrix $S(λ)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(λ)$ is empty.
We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates. These results extend, and improve, classical perturbation results by Kato and by Gohberg/Krein. Further, we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator; in the latter case, we can even allow for perturbations with relative bound $\ge 1$. The generality of our results is…
We consider the self-adjoint Dirac operators on a finite interval with summable matrix-valued potentials and general boundary conditions. For such operators, we study the inverse problem of reconstructing the potential and the boundary conditions of the operator from its eigenvalues and suitably defined norming matrices.
This paper is devoted to the mathematical investigation of the MIT bag model, that is the Dirac operator on a smooth and bounded domain with certain boundary conditions. We prove that the operator is self-adjoint and, when the mass goes to infinity, we provide spectral asymptotic results.
We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover we establish isoperimet-ric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian on heavy end directed graphs.
We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space. One of the main goals here is to obtain the representation theorem for such operators in a form analogous to the one well-known in the cyclic self-adjoint operators case. To do this, we present here a detailed introduction to matrix measures, to the matrix measure $L^2$ spaces, and to the multiplication by scalar functions operators in such spaces. This all…
The inverse problem for the differential operator pencil with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem
The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. We study the inverse spectral problem, which consists in recovering of this operator by the Weyl matrix. The main result of the paper is the necessary and sufficient conditions on the Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.
Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl--von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint operator is the sum of a diagonalizable operator and of a compact operator with arbitrarily small Schatten $p$-norm. In doing so, we discuss conjugations and their properties. A spectral integral representation for antilinear self-adjoint operators is constructed.
This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems.
We prove an analogue to the Cayley identity for an arbitrary self-adjoint operator in a Hilbert space. We also provide two new ways to characterize vectors belonging to the singular spectral subspace in terms of the analytic properties of the resolvent of the operator, computed on these vectors. The latter are analogous to those used routinely in the scattering theory for the absolutely continuous subspace.
Sufficient and necessary conditions on the spectral measure of a self-adjoint operator $A$, acting in a Hilbert space, are obtained, under which for any continuous scalar function the operator function $ϕ(A+γB)$ is holomorphic with rspect to the coupling constant $γ$ in a neighborhood of $γ=0$, where $B$ is a self-adjoint operator. The sharpest results are obtained in the case where $B$ is a rank-one operator.
We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl-Titchmarsh half-line m-coefficients (and 2 x 2 matrix-valued M-matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoi…
We consider self-adjoint extensions of differential operators of the type $ (-\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2})^3 $ on the real semi-axis for l=1,2 with two kinds of boundary conditions: first that nullify the value of a function and its first derivative and second that nullify the 4th (l=1) or the 3rd (l=2) derivative. We calculate the expressions for the correponding resolvents and derive spectral decompositions. These types of boundary conditions are interesting from the physical point of view, especially the second ones, which give an example of emergence of long-range action in excha…
One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self-adjoint bounded Breuer-Fredholm operators in a semi-finite von Neumann algebra. These formulae have a geometric interpretation which derives from the proof. Namely we define a family of Banach submanifolds of all bounded self-adjoint Breuer-Fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path …
We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H=-d^2/dx^2+V in L^2(R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V.
We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes…
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.
