We study various spectral theoretic aspects of non-self-adjoint operators. Specifically, we consider a class of factorable non-self-adjoint perturbations of a given unperturbed non-self-adjoint operator and provide an in-depth study of a variant of the Birman-Schwinger principle as well as local and global Weinstein-Aronszajn formulas.
Our applications include a study of suitably symmetrized (modified) perturbation determinants of Schrödinger operators in dimensions n=1,2,3 and their connection with Krein's spectral shift function in two- and three-dimensional scattering theory. Moreover, we study an appropriate multi-dimensional analog of the celebrated formula by Jost and Pais that identifies Jost functions with suitable Fredholm (perturbation) determinants and hence reduces the latter to simple Wronski determinants.
We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed.
Let X be a compact Riemannian manifold of dimension two or three and let P be a point of X. We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of (symmetric) Laplace operator with domain, consisting of smooth functions with compact supports which does not contain P, to the zeta-regularized determinant of the self-adjoint Laplacian on X.
Let $L$ be a self-adjoint invertible operator in a Hilbert space such that $L^{-1}$ is $p$-summable. Under a certain discrete dimension spectrum assumption on $L$, we study the relation between the (regularized) Fredholm determinant, $\det_{p}(I+z\cdot L^{-1})$, on the one hand and the zeta regularized determinant, $\det_ζ(L+z)$, on the other. One of the main results is the formula \begin{equation*} \frac{\det_ζ (L + z)}{\det_ζ (L)} = \exp \left( \sum_{j=1}^{p-1} \frac{z^j}{j!} \cdot \frac{d^j}{dz^j} \log \det\nolimits_ζ (L+z) |_{z=0} \right) \cdot \det\nolimits_{p}(I + z \cdot L^{-1} ). \end{…
We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for such operators is also discussed.
For a purely imaginary sign-definite perturbation of a self-adjoint operator, we obtain exponential representations for the perturbation determinant in both upper and lower half-planes and derive respective trace formulas.
We study comparison formulas for $ζ$-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus $g\geq 2$. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly.
We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1} -\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big) \right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\, \frac{ξ(x)ξ(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the spectra…
We use the boundary triplet approach to extend the classical concept of perturbation determinants to a more general setup. In particular, we examine the concept of perturbation determinants to pairs of proper extensions of closed symmetric operators. For an ordered pair of extensions we express the perturbation determinant in terms of the abstract Weyl function and the corresponding boundary operators. A crucial role in our approach plays so-called almost solvable extensions. We obtain trace formulas for pairs of self-adjoint, dissipative and other pairs of extensions and express the spectral …
We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.
The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin…
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with …
This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral propert…
After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger operators $(- d^2/dx^2) + q$ on $(0,\infty)$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials $q$ that, for some fixed $C_0 > 0$, $\varepsilon > 0$, $x_0 \in (0, \infty)$, diverge at infinity in the manner that $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treat all self-adjoint boundary conditions at the left endpoint $0$.
We consider the four-component two-valley Dirac operator on a wedge in $\mathbb{R}^2$ with infinite mass boundary conditions, which enjoy a flip at the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize all its self-adjoint extensions, relying on the fact that the underlying two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The respective defect element is computed explicitly. We observe that there exists no self-adjoint extension, which can be decomposed into an orthogonal sum of two two-component operators. In physics, this effect is called mix…
We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm-Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the zeta-determinant of an infinite direct sum of Sturm-Liouville operators. The Laplace-Beltrami operator on a surface of revo…
We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(Ω; d^n x)$, $n\geq 2$, where $Ω$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in $L^2(Ω; d^n x)$ to Fredholm perturbation determinants associated with operators in $L^2(\partialΩ; d^{n-1}σ)$. This leads to an extension of a variant of a celebrated formula due to Jost and Pais, whi…
We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators in $L^2(Ω; d^n x)$, $n=2,3$, where $Ω$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(Ω; d^n x)$ to modified Fredholm perturbation determinants associated with operators in $L^2(\partialΩ; d^{n-1}σ)$, $n=2,3$. This leads to a two- and three-dimensional extension of a variant of a celebrated formula due to …
We investigate the relationship between one of the classical notions of boundaries for infinite graphs, \emph{graph ends}, and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of \emph{finite volume} for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self-adjointness of the Gaffney…
We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $\pm \infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.
We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansions, we derive adiabatic splitting formulas for regularized determinants of Laplace type operators.
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…
The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent 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 Schrödinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. One of the principal new results in this paper reduces a…
The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of $\ell^\infty$ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an $\ell^\infty$ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spect…
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 $x=R$.
A theorem that is of aid in computing the domain of the adjoint operator is provided. It may serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for $H$--selfadjointness of an $H$--symmetric operator. Differential operators and operators given by an infinite matrix are considered as examples.
We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank one imaginary part. It is shown that given a set of $n$ not necessarily distinct non-real numbers in the open upper (lower) half-plane uniquely determines a n x n Jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting multiplicity. An algorithm for reconstruction for such finite Jacobi matrices is presented. A new model complementing the well known Livsic triangular model for bounded linear operators with rank one imaginary part is obt…
We propose a general condition, to ensure essential self-adjointness for the Gauß-Bonnet operator, based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions or 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff's laws.
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $ζ$-functions to efficiently compute values of spectral $ζ$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $τ$. Depending on the underlying boundary conditions, we express the $ζ$-function values in terms of a fundamental system of solutions of $τy = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $ζ$-function through …
Given an infinite graph $G$ on countably many vertices, and a closed, infinite set $Λ$ of real numbers, we prove the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $Λ$.
We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known that the $ζ$-function of this operator $ζ_L(s)=\sum_{λ\in\spec(L)\setminus\{0\}} λ^{-s}$ has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to Ray and Singer the $ζ$-regularized determinant of $L$ is defined by $\detz(L):=\exp(-ζ_L'(0)).$ In this paper we are going to express this determinant in terms o…
We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate ou…
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.
In this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schrödinger operator $(-i\nabla - \textbf{\textup{A}})^{2} - b$ in dimension $3$ with constant magnetic field of strength $b>0$. The situation near the Landau levels $2bq$, $q \in \mathbb{N}$, is more interesting since they play the role of thresholds of the spectrum of the free operator. First, we obtain sharp upper bounds on the number of the complex eigenvalues near the Landau levels. Under appropriate hypothesis, we then prove t…
We study the analog of semi-separable integral kernels in $\mathcal{H}$ of the type $$ K(x,x')=\begin{cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, \end{cases} $$ where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \mathcal{B}_2(\mathcal{H}_j,\mathcal{H})$ and $G_j(x) \in \mathcal{B}_2(\mathcal{H},\mathcal{H}_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and $$ \|F_j(\cdot)\|_{\mathcal{B}_2(\mathcal{H}_j,\mathcal{H})} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\mathcal{B}_2(\mathcal{H},\mathcal{H}_j)} \in L^2((a,b)), \quad j=1,2, $…
This paper considers $N\times N$ matrices of the form $A_γ=A+ γB$, where $A$ is self-adjoint, $γ\in C$ and $B$ is a non-self-adjoint perturbation of $A$. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes $|γ|\to\infty$ and $|γ|\to 0$ under certain assumptions on $B$. We also explain some properties of the spectrum of $A_γ$ for intermediate sized $γ$ by considering the limit $N\to\infty$, concentrating on properties that have no self-adjoint analogue. A substantial number of the results extend to operators on infinite-dimensiona…
We study the spectral properties of the Laplace type operator on the circle. We discuss various approximations for the heat trace, the zeta function and the zeta-regularized determinant. We obtain a differential equation for the heat kernel diagonal and a recursive system for the diagonal heat kernel coefficients, which enables us to find closed approximate formulas for the heat trace and the functional determinant which become exact in the limit of infinite radius. The relation to the generalized KdV hierarchy is discussed as well.
