The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with $C^{1,1}$ Hölder smoothness, by use of a recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols.
We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$, with generalized Robin boundary conditions.
We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$.
This paper has two main goals. First, we are concerned with the classification of self-adjoint extensions of the Laplacian $-Δ\big|_{C^\infty_0(Ω)}$ in $L^2(Ω; d^n x)$. Here, the domain $Ω$ belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), which contain all convex domains, as well as all domains of class $C^{1,r}$, for $r\in(1/2,1)$. Second, we establish Krein-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasi-convex domains and study the properties of the corresponding Weyl--Titchmarsh operators (or energy-d…
We study spectral properties for $H_{K,Ω}$, the Krein--von Neumann extension of the perturbed Laplacian $-Δ+V$ defined on $C^\infty_0(Ω)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Ω\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula \[ #\{j\in\mathbb{N} | λ_{K,Ω,j}\leqλ\} = (2π)^{-n} v_n |Ω| λ^{n/2}+O\big(λ^{(n-(1/2))/2}\big) {as} λ\to\infty, \] where $v_n=π^{n/2}/ Γ((n/2)+1)$ deno…
We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if $H$ is a selfadjoint operator on a Krein space $\cH$, equipped with the Krein scalar product $\langle \cdot| \cdot \rangle$, $A$ is the generator of a $C_{0}-$group on $\cH$ and $I\subset \rr$ is an interval such that: \begin{itemize} \item[]1) $H$ admits a Borel functional calculus on $I$, \item[]2) the spectral projection $\one_{I}(H)$ is positive in the Krein sense, \item[]3) the following {\em positive commutator estimate} holds: \[ \Re \langle u| [H, ıA]u\rangle\g…
Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with $C^{1,1}$ boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.
For a strongly elliptic second-order operator $A$ on a bounded domain $Ω\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(Ω)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of Hölder $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show…
For a second-order symmetric strongly elliptic operator A on a smooth bounded open set Ωin R^n with boundary Σ, the mixed problem is defined by a Neumann-type condition on a part Sigma_+ of the boundary and a Dirichlet condition on the other part Sigma_-. We show a Krein resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Sigma_+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely s_j j^{2/(n-1)}\t…
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.
Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω\subset R^n, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) s_j(G) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**)…
I offer a simple and useful formula for the resolvent of a small rank perturbation of large matrices. I discuss applications of this formula, in particular, to analytical and numerical solving of difference boundary value problems. I present examples connected with such problems for the difference Laplacian and estimate numerical efficiency of the corresponding algorithms.
We consider self-adjoint realizations of a second-order elliptic differential expression on ${\mathbb R}^n$ with singular interactions of $δ$ and $δ^\prime$-type supported on a compact closed smooth hypersurface in ${\mathbb R}^n$. In our main results we prove spectral asymptotics formulae with refined remainder estimates for the singular values of the resolvent difference between the standard self-adjoint realizations and the operators with a $δ$ and $δ^\prime$-interaction, respectively. Our technique makes use of general pseudodifferential methods, classical results on spectral asymptotics o…
In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $Ω$ with smooth boundary $\partialΩ$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partialΩ$. It will be shown that the resolvent difference of two generalized Rob…
In this paper the spectral properties of Dirac operators $A_η$ with electrostatic $δ$-shell interactions of constant strength $η$ supported on compact smooth surfaces in $\mathbb{R}^3$ are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of $A_η$ are investigated. In particular, it turns out that the discrete spectrum of $A_η$ inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of $A…
We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}ϕ(t)- 2ık \p_{t}ϕ(t)+ h ϕ(t)=0, \] where $ϕ: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint oper…
For $2a$-order strongly elliptic operators $P$ generalizing $(-Δ)^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $Ω\subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\infty $, when $Ω$ is $C^{τ+1}$ with a finite $τ>2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar resu…
The spectral behavior of the difference between the resolvents of two realizations $\tilde A_1$ and $\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $νu=b_1γ_0u$ and $νu=b_2γ_0u$, can in the case where all coefficients are $C^\infty$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ of $(\tilde A_1 -λ)^{-1}-(\tilde A_2 -λ)^{-1}$ s…
The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain $H^{2}(\mathbb{R}^{n})$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditio…
This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of…
We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the fre…
In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Ω; \mathbb{C}^4)$, where $Ω\subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract bound…
In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{χ,Σ_+}$ in $L_2(Ω)$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $Ω\subset R^n$; here the boundary $\partialΩ=Σ$ is partioned …
Let $Ω\subset{\mathbb R}^n$ be bounded with a smooth boundary $Γ$ and let $S$ be the symmetric operator in $L^2(Ω)$ given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of $S$ by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms in $L^2(Γ)$ which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one …
Assuming $T_0$ to be an m-accretive operator in the complex Hilbert space $\mathcal{H}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T = T_0 + W$ and prove stability of square root domains, that is, $$ dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big). $$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big) = dom\big(((T_0 + W)^*)^{1/2}\big), $$ which is most suitable for…
