We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type $\cA=\sum_{0\le |α|,|β|\le m}(-1)^αD^αa_{α, β}(x)D^β$, $a_{α, β}(\cdot)\in C^{\infty}({\overlineΩ})$, on smooth (bounded or unbounded) domains in $\bbR^n$ with compact boundary. Using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, we prove various trace ideal properties of powers of resolvent differences of these closed realizations of $\cA$ and derive estimates on eigenvalues of certain self-adjoint realizations in spectral gaps of the Dirichlet realization.
Our results extend classical theorems due to Visik, Povzner, Birman, and Grubb.
For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it i…
We consider exterior elliptic problems with coefficients stabilizing at infinity and study the critical value $β_{cr}$ of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. The dependence of $β_{cr}$ on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated.
We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spec…
The spectrum of a selfadjoint second order elliptic differential operator in $L^2(\mathbb{R}^n)$ is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and correspond to an interior and an exterior boundary value problem. This leads to PDE analogs of renowned facts in spectral theory of ODEs. The main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding Weyl functions, and are applied to the PDE setting afterwards.
We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$ for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast with exterior domains depends both on the geometry of the domain $G$ and the coefficients $a$ of the equation.
We analyze the behavior of the eigenvalues and eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions when the domain is perturbed. We focus on exterior perturbations of the domain, that is, the limit domain is contained in each of the perturbed domains. Moreover, the family of perturbed domains is made up of bounded domains although not necessarily uniformly bounded. We show that if we know that the eigenvalues and eigenfunctions of the perturbed domains converge to the ones of the limit domain, then necessarily we have that the measure of the perturbation tends t…
We study Schroedinger operators with Robin boundary conditions on exterior domains in $\R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.
We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.
In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schrödinger operators, Laplacians on exterior domains, and Laplacians on manifolds which are hyperbolic near infinity.
In this work, we investigate the spectral problem $Au = λu$ where $A$ is a fractional elliptic operator involving left- and right-sided Riemann-Liouville derivatives. These operators are nonlocal and nonsymmetric, however, share certain classic elliptic properties. The eigenvalues correspond to the roots of a class of certain special functions. Compared with classic Sturm-Liouville problems, the most challenging part is to set up the framework for analyzing these nonlocal operators, which requires developing new tools. We prove the existence of the real eigenvalues, find the range for all poss…
We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the euclidean N-dimensional space. We prove stability results for the dependence of the eigenvalues upon variation of the mass density and we prove a maximum principle for extremum problems related to mass density perturbations which preserve the total mass.
The spectral properties of a class of non-selfadjoint second order elliptic operators with indefinite weight functions on unbounded domains $Ω$ are investigated. It is shown that under an abstract regularity assumption the nonreal spectrum of the associated elliptic operator in $L^2(Ω)$ is bounded. In the special case that $Ω=R^n $decomposes into subdomains $Ω_+$ and $Ω_-$ with smooth compact boundaries and the weight function is positive on $Ω_+$ and negative on $Ω_-$, it turns out that the nonreal spectrum consists only of normal eigenvalues which can be characterized with a Dirichlet-to-Neu…
Let $M$ be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a torus $T$, and $P_0$ an invariant elliptic classical pseudodifferential operator on $M$. In this note, we strengthen asymptotics for the equivariant (or reduced) spectral function of $P_0$ derived previously, which are already sharp in the eigenvalue aspect, to become almost sharp in the isotypic aspect. In particular, this leads to hybrid equivariant $L^p$-bounds for eigenfunctions that are almost sharp in the eigenvalue and isotypic aspect.
Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\DD +\nn V$ for some $V\in C^2(M)$ with $\exp[V]$ integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of $L$. As a consequence of the main result, let $\rr$ be the distance function from a point $o$, then the spectral gap exists provided $\lim_{\rr\to\infty}\sup L\rr<0$ while the spectral gap does not exist if $o$ is a pole and $\lim_{\rr\to\infty}\inf L\rr\ge 0.$ Moreover, the elliptic operators on $\mathbb R^d$ are also studied.
We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space $L_2(\R^m)$, for $m \ge 2$. We specifically consider situations where high contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schrödinger operators, elliptic operators in divergence form, Laplace-Beltrami-operators, Schrödinger and Pauli operators with periodic magnetic fields. There are corresponding applications in heat and wave …
In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $Ω$ corresponding to a negative Robin parameter does not exceed the analogous quantity …
