In this paper, we combine results on extensions of operators with recent results on the relation between the M-function and the spectrum, to examine the spectral behaviour of boundary value problems. M-functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that can not be detected by the M-function.
In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…
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.
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…
In this paper we provide a complete study of the spectrum of a constant coefficients differential operator on a scale of localized Sobolev spaces, $H^{s}_{loc}(I),$ which are Fréchet spaces. This is quite different from what we find in the literature, where all the relevant results are concerned with spectrum on Banach spaces. Our aim is to understand the behavior of all the three types of spectrum (point, residual and continuous) and the relation between them and those of the dual operator. The main result we present shows that there is no complex number in the resolvent set of such operators…
We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.
We consider second order elliptic differential operators on a bounded Lipschitz domain $Ω$. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in $H^1(Ω)$, and Lagrangian planes in $H^{1/2}(\partialΩ)\times H^{-1/2}(\partialΩ)$. Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in $H^{1/2}(\partialΩ)\times H^{-1/2}(\partialΩ)$. Further…
We perform the spectral analysis of a family of Jacobi operators $J(α)$ depending on a complex parameter $α$. If $|α|\neq1$ the spectrum of $J(α)$ is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If $|α|=1$, $α\neq \pm 1$, the essential spectrum of $J(α)$ covers the entire complex plane. In addition, a formula for the Weyl $m$-function as well as the asymptotic expansions of solutions of the difference equation corresponding to $J(α)$ are obtained. Finally, the completeness of eigenvectors and Rodriguez-li…
In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $Ω\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is repla…
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.
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 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…
This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all se…
We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well-known in some special cases) by an abstract rank condition. The latter seems to apply particularly well to a specific boundary condition, sometimes dubbed "anti-Kirchhoff" in the literature, that arise in the theory of differential operators on graphs; it also permits to discuss interest…
In this article, we state the Bohr-Sommerfeld conditions around a global minimum of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface, using an argument of normal form which is obtained thanks to Fourier integral operators. These conditions give an asymptotic expansion of the eigenvalues of the operator in a neighbourhood of fixed size of the singularity. We also recover the usual Bohr-Sommerfeld conditions away from the critical point. We end by investigating an example on the two-dimensional torus.
We study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.
In this note semibounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second order PDEs on domains with non-compact boundaries.
In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $δ$- and $δ'$-interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples.
This paper concerns spectral invariants of the Laplacian on a compact Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave group of (M,g), then the trace Tr U(t) is singular when t = 0 or when ti is the length of a closed geodesic. It has a special type of singularity expansion at each length and the coefficients are known as the wave invariants. Our main purpose is to calculate the wave invariants explicitly in terms of curvature, Jacobi fields etc. when the closed geodesic is non-degenerate elliptic. We do this by putting the Laplacian into quantum Birkhoff normal for…
