Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains

Abstract

We study spectral properties for $H_{K,\Omega }$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta +V$ defined on $C_0^{\infty }(\Omega )$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega \subset {\mathbb{R}}^n$ 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 $v_n={\pi }^{n/2}/\Gamma ((n/2)+1)$ denotes the volume of the unit ball in ${\mathbb{R}}^n$, and ${\lambda }_{K,\Omega ,j}$, $j\in \mathbb{N}$, are the non-zero eigenvalues of $H_{K,\Omega }$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-\Delta +V$ defined on $C_0^{\infty }(\Omega )$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980's. Our work builds on that of Grubb in the early 1980's, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.

60 pages

References in corpus (7)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators
• Krein resolvent formulas for elliptic boundary problems in nonsmooth domains
• The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem
• M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems
• Non-negative perturbations of non-negative self-adjoint operators
• The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients

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• Spectral estimates for resolvent differences of self-adjoint elliptic operators
• QFT over the finite line. Heat kernel coefficients, spectral zeta functions and selfadjoint extensions
• The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem
• The Maslov Index and the Spectra of Second Order Elliptic Operators
• Self-adjoint extension schemes and modern applications to quantum Hamiltonians
• Local Density Approximation for Almost-Bosonic Anyons
• Lieb-Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems
• Spectral Theory as Influenced by Fritz Gesztesy
• Heat invariants of the perturbed polyharmonic Steklov problem
• The spectral $\zeta$-function for quasi-regular Sturm--Liouville operators
• Krein-like extensions and the lower boundedness problem for elliptic operators
• One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues
• Geometric invariants of spectrum of the Navier-Lamé operator
• Some remarks on the Krein--von Neumann extension of different Laplacians
• On the spectral asymptotics for the buckling problem
• Krein extension of a differential operator of even order
• The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues with Dirichlet boundary condition on Riemannian manifolds
• Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators
• A Note on Operator-Theoretic Approach to Classic Boundary Value Problems for Harmonic and Analytic Functions in the Complex Plane Domain
• The Krein-von Neumann extension revisited
• Sharp Weyl-Type Formulas of the Spectral Functions for Biharmonic Steklov Eigenvalues