A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

Abstract

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\ge \epsilon I_{\mathcal{H}}$ for some $\epsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega }$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta +V$ (in short, the perturbed Krein Laplacian) 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 $C^{1,r}$, $r>1/2$.

68 pages. arXiv admin note: extreme text overlap with arXiv:0907.1442

References in corpus (5)

• 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
• 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

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• Functional model for boundary-value problems
• The heat equation under linear conditions on the moments in higher dimensions
• Some remarks on the Krein--von Neumann extension of different Laplacians
• The Krein-von Neumann extension revisited
• An improved version of a spectral inequality by Payne
• Selfadjoint and $m$ sectorial extensions of Sturm-Liouville operators
• Extensions of symmetric operators that are invariant under scaling and applications to indicial operators
• The Krein-von Neumann Extension of a Regular Even Order Quasi-Differential Operator