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

摘要

In the first (and abstract) part of this survey we prove the unitaryequivalence of the inverse of the Krein--von Neumann extension (on theorthogonal complement of its kernel) of a densely defined, closed, strictlypositive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon>0$ 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 inquantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectralproperties for $H_{K,\Omega}$, the Krein--von Neumann extension of theperturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian)defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded andnonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to aclass of nonsmooth domains which contains all convex domains, along with alldomains of class $C^{1,r}$, $r>1/2$.