Krein-like extensions and the lower boundedness problem for elliptic operators

TL;DR

This paper investigates the lower boundedness of selfadjoint extensions of elliptic operators, extending known results to exterior domains and analyzing the spectral properties of Krein-like extensions.

Contribution

It generalizes the lower boundedness criteria for selfadjoint extensions of elliptic operators to exterior domains and studies spectral asymptotics of Krein-like extensions.

Findings

Lower boundedness of extensions is characterized for general boundary operators.

Spectral asymptotics for Krein-like extensions are established on bounded domains.

Results extend classical theory to non-compact inverse cases.

Abstract

For selfadjoint extensions tilde-A of a symmetric densely defined positive operator A_min, the lower boundedness problem is the question of whether tilde-A is lower bounded {\it if and only if} an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets 1976); this applies to elliptic operators A on bounded domains. For exterior domains, A_gamma ^{-1} is not compact, and whereas the lower bounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from T to tilde-A has only been known when m(T)>-m(A_gamma). We now show it for general T. The operator A_a corresponding to T=aI, generalizing the Krein-von Neumann extension A_0, appears here; its possible lower boundedness for all real a is decisive. We study this Krein-like extension, showing for bounded domains that the discrete eigenvalues satisfy N_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .