A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions

Abstract

For an arbitrary open, nonempty, bounded set $\Omega \subset {\mathbb{R}}^n$, $n\in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{\Omega ,2m}(a,b,q)$ in $L^2(\Omega )$ defined on $W_0^{2m,2}(\Omega )$, associated with the higher-order differential expression $${\tau }_{2m}(a,b,q):=(\sum _{j,k=1}^n(-i{\partial }_j-b_j)a_{j,k}(-i{\partial }_k-b_k)+q{)}^m,m\in \mathbb{N},$$ and its Krein--von Neumann extension $A_{K,\Omega ,2m}(a,b,q)$ in $L^2(\Omega )$. Denoting by $N(\lambda ;A_{K,\Omega ,2m}(a,b,q))$, $\lambda >0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,\Omega ,2m}(a,b,q)$, we derive the bound $$N(\lambda ;A_{K,\Omega ,2m}(a,b,q))\le C{v}_n(2\pi {)}^{-n}(1+\frac{2m}{2m+n}{)}^{n/(2m)}{\lambda }^{n/(2m)},\lambda >0,$$ where $C=C(a,b,q,\Omega )>0$ (with $C(I_n,0,0,\Omega )=|\Omega |$) is connected to the eigenfunction expansion of the self-adjoint operator ${\tilde{A}}_{2m}(a,b,q)$ in $L^2({\mathbb{R}}^n)$ defined on $W^{2m,2}({\mathbb{R}}^n)$, corresponding to ${\tau }_{2m}(a,b,q)$. Here $v_n:={\pi }^{n/2}/\Gamma ((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in ${\mathbb{R}}^n$. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of ${\tilde{A}}_2(a,b,q)$ in $L^2({\mathbb{R}}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,\Omega ,2m}(a,b,q)$ in $L^2(\Omega )$ of $A_{\Omega ,2m}(a,b,q)$. No assumptions on the boundary $\partial \Omega$ of $\Omega$ are made.

39 pages. arXiv admin note: substantial text overlap with arXiv:1403.3731