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 $$ \tau_{ 2m} (a,b,q) := \bigg(\sum_{ j,k=1}^{ n} (-i \partial_j - b_j) a_{ j,k} (-i \partial_k - b_k)+q\bigg)^m, \quad m \in \mathbb{ N}, $$ 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 $$ N(\lambda; A_{ K, \Omega, 2m} (a,b,q)) \leq C v_n (2\pi)^{ -n} \bigg(1+\frac{ 2m}{ 2m+n}\bigg)^{ n/(2m)} \lambda^{ n/(2m)} , \quad \lambda > 0, $$ 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 $\widetilde A_{ 2m} (a,b,q)$ 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 $\widetilde A_{ 2} (a,b,q)$ 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.
Categories: math.AP math-ph math.MP math.SP