For an arbitrary open, nonempty, bounded set Ω⊂Rn,
n∈N, and sufficiently smooth coefficients a,b,q, we consider
the closed, strictly positive, higher-order differential operator AΩ,2m(a,b,q) in L2(Ω) defined on W02m,2(Ω), associated with
the higher-order differential expression τ2m(a,b,q):=(j,k=1∑n(−i∂j−bj)aj,k(−i∂k−bk)+q)m,m∈N, and its Krein--von Neumann extension
AK,Ω,2m(a,b,q) in L2(Ω). Denoting by N(λ;AK,Ω,2m(a,b,q)), λ>0, the eigenvalue counting function
corresponding to the strictly positive eigenvalues of AK,Ω,2m(a,b,q), we derive the bound N(λ;AK,Ω,2m(a,b,q))≤Cvn(2π)−n(1+2m+n2m)n/(2m)λn/(2m),λ>0, where C=C(a,b,q,Ω)>0 (with C(In,0,0,Ω)=∣Ω∣) is connected to the eigenfunction expansion of the self-adjoint
operator A2m(a,b,q) in L2(Rn) defined on
W2m,2(Rn), corresponding to τ2m(a,b,q). Here vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in
Rn.
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 A2(a,b,q) in
L2(Rn).
We also consider the analogous bound for the eigenvalue counting function for
the Friedrichs extension AF,Ω,2m(a,b,q) in L2(Ω) of
AΩ,2m(a,b,q).
No assumptions on the boundary ∂Ω of Ω are made.Comment: 39 pages. arXiv admin note: substantial text overlap with
arXiv:1403.373