Counting eigenvalues of biharmonic operators with magnetic fields

Abstract

An analysis is given of the spectral properties of perturbations of the magnetic bi-harmonic operator ∆ 2 A in L 2 (R n ), n = 2, 3, 4, where A is a magnetic vector potential of Aharonov-Bohm type, and bounds for the number of negative eigenvalues are established. Key elements of the proofs are newly derived Rellich inequalities for ∆ 2 A which are shown to have a bearing on the limiting cases of embedding theorems for Sobolev spaces H 2 (R n ).

Sharp bounds for eigenvalues of biharmonic operators with complex potentials in low dimensions

“…On the other hand, related issues for higher‐order partial differential operators seem to be just scarcely considered in the literature, even in the self‐adjoint setting. The reader is referred to [7, 9, 10, 17, 20, 22, 26]; see also [4–6, 11, 14] for other spectral questions.…”

Sharp bounds for eigenvalues of biharmonic operators with complex potentials in low dimensions

“…On the other hand, related issues for higher-order partial differential operators seem to be just scarcely considered in the literature, even in the self-adjoint setting. The reader is referred to [22,25,9,20,10,7,17]; see also [11,4,5,14,6] for other spectral questions.…”