W.D. Evans and Roger T. Lewis
Counting Eigenvalues of Biharmonic Operators with Magnetic Fields
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ABSTRACT. An analysis is given of the spectral properties of perturbations of the magnetic bi-harmonic operator $\Delta_A^2$ 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 $\Delta_\ab^2$ which are shown to have a bearing on the limiting cases of embedding theorems for Sobolev spaces $H^2(\R^n)$.