04-338 W.D. Evans and Roger T. Lewis
Counting Eigenvalues of Biharmonic Operators with Magnetic Fields (234K, PDF) Oct 28, 04
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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)\$.

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