Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree ![$ -2$](/tran/2008-360-08/S0002-9947-08-04479-6/gif-title0/img1.gif) |
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Authors: | Andrew Hassell Simon Marshall |
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Institution: | Department of Mathematics, The Australian National University, ACT 0200, Australia ; Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand |
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Abstract: | We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function , the number of bound states of the operator in below . Here is a bounded potential behaving asymptotically like where is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator on the sphere has negative eigenvalues less than , we prove that may be estimated as Thus, in particular, if there are no such negative eigenvalues, then has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that and that there is exactly one eigenvalue less than , with all others , we show that the negative spectrum is asymptotic to a geometric progression with ratio . |
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Keywords: | |
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