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Analysis of Schrodinger operators with inverse square potentials I: regularity results in 3D
Let V be a potential on R3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ 2, with (x) = x − p for x close to p and Z continuous on R3 with Z(p) > −1/4 for p 2 S. Also assume that and Z are smooth outside S and Z is smooth in polar coord...
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Main Authors:  , , , 

Format:  Default Article 
Published: 
2012

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Online Access:  https://hdl.handle.net/2134/17171 
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Summary:  Let V be a potential on R3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ 2, with (x) = x − p for x close to p and Z continuous on R3 with Z(p) > −1/4 for p 2 S. Also assume that and Z are smooth outside S and Z is smooth in polar coordinates around each singular point. We either assume that V is periodic or that the set S is finite and V extends to a smooth function on the radial compactification of R3 that is bounded outside a compact set containing S. In the periodic case, we let be the periodicity lattice and define T := R3/ . We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr¨odingertype operator H = − + V acting on L2(T), as well as for the induced k–Hamiltonians Hk obtained by restricting the action of H to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the nonperiodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper. 
