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On a sum rule for Schrödinger operators with complex potentials

Author: Oleg Safronov
Journal: Proc. Amer. Math. Soc. 138 (2010), 2107-2112
MSC (2000): Primary 47F05
Published electronically: January 22, 2010
MathSciNet review: 2596049
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Abstract: We study the distribution of eigenvalues of the one-dimensional Schrödinger operator with a complex valued potential $ V$. We prove that if $ \vert V\vert$ decays faster than the Coulomb potential, then the series of imaginary parts of square roots of eigenvalues is convergent.

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  • 1. A. A. Abramov, A. Aslanyan, and E. B. Davies, Bounds on complex eigenvalues and resonances, J. Phys. A 34 (2001), no. 1, 57–72. MR 1819914, 10.1088/0305-4470/34/1/304
  • 2. E. B. Davies and Jiban Nath, Schrödinger operators with slowly decaying potentials, J. Comput. Appl. Math. 148 (2002), no. 1, 1–28. On the occasion of the 65th birthday of Professor Michael Eastham. MR 1946184, 10.1016/S0377-0427(02)00570-8
  • 3. Demuth, M., Hansmann, M. and Katriel G.: On the distribution of eigenvalues of non-selfadjoint operators, preprint.
  • 4. Rupert L. Frank, Ari Laptev, Elliott H. Lieb, and Robert Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials, Lett. Math. Phys. 77 (2006), no. 3, 309–316. MR 2260376, 10.1007/s11005-006-0095-1
  • 5. Leonid Golinskii and Stanislav Kupin, Lieb-Thirring bounds for complex Jacobi matrices, Lett. Math. Phys. 82 (2007), no. 1, 79–90. MR 2367876, 10.1007/s11005-007-0189-4
  • 6. Hansmann, M. and Katriel, G.: Inequalities for the eigenvalues of non-selfadjoint Jacobi operators, preprint.
  • 7. Rowan Killip, Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum, Int. Math. Res. Not. 38 (2002), 2029–2061. MR 1925875, 10.1155/S1073792802204250
  • 8. Laptev, A. and Safronov, O.: Eigenvalue estimates for Schrödinger operators with complex potentials, Comm. Math. Phys. 292 (2009), no. 1, 29-54.
  • 9. Lieb, E. H. and Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in Studies in Mathematical Physics (Essays in Honor of Valentine Bargmann), 269-303. Princeton Univ. Press, Princeton, NJ, 1976.
  • 10. Safronov, O.: Estimates for eigenvalues of the Schrödinger operator with a complex potential, Bull. London Math. Soc., to appear.

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Additional Information

Oleg Safronov
Affiliation: Department of Mathematics, University of North Carolina, Charlotte, 9201 University City Boulevard, Charlotte, North Carolina 28223-0001

Keywords: Eigenvalue estimates, Schr\"odinger operators, complex potentials, sum rules
Received by editor(s): April 10, 2009
Received by editor(s) in revised form: October 4, 2009
Published electronically: January 22, 2010
Additional Notes: The author would like to thank B. Vainberg, S. Molchanov, A. Gordon and P. Grigoriev for inspiring and motivating discussions
Communicated by: Varghese Mathai
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.