Approximation of eigenvalues of Schrodinger operators¨
https://doi.org/10.17586/2220-8054-2018-9-2-145-161
Abstract
It is known that convergence of l. s. b. closed symmetric sesquilinear forms implies norm resolvent convergence of the associated self-adjoint operators and thus, in turn, convergence of discrete spectra. In this paper, in both cases, sharp estimates for the rate of convergence are derived. An algorithm for the numerical computation of eigenvalues of generalized Schrodinger operators in¨ L2(R) is presented and illustrated by explicit examples; the mentioned general results on the rate of convergence are applied in order to obtain error estimates for these computations. An extension of the results to Schrodinger operators on metric graphs is sketched.
Supporting agencies: Work supported by DFG grant Br1686/3-1. J. Brasche thanks the ITMO University of St. Petersburg for their hospitality.
About the Authors
J. F. Brasche
Institut fu¨r Mathematik, Technische Universitat Clausthal
Germany
Germany
Erzstraße 1, 30867 Clausthal-Zellerfeld
R. Fulsche
Institut fu¨r Analysis, Leibniz Universitat Hannover
Germany
Germany
Welfengarten 1, 30167 Hannover
References
1. BelHadjAli H., BenAmor A., Brasche J. Large Coupling Convergence: Overview and New Results. In Partial Differential Equations and Spectral Theory (Demuth M., Schulze B.-W., Witt I., editors), volume 211 of Operator Theory: Advances and Applications, pp. 73–117. Springer Basel, Basel, 2011.
2. Brasche J. On large coupling convergence within trace ideals. Methods Funct. Anal. Topology, 2014, 20, P. 3–9.
3. Belgacem F., BelHadjAli H., BenAmor A., Thabet A. The Robin Laplacian in the large coupling limit: Convergence and spectral asymptotic. available on arXiv: 1511.06086.
4. BelHadjAli H., BenAmor A., Brasche J. On trace and Hilbert-Schmidt norm estimates. Bull. London Math. Soc., 2012, 44, P. 661–674.
5. BelHadjAli H., BenAmor A., Brasche J. Large coupling convergence with negative perturbations. J. Math. Anal. Appl., 2014, 409, P. 582–597.
6. Ben Amor A., Brasche J. Sharp estimates for large coupling convergence with applications to Dirichlet operators. J. Funct. Anal., 2008, 254, P. 454–475.
7. Brasche J., Demuth M. Dynkin’s formula and large coupling convergence. J. Funct. Anal., 2005, 219, P. 34–69.
8. Grubisiˇ c L. Relative Convergence Estimates for the Spectral Asymptotic in the Large Coupling Limit.´ Integral Equations Operator Theory, 2009, 65, P. 51–81.
9. Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, 2nd edition, 2005.
10. Brasche J., Figari R., Teta A. Singular Schrodinger Operators as Limits Point Interaction Hamiltonians.¨ Potential Anal., 1998, 8, P. 163– 178.
11. Brasche J., Oz˘anova K. Convergence of Schr´ odinger Operators.¨ SIAM J. Math. Anal., 2007, 39, P. 281–297.
12. Oz˘anova K. Approximation by point potentials in a magnetic field.´ J. Phys. A: Math. Gen., 2006, 39, P. 3071–3083.
13. Kato T. Perturbation Theory for Linear Operators. Springer, 2nd edition, 1976.
14. Reed M., Simon B. Methods of Modern Mathematical Physics 1: Functional Analysis. Academic Press, 1972.
15. Brasche J., Exner P., Kuperin Y., Seba P. Schr˘ odinger Operators with Singular Interactions.¨ J. Math. Anal. Appl., 1994, 184, P. 112–139.
16. Schmincke U.-W. On Schrodinger’s factorization method for Sturm-Liouville operators.¨ Proc. Roy. Soc. Edinburgh Sect. A, 1978, 80, P. 67–84.
17. Simon B. The Bound State of Weakly Coupled Schrodinger Operators in One and Two Dimensions.¨ Ann. Phys., 1976, 97, P. 279–288.
18. Gustafson S., Sigal I. Mathematical Concepts of Quantum Mechanics. Springer, 2nd edition, 2011.
19. Berkolaiko G., Kuchment P. Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs. American Mathematical Society, 2013.
Review
For citations:
Brasche J.F., Fulsche R. Approximation of eigenvalues of Schrodinger operators¨. Nanosystems: Physics, Chemistry, Mathematics. 2018;9(2):145–161. https://doi.org/10.17586/2220-8054-2018-9-2-145-161
Views:
2
Email this article 