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Optimal Eigenvalues for Some Laplacians and Schrödinger Operators Depending on Curvature

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Mathematical Results in Quantum Mechanics

Part of the book series: Operator Theory Advances and Applications ((OT,volume 108))

Abstract

We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round.

We also comment on the use of coordinate transformations for these operators and mention some open problems.

Work supported by GA AS No.1048801

Work supported by N.S.F. grant DMS-9622730

Work supported by N.S.F. grant DMS-9500840

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Author information

Authors and Affiliations

  1. Nuclear Physics Institute, Academy of Sciences, Řež, CZ-25068, Japan

    Pavel Exner

  2. School of Mathematics, Georgia Tech, Atlanta, 30332-0160, GA, USA

    Evans M. Harrell & Michael Loss

  3. Doppler Institute, Czech Technical Univ., Břehová 7, Prague, CZ-11519, Czech Republic

    Pavel Exner

Authors
  1. Pavel Exner
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  2. Evans M. Harrell
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  3. Michael Loss
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, Nuclear Physics Institute Academy of Science, Rez near Prague, 25068, Czech Republic

    Jaroslav Dittrich , Pavel Exner  & Miloš Tater ,  & 

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© 1999 Springer Basel AG

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Exner, P., Harrell, E.M., Loss, M. (1999). Optimal Eigenvalues for Some Laplacians and Schrödinger Operators Depending on Curvature. In: Dittrich, J., Exner, P., Tater, M. (eds) Mathematical Results in Quantum Mechanics. Operator Theory Advances and Applications, vol 108. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8745-8_4

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  • DOI: https://doi.org/10.1007/978-3-0348-8745-8_4

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9754-9

  • Online ISBN: 978-3-0348-8745-8

  • eBook Packages: Springer Book Archive

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