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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A lower bound for the ground state energy of a Schrödinger operator on a loop


Author: Helmut Linde
Journal: Proc. Amer. Math. Soc. 134 (2006), 3629-3635
MSC (2000): Primary 81Q10; Secondary 53A04
Published electronically: May 31, 2006
MathSciNet review: 2240676
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Abstract: Consider a one-dimensional quantum mechanical particle described by the Schrödinger equation on a closed curve of length $ 2\pi$. Assume that the potential is given by the square of the curve's curvature. We show that in this case the energy of the particle cannot be lower than $ 0.6085$. We also prove that it is not lower than $ 1$ (the conjectured optimal lower bound) for a certain class of closed curves that have an additional geometrical property.


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Additional Information

Helmut Linde
Affiliation: Department of Physics, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22 Santiago, Chile
Email: Helmut.Linde@gmx.de

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08483-8
PII: S 0002-9939(06)08483-8
Received by editor(s): June 21, 2005
Published electronically: May 31, 2006
Additional Notes: This work was supported by DIPUC (Pontificia Universidad Católica de Chile).
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.