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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
Posted:
May 31, 2006
MathSciNet review:
2240676
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Abstract |
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Abstract: Consider a one-dimensional quantum mechanical particle described by the Schrödinger equation on a closed curve of length  . 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  . We also prove that it is not lower than  (the conjectured optimal lower bound) for a certain class of closed curves that have an additional geometrical property.
References
- 1.
R.D. Benguria and M. Loss: Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals in the plane, Contemporary Math. 362 (2004), 53-61. MR 2091490 (2005f:81057)
- 2.
A. Burchard and L.E. Thomas: On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop, Journal of Geometric Analysis 15 (2005), 543-563. MR 2203162
- 3.
P. Exner, E.M. Harrell and M. Loss: Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature, Mathematical results in quantum mechanics (Prague 1998), Oper. Theory Adv. Appl. 108 (1999), 47-58. MR 1708787 (2000e:58045)
- 4.
E.M. Harrell: Gap estimates for Schrödinger operators depending on curvature, talk delivered at the 2002 UAB International Conference on Differential Equations and Mathematical Physics. Available electronically at http://www.math.gatech.edu/~harrell/
- 5.
E.M. Harrell and M. Loss: On the Laplace operator penalized by mean curvature, Commun. Math. Phys. 195 (1998), 643-650. MR 1641019 (99f:58213)
- 6.
E.H. Lieb and W. Thirring: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Math. Phys., Essays in Honor of Valentine Bargmann, Princeton (1976), 269-303.
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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
Posted:
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 after
28 years from publication.
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