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A lower bound for the ground state energy of a Schrödinger operator on a loop
Author(s):
Helmut
Linde
Journal:
Proc. Amer. Math. Soc.
134
(2006),
3629-3635.
MSC (2000):
Primary 81Q10;
Secondary 53A04
Posted:
May 31, 2006
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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:
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- 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)
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- 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/
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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:
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
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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