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

DOI:
https://doi.org/10.1090/S0002-9939-06-08483-8

Published electronically:
May 31, 2006

MathSciNet review:
2240676

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
81Q10,
53A04

Retrieve articles in all journals with MSC (2000): 81Q10, 53A04

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:
https://doi.org/10.1090/S0002-9939-06-08483-8

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.