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

Linde, Helmut

doi:10.1090/S0002-9939-06-08483-8

A lower bound for the ground state energy of a Schrödinger operator on a loop
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Proc. Amer. Math. Soc. 134 (2006), 3629-3635 Request permission

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.

References

Rafael D. Benguria and Michael Loss, Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane, Partial differential equations and inverse problems, Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 53–61. MR 2091490, DOI 10.1090/conm/362/06604
Almut Burchard and Lawrence E. Thomas, On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop, J. Geom. Anal. 15 (2005), no. 4, 543–563. MR 2203162, DOI 10.1007/BF02922244
Pavel Exner, Evans M. Harrell, and Michael Loss, Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature, Mathematical results in quantum mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 47–58. MR 1708787
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/
Evans M. Harrell II and Michael Loss, On the Laplace operator penalized by mean curvature, Comm. Math. Phys. 195 (1998), no. 3, 643–650. MR 1641019, DOI 10.1007/s002200050406
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.