# Optimal measures for the fundamental gap of Schrödinger operators

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 1, page 194-205
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topVarchon, Nicolas. "Optimal measures for the fundamental gap of Schrödinger operators." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 194-205. <http://eudml.org/doc/250722>.

@article{Varchon2010,

abstract = {
We study the potential which minimizes the fundamental gap of the
Schrödinger operator under the total mass constraint. We consider
the relaxed potential and prove a regularity result for the optimal
one, we also give a description of it. A consequence of this result
is the existence of an optimal potential under L1 constraints.
},

author = {Varchon, Nicolas},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Schrödinger operator; eigenvalue problems; measure
theory; shape optimization; measure theory},

language = {eng},

month = {1},

number = {1},

pages = {194-205},

publisher = {EDP Sciences},

title = {Optimal measures for the fundamental gap of Schrödinger operators},

url = {http://eudml.org/doc/250722},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Varchon, Nicolas

TI - Optimal measures for the fundamental gap of Schrödinger operators

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/1//

PB - EDP Sciences

VL - 16

IS - 1

SP - 194

EP - 205

AB -
We study the potential which minimizes the fundamental gap of the
Schrödinger operator under the total mass constraint. We consider
the relaxed potential and prove a regularity result for the optimal
one, we also give a description of it. A consequence of this result
is the existence of an optimal potential under L1 constraints.

LA - eng

KW - Schrödinger operator; eigenvalue problems; measure
theory; shape optimization; measure theory

UR - http://eudml.org/doc/250722

ER -

## References

top- M.S. Ashbaugh, E.M. Harrell and R. Svirsky, On minimal and maximal eigenvalue gaps and their causes. Pacific J. Math.147 (1991) 1–24.
- D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Their Applications65. Birkhäuser, Basel, Boston (2005).
- D. Bucur and T. Chatelain, Strict monotonicity of the second eigenvalue of the Laplace operator on relaxed domain. Bull. Appl. Comp. Math.1510–1566 (1998) 115–122.
- D. Bucur and A. Henrot, Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. Roy. Soc. London456 (2000) 985–996.
- G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim.23 (1991) 17–49.
- G. Buttazzo, N. Varchon and H. Zoubairi, Optimal measures for elliptic problems. Annali Mat. Pur. Appl.185 (2006) 207–221.
- R. Courant and D. Hilbert, Methods of Mathematical Physics. Interscience Publishers (1953).
- G. Dal Maso, Γ-convergence and µ-capacities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)14 (1987) 423–464.
- G. Dal Maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993).
- G. Dal Maso and U. Mosco, Wiener's criterion and Γ-convergence. Appl. Math. Optim.15 (1987) 15–63.
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992).
- A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser Verlag, Basel, Boston, Berlin (2006).
- T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag (1980).
- N. Varchon, Optimal measures for nonlinear cost functionals. Appl. Mat. Opt.54 (2006) 205–221.
- W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.