Spectral isoperimetric inequalities for singular interactions on open arcs

ABSTRACT

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive $\delta$-interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. As a consequence of the result for $\delta$-interaction, we obtain that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length.

KEYWORDS:

• $\delta$-interaction on an open arc
• Robin Laplacian on planes with slits
• lowest eigenvalue
• spectral isoperimetric inequality
• Birman–Schwinger principle

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35P15
• Secondary: 58J50

Acknowledgements

The author was very grateful to Pavel Exner and David Krejčiřík for fruitful discussions on the subject.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. We denote by $\#{\sigma }_{\mathrm{d}}(T)$ the number of discrete eigenvalues with multiplicities taken into account for a self-adjoint operator T.

Additional information

Funding

The work was supported by the [grant number 14-06818S] of the Czech Science Foundation (GAČR).