An Isoperimetric-type inequality for electrostatic shell interactions for Dirac operators
In this talk I will present a recent work with N. Arrizabalaga and A. Mas about
spectral properties of the coupling $H +V$, where
$H$ is the free Dirac operator in 3D, $m > 0$ and $V$ is an electrostatic shell
potential which depends on a coupling constant and it is located on the boundary of a smooth
domain. Our main result is an isoperimetric-type inequality for the admissible range
of the values of the coupling constant for which the coupling $H + V$ generates pure point spectrum in $(-m, m)$.
That the ball is the unique optimizer of this inequality is also shown. In the proof
we make use of the Birman-Schwinger principle adapted to our setting in order
to obtain some crucial monotonicity property.