An Isoperimetric-type inequality for electrostatic shell interactions for Dirac operators

Luis Vega
Date et heure
Workshop - Champs magnétiques et analyse semi-classique

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.

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