Spectral properties of operators of the theory of harmonic potential

J. Ahner ; V. V. Dyakin ; V. Ya. Raevskii ; S. Ritter

Matematičeskie zametki, Tome 59 (1996) no. 1, pp. 3-11

Résumé

We classify the points of the spectrum of the operators $B$ and $B^*$ of the theory of harmonic potential on a smooth closed surface $S\subset\mathbb R^3$. These operators give the direct value on $S$ of the normal derivative of the simple layer potential and the double layer potential. We show that zero can belong to the point spectrum of both operators in $L_2(S)$. We prove that the half-interval $[-2,2)$ is densely filled by spectrum points of the operators for a varying surface; this is a generalization of the classical result of Plemelj. We obtain a series of new spectral properties of the operators $B$ and $B^*$ on ellipsoidal surfaces.