Spectral asymptotics for Robin problems with a discontinuous coefficient

Abstract

The spectral behavior of the difference between the resolvents of two realizations of a second-order strongly elliptic symmetric differential operator $A$ defined by different Robin conditions $\chi u=b_1{\gamma }_{0}u$ and $\chi u=b_2{\gamma }_{0}u$, can in the case where all coefficients are $C^{\infty }$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$, showing that if $b_1$ and $b_2$ are in $L_{\infty }$, the s-numbers $s_j$ satisfy $s_j{j}^{3/(n-1)}\le C$ for all $j$. This improves a recent result for $A=-\Delta$ by Behrndt et al., that ${\sum }_j{s}_j^p<\infty$ for $p>(n-1)/3$, under a hypothesis of boundedness of $b_i^{-1}$. Moreover, we show that if $b_1$ and $b_2$ are in $C^{\epsilon }$ for some $\epsilon >0$, with jumps at a smooth hypersurface, then $s_j{j}^{3/(n-1)}\to c$ for $j\to \infty$, with a constant defined from the principal symbol of $A$ and $b_2-b_1$. We also show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators of negative order interspersed with piecewise continuous functions.