Spectral asymptotics for Robin problems with a discontinuous coefficient

Abstract

The spectral behavior of the difference between the resolvents of two realizations $\tilde A_1$ and $\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $\nu u=b_1\gamma_0u$ and $\nu u=b_2\gamma_0u$, 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 $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ of $(\tilde A_1 -\lambda)^{-1}-(\tilde A_2 -\lambda)^{-1}$ 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 $\sum_js_j ^p<\infty$ for $p>(n-1)/3$. A sharper estimate is obtained when $b_1$ and $b_2$ are in $C^\epsilon$ for some $\epsilon >0$, with jumps at a smooth hypersurface, namely that $s_j j^{3/(n-1)}\to c$ for $j\to \infty$, with a constant $c$ defined from the principal symbol of $A$ and $b_2-b_1$. As an auxiliary result we show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators interspersed with piecewise continuous functions.