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 $\nu u=b_1{\gamma }_{0}u$ and $\nu u=b_2{\gamma }_{0}u$, 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 $b_i$. 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 ${\sum }_j{s}_j^p<\infty$ 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.

20 pages, notation simplified. To appear in J. Spectral Theory

Cited by in corpus (7)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Spectral results for mixed problems and fractional elliptic operators
• Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary
• Spectral estimates for resolvent differences of self-adjoint elliptic operators
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• Functional model for boundary-value problems
• Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains