The spectral behavior of the difference between the resolvents of two realizations A~1 and A~2
Nearby in the stack
of a second-order strongly elliptic symmetric differential operator
A
, defined by different Robin conditions
νu=b1γ0u
and
νu=b2γ0u
, can in the case where all coefficients are
C∞
be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth
bi
. Using a Krein resolvent formula, we show that if
b1
and
b2
are in
L∞
, the s-numbers
sj
of
(A~1−λ)−1−(A~2−λ)−1
satisfy
sjj3/(n−1)≤C
for all
j
; this improves a recent result for
A=−Δ
by Behrndt et al., that
∑jsjp<∞
for
p>(n−1)/3
. A sharper estimate is obtained when
b1
and
b2
are in
Cε
for some
ε>0
, with jumps at a smooth hypersurface, namely that
sjj3/(n−1)→c
for
j→∞
, with a constant
c
defined from the principal symbol of
A
and
b2−b1
. 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.