Spectral asymptotics of periodic elliptic operators
Abstract
We demonstrate that the structure of complex second-order strongly elliptic operators
H
on
R
d
with coefficients invariant under translation by
Z
d
can be analyzed through decomposition in terms of versions
H
z
,
z∈
T
d
, of
H
with
z
-periodic boundary conditions acting on
L
2
(
I
d
)
where
I=[0,1>
. If the semigroup
S
generated by
H
has a Hölder continuous integral kernel satisfying Gaussian bounds then the semigroups
S
z
generated by the
H
z
have kernels with similar properties and
z↦
S
z
extends to a function on
C
d
∖{0}
which is analytic with respect to the trace norm. The sequence of semigroups
S
(m),z
obtained by rescaling the coefficients of
H
z
by
c(x)→c(mx)
converges in trace norm to the semigroup
S
^
z
generated by the homogenization
H
^
z
of
H
z
. These convergence properties allow asymptotic analysis of the spectrum of
H
.