Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schrödinger Operators

The aim of this paper is to study relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on

L

p

(ℝ

n

) perturbed by singular potentials.

Our approach to this problem is via the limit operators method. This method associates with each band-dominated operator

A

a family

op

(

A

) of so-called limit operators which reflect the properties of

A

at infinity. Consider the compactification of ℝ

n

by the “infinitely distant” sphere

S

n

−1

. Then the set

op

(

A

) can be written as the union of its components

op

ηω

(

A

) where

ω

runs through the points of

S

n

−1

and where

op

ηω

(

A

) collects all limit operators of

A

which reflect the properties of

A

if one tends to infinity “in the direction of

ω

”. Set

$$ sp_{n_\omega } A: = \cup _{A_h \in op_{\eta \omega } (A)} spA_h $$

.

We show that the distance of an eigenvalue

λ

∉

sp

ess

A

to

sp

ηω

A

determines the exponential decay of the

λ

-eigenfunctions of

A

in the direction of

ω

. We apply these results to estimate the exponential decay of eigenfunctions of electro-magnetic Schrödinger operators for a large class of electric potentials, in particular, for multiparticle Schrödinger operators and periodic Schrödinger operators perturbed by slowly oscillating at infinity potentials.