234 P. Boggiatto - E. Schrohe
Let px be a positive multi-quasi-elliptic polynomial in the variables x ∈ R
n
. Then the Schr¨odinger operator P = −1
x
+ px , a generalization of the harmonic oscillator of quantum mechanics, is multi-quasi-elliptic in the sense of our previous
definition. A slight modification of the above operators P = x
2h
1
+ x
2h
D
2k
+ D
2k
1
yields self-adjoint multi-quasi-elliptic operators of the form:
P = x
2h
1
+ D
k
x
2h
D
k
+ D
2k
1
. In both cases one would like to have spectral asymptotic estimates for these operators,
in particular for the function N λ = P
λ
j
≤λ
1 counting the number of the eigenvalues λ
j
not exceeding λ [3] can be considered a first step in this direction. The idea is to follow the approach taken by Helffer in [14] which is based on a thorough analysis
of Shubin’s classes, and this makes the results of spectral invariance and their conse- quences particularly interesting.
2. Pseudodifferential operators in L
m ρ,P
We now review the main properties of the multi-quasi-elliptic calculus assuming famil- iarity with the standard pseudodifferential calculus, cf. [16].
P
ROPOSITION
1. Let m, m
1
, m
2
∈ R. a 3
m ρ,P
R
d
is a vector space. b If a
1
∈ 3
m
1
ρ
1
,P
R
d
, a
2
∈ 3
m
2
ρ
2
,P
R
d
, then a
1
a
2
∈ 3
m
1
+m
2
min ρ
1
,ρ
2
,P
R
d
. c For every multi-index α ∈ N
d
we have ∂
α ζ
a ∈ 3
m−ρ|α| ρ,P
R
d
. d
T
m∈ R
3
m ρ,P
R
d
= SR
d
, the Schwartz space of rapidly decreasing functions. D
EFINITION
5. Let a
j
∈ 3
m
j
ρ,P
R
d
and m
j
→ −∞ for j → +∞. We write a ∼
P
∞ j =1
a
j
if a ∈ C
∞
R
d
and a − P
r−1 j =1
a
j
∈ 3
˜ m
r
ρ,P
R
d
where ˜ m
r
= max
j ≥r
m
j
. We then have a ∈ 3
m ρ,P
R
d
, m = max
j ≥1
m
j
. P
ROPOSITION
2. Given a
j
∈ 3
m
j
ρ,P
R
d
with m
j
→ −∞ as j → +∞ there exists a ∈ C
∞
R
d
such that a ∼ P
∞ j =1
a
j
. Furthermore, if b is another function such
that b ∼ P
∞ j =1
a
j
, then a − b ∈ SR
d
. P
ROPOSITION
3. Let A
1
= Op a
1
∈ L
m
1
ρ,P
and A
2
= Op a
2
∈ L
m
2
ρ,P
. Then A
1
A
2
∈ L
m
1
+m
2
ρ,P
, and the symbol σ A
1
A
2
of A
1
A
2
has the asymptotic expansion σ A
1
A
2
∼ P
α 1
α
∂
α ξ
a
1
x , ξ D
α x
a
2
x , ξ .
Characterization of multi-quasi-elliptic operators 235
P
ROPOSITION
4. a If a ∈ E3
m ρ,P
then a
−1
∈ E3
−m ρ,P
and a
−1
∂
α
a ∈ E3
−ρ|α| ρ,P
for all α possibly after a modification of aζ on a compact set. b If a
1
∈ E3
m
1
ρ,P
and a
2
∈ E3
m
2
ρ,P
then a
1
a
2
∈ E3
m
1
+m
2
ρ,P
. c If A
1
∈ EL
m
1
ρ,P
and A
2
∈ EL
m
2
ρ,P
then A
1
A
2
∈ EL
m
1
+m
2
ρ,P
. D
EFINITION
6. An operator R ∈ T
m∈ R
L
m ρ,P
is called regularizing or smooth- ing. Regularizing operators define continuous maps R : S
′
R
n
→ SR
n
. They have
integral kernels R = Rx , y ∈ SR
n x
× R
n y
. P
ROPOSITION
5 E
XISTENCE OF THE
P
ARAMETRIX
. Let A ∈ EL
m ρ,P
; then there exists an operator B ∈ EL
−m ρ,P
such that the operators R
1
= A B − I and R
2
= B A − I both are regularizing. B is said to be a parametrix to A. If B
′
is another parametrix to the same operator A, then B − B
′
is a regularizing operator. P
ROPOSITION
6 E
LLIPTIC
R
EGULARITY
. Let A ∈ EL
m ρ,P
. If Au ∈ SR
n
for some u ∈ S
′
R
n
then necessarily u ∈ SR
n
. Definition 4 of the Sobolev spaces H
m P
R
n
can be rephrased. P
ROPOSITION
7. Let P be a fixed complete Newton polyhedron and let A
m
∈ E L
m ρ,P
be a multi-quasi-elliptic operator; then H
m P
R
n
= A
−1 m
L
2
R
n
. Note that H
m P
R
n
depends neither on ρ nor on the particular operator A
m
, but only on m and P.
The main features of these spaces are the following. P
ROPOSITION
8. H
m P
R
n
has a Hilbert space structure given by the inner prod- uct u, v
P
= A
m
u, A
m
v
L
2
+ Ru, Rv
L
2
. Here A
m
is an elliptic operator defining the space H
m P
R
n
according to Proposition 7, and R = I − e A
m
A
m
, with a parametrix e
A
m
of A
m
. We denote by ||u||
m
the norm of an element u in the space H
m P
R
n
. Equiv-
alently, we could define H
1 P
R
n
= { u ∈ S
′
R
n
: x
α
D
β
u ∈ L
2
R
n
for α, β ∈ P}. with the inner product u, v
3
P
= P
α,β∈P
x
α
D
β
u, x
α
D
β
v
L
2
. P
ROPOSITION
9. a The topological dual H
m
′
P
R
n
of H
m P
R
n
is H
−m P
R
n
. b We have continuous imbeddings SR
n
֒→ H
m P
R
n
֒→ S
′
R
n
. c We have compact imbeddings H
t P
R
n
֒→ H
s P
R
n
if t s. d proj − lim
m∈ R
H
m P
R
n
= SR
n
, ind − lim
m∈ R
H
m P
R
n
= S
′
R
n
.
236 P. Boggiatto - E. Schrohe
3. Abstract characterization, order reduction, spectral invariance and the Fred- hold property
