1. Introduction
A great number of researchers referred to the connection between, time-dependent, time-independent, Laplacian, manifold, wave operators, matrices, Riemannian metric, and Schrödinger equation linked to the theory of scattering.
For example, Itoa, K. and Skibsted, E. in [1] included time-dependent scattering theory along with allowed range perturbation and scattering by obstacles. The “independent” and “dependent” scattering by particles has been studied in appropriate single-particle, and examples of independent scattering are described by Michael I. Mishchenko, see [2]. The scattering theory for the Laplacian on symmetric spaces of a non-compact type in the frame work of Agmon-Hörmander has been updated by Koichi Kaizuka in [3]. Thierry Cazenave and Ivan Naumk in [4] modified scattering for the critical nonlinear Schrödinger equation. The exhibited conditions under which the stationary wave operators and the strong wave operators exist and coincide have been discussed by R. Tiedra de Aldecoa [5]. The scattering matrices for dissipative quantum system and Neumann maps have been studied by many authors see [6] [7]. Subsequently, Rainer Hempel, Olaf Post, and Ricardo Weder [8] obtained the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension.
In this paper, we follow the exact reviews and approaches of Werner Muller and Corm Salomonsen in [9] with a slight change. The current study contributes to the expansion of the knowledge in this field by addressing the scattering theory for the Laplacian spectrum (
and
) on the manifold with bounded curvature comparison dynamics.
Definition 1. Let
be a positive, continuous, non-increasing function. Then
is called a function of moderate decay, if it satisfies the following condition:
(i)
;
(ii)
(1)
Further
is called of sub-exponential decay if for any
,
. As
.
Definition 2. Let
be a function of moderate decay. Two metrics
are said to be
-equivalent up to order k if There exist
and
such that for all
we have
holds.
In this case, we write
.
Definition 3. Let
. For
let
be the smallest number such that there exists a sequence
such that
Further, let
put
Definition 4. Let
be a complete. Then
is essentially self-adjoint and function
can be defined by the spectral theorem for unbounded self-adjoint operators by
, where
is the projection spectral measure associated with
. Let
be even and let
. Then
can also be defined by
(2)
Eichhorn, Proposition 2.1 in [10] has shown that M can be endowed with a canonical topology given by a metrizable uniform structure. For a given Riemannian metric
on M, denote by
the Levi-Civita connection 2.5 in [11] of g and by
the norm induced by g in the fibers of
. Let h be any other Riemannian metric on M. For
set
(3)
and
. Recall that two metrics
are said to be quasi-isometric if there exist
such that
, for all
(4)
in the sense of positive definite quadratic forms. We shall write
for quasi-isometric metrics
and
. If g and h are quasi-isometric, then (4) implies that for all
, there exist
such that for every tensor field T on M of bidegree
we have
(5)
2. Theorems and Lemmas
Lemma 1. Let
be of moderate decay. Then there exist a constants
and
such that,
(6)
Lemma 2. Let
be quasi-isometric. For every
, there exists a polynomial
depending on the quasi-isometry constants, with nonnegative coefficients and vanishing constant term, such that
Proof. From (4) follows that
and
. (7)
This is as important as the first two terms in (3) and deals with the question for
. Now we shall proceed by induction. Let
and suppose that the lemma holds for
. For each,
we have
(8)
Let
using (7), (6) and the hypothesis, we can estimate the point wise h norm the second term on the right-hand side of (8) in desired way deal with the first term. We use the formula
Applying the Leibniz rule, we get
for some
and all
. Inserting (8) and iterating these formulas reduces everything to the induction hypothesis.
Lemma 3. Let
be a function of moderate decay. Then for all
, we have
(9)
Moreover, for every
there exists a constant
, depending only on q and
such that
.
Lemma 4. There exists a constant
depend only on K such that
(10)
for all
.
Lemma 5. For
,
We note that the inequality on the right-hand side holds for all
. In particular
as
.
It is also important to know the maximal possible decay of the injectivity radius.
Lemma 6.
finite for all
. Moreover, there exist constants
, which depend only on K, such that for
, we have
.
Lemma 7. Let
be even. Assume that M has bounded curvature of order k. Let
be such that
, there exist constants
and
such that for all
and
one has
for all
.
Lemma 8. Let
be even. Suppose that
has bounded curvature of order 2k Let
be a function of moderate decay. Then there exists a canonical bounded inclusions
and
Proof. By Theorem (2.6) in [9] in M there exist a covering
of M by balls and a constant
such that
(11)
Let
go be such that
on
and
on
for
and
, we define
then
. Let
. Using Lemma 6, it follows that
. Then by Lemma 7, we get
and by the Leibniz rule there is
such that
.
By estimating the supremum-norm of the derivatives of
and using Lemma 7, we get
By induction, this yields
(12)
Let
. By Lemma 7, (11) and (12) we get
By (10) there exists
such that
for all
and
. This implies
. Assume that
is complete. Then
is essentially self-ad joint and function
can be defined by the spectral theorem for unbounded self-ad joint operators by
, where
is
the projection spectral measure associate with
. Let
be even and let
, then
can also be defined by
(13)
This representation has been used in [12] to study the kernel of
we will used (13) to study
as operator in weighted
-spaces. To this end we need to study
as operator in
given
, let
be the constant introduced in Definition (1.3).
Theorem 1. Assume that
has bounded curvature. Let
be a function of moderate decay. Then
extends to a bounded operator in
for all
and there exist
, such that
(14)
Moreover
is strongly continuous in S.
Proof. Let
Choose a sequence
which minimizes.
. For
let
denote the multiplication by the characteristic function of
. Then each
is an orthogonal projection in
and
respectively. Moreover the projections satisfy
for
and
where the series is strongly convergent. Obviously the image of
consists of functions with support in
. Now recall that
has unit propagation speed [13], i.e.,
for all
and
. Let
. Then it follows that
and
Hence
(15)
Now observe that the norm of
as an operation in
is bounded by 1. This implies
To estimate the right-hand side, we write
Since the support of
is contained in
we can use (9) to estimate the right-hand side. This gives
. A similar inequality holds with respect to
putting the estimations together, we get
Now recall that by Lemma 6, we have
. Hence together with (14) and (15) we obtain
Recall that by (1) we have
. Therefore,
, and
is a dense subspace of
. This implies that
extends to a bounded operator in
. Moreover by (7) and Lemma 6, it follows that there exist constants
such that
. Since
this extends to all
such that holds. The strong continuity is a consequence of the local bound of the norm and the strong continuity on the dense subspace
. Using Theorem 1, we can study
as an operator in
given
, let
.
Lemma 9. Let
a function of moderate decay. If
and
satisfy conditions (b) of Corollary 4.3 in [9] then
Proof. First: note that
is dense in
. Indeed
is dense in
and
is dense in
. Let
. Then there exists a sequence
which converges to
in
and
converges to f in
. Let
. Then
.
Thus
and hence
now suppose that
and set
. Then
and we need to show that
. Let
. By definition of
, there exists a sequence
such that
converges to
in
as
. Using this fact, we get
(16)
Now, observe that
belongs to
. By Lemma (3.1) in [9] there exists a sequence
which converges to
in
. Thus
Together with (16) this implies that
.
Lemma 10. Let
be of moderate decay. Assume that
then the Sobolev spaces
and
are equivalent.
Proof. First note that by Lemma 1.7 in [9] the metrics g and h are quasi-isometric. This implies that
and
are equivalent. So the statement of the lemma holds for
. Let
and
by induction we will prove that for
there exists
such that for
,
,
(17)
Let
. Since on functions the connections equal, (17) follows from quasi-isometry of g and
. Next suppose that (17) holds for
. To establish (17) for
, we proceed by induction with respect to a. Let
with
. We may assume that
. Using
and
, it follows that (17) holds for
. Especially, putting
we get
(18)
Suppose that
then (18) implies that
and
.
By Lemma (3.1) in [9]
is dense in
. Therefore this inequality holds for all
. By symmetry, a similar inequality holds with the roles of
and
inter-changed. This concludes the proof.
Next we compare the Sobolev spaces
and
. Let
denote the Laplace operator with respect to the metric g. Recall, that
, and that the formal ad joint
of
is given by
. Where
is the isomorphism induced by the metric and
denotes
contraction. Since contraction commutes with covariant differentiation and
, we get the well-known formula
. This can be iterated. For
define
, and let
denote,
followed by the contraction of the ith and jth component using. That contraction commutes with covariant differentiation and
, we get
(19)
In more traditional notation this mean
. For short notation we will write
.
Lemma 11. Assume that
. Then for each
and
, there exist section
such that
and there exists
such that for
,
,
,
.
Lemma 12. Assume that
is a function of moderate decay and there exist real numbers
such that
(i)
, and
,
(ii)
,
(iii)
.
Let
be the operator of multiplication by
. Then the operator all
is a trace-class operator for
and t in a compact interval, the trace-class norm is bounded.
3. Main Results
The main verification results are the following corollaries and lemma.
Corollary 1. Let
be given. There exists
and
such that for all
,
and
.
for all
.
Proof. Let
and let
. Put
. By Lemma 17.1.2 in [14] there exists
which depends only on
such that for all
:
. (20)
Now
. Thus
. Next observe that
Hence by lemma 17.1.2 in [14]:
(21)
By the Poincare inequality there exists
which is independent of
such that for all
:
. Using this inequality, it's follows from (21) that
. Together with (20) we get
Set
then it follows that for all
and
Corollary 2. Assume
has bounded curvature and let
be functions of moderate decay. Then there exists a constant
such that for all functions
, the operator
extends to abounded operator in
. Moreover, there exists a constant
such that
for all
as above. If
is at most sub-exponentially increasing, then
can be chosen arbitrarily.
Proof. By Theorem 1, there exist constants
, depending on
such that
, for all
. Let
using (15), it follows that
. Since
, it follows from (2) that
extends to a bounded operator in
. The last statement is obvious.
Corollary 3. Let
be a function of moderate decay. Assume that there exist real numbers
such that:
(i)
,
(ii)
,
(iii)
.
Let
the operator of multiplication by
. Then for every
the operator
is Hilbert-Schmidt. For
in a compact interval in
the Hilbert-Schmidt norm is bounded.
Proof. We have
. Note that the operator norm of
is bounded on compact subsets of
. Hence we assume that
. Lemma 11, (i) implies that
. Let
be the kernel
then
. The integral converges since
we get
This proves the corollary.
Lemma 13. Let
be a function of moderate decay, satisfying the conditions of Lemma 11. Let
be two complete metrics on M such that
. Let
and
be the Laplacians of
and
, respectively. Then
and
are trace class operators, and the trace norm is uniformly bounded for
in a compact subset of
.
Proof. We decompose
as
. By Lemma 11, the second factor is a Hilbert-Schmidt operator and it suffices to show that
is Hilbert-Schmidt and that the Hilbert-Schmidt norm is bounded for t in a compact interval, using Lemmas 8, and Lemmas 10, it follows that the Hilbert-Schmidt norm can be estimated by
By Lemma 13, the right-hand side is finite and bounded for t in a compact interval of
prove that
is a trace class operator, it suffices to establish it for its adjoint
with respect to t. By (19) and (18) we have
using (14) and (16), it follows that there exists
such that
and these sections satisfy
(22)
By principle we have
(23)
Using (22) and (23) we can proceed as above and prove that
is a trace class operator.