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  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 . 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 . Thierry Cazenave and Ivan Naumk in  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 . The scattering matrices for dissipative quantum system and Neumann maps have been studied by many authors see  . Subsequently, Rainer Hempel, Olaf Post, and Ricardo Weder  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  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:
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
Eichhorn, Proposition 2.1 in  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  of g and by the norm induced by g in the fibers of . Let h be any other Riemannian metric on M. For set
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
2. Theorems and Lemmas
Lemma 1. Let be of moderate decay. Then there exist a constants and such that,
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
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
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
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
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  in M there exist a covering of M by balls and a constant such that
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
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
This representation has been used in  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
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 , i.e., for all and . Let . Then it follows that
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  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
Now, observe that belongs to . By Lemma (3.1) in  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  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 , ,
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
Suppose that then (18) implies that and .
By Lemma (3.1) in  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
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 ,
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  there exists which depends only on such that for all :
Now . Thus . Next observe that
Hence by lemma 17.1.2 in :
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:
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
By principle we have
Using (22) and (23) we can proceed as above and prove that is a trace class operator.
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