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Scattering theory plays the main role in the study of manifolds and the Laplacian spectrum. In this article, we process justifying the continuous Laplacian spectrum
*M*,
*g _{i}*) is categorized by the use of bounded curvature of the metric. In particular, the covariant derivative is limitedly considered as an application in the geodesic distance from a fixed point.

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 [

In this paper, we follow the exact reviews and approaches of Werner Muller and Corm Salomonsen in [

Definition 1. Let β : [ 0 , ∞ ) → ℝ be a positive, continuous, non-increasing function. Then β is called a function of moderate decay, if it satisfies the following condition:

(i) sup x ∈ [ 1 , ∞ ) x β ( x ) < ∞ ;

(ii) ∃ C β > 0 : β ( x + y ) ≥ C β β ( x ) β ( y ) , x , y ≥ 1 (1)

Further β is called of sub-exponential decay if for any c > 0 , e c x β ( x ) → ∞ . As x → ∞ .

Definition 2. Let β be a function of moderate decay. Two metrics g , h ∈ M are said to be β -equivalent up to order k if There exist q ∈ M and C > 0 such that for all x ∈ M we have | g − h | k g ( x ) ≤ C β ( 1 + d g ( x , q ) ) holds.

In this case, we write g ~ β k h .

Definition 3. Let s > 0 . For s > ε ≥ 0 let K ε ( M , g ; s ) ∈ ℕ ∪ { ∞ } be the smallest number such that there exists a sequence { x i } i = 1 ∞ such that sup x ∈ M # { i ∈ ℕ | x ∈ B 3 S + ε ( x i ) } ≤ K ε ( M , g ; s ) Further, let K ( M , g ; s ) = K 0 ( M , g ; s ) put

k ( M , g ; s ) = 1.

Definition 4. Let ( M , g ) be a complete. Then Δ : C c ∞ ( M ) → L 2 ( M ) is essentially self-adjoint and function f ( Δ ) can be defined by the spectral theorem for unbounded self-adjoint operators by f ( Δ ) = ∫ 0 ∞ f ( λ ) d E ( λ ) , where

d E ( λ ) is the projection spectral measure associated with Δ . Let f ∈ L 1 ( ℝ ) be even and let f ^ ( λ ) = ∫ − ∞ ∞ f ( x ) cos ( λ x ) d x . Then f ( Δ ) can also be defined by

f ( Δ ) = 1 2 π ∫ − ∞ ∞ f ˜ ( λ ) cos ( λ Δ ) d λ . (2)

Eichhorn, Proposition 2.1 in [

∑ i = 1 ∞ | g i − h i | k g i ( x ) = ∑ i = 1 ∞ ( | g i − h i | g ( x ) ) + ∑ j = 0 k − 1 ( | ( ∇ g ) j ( ∇ g − ∇ h ) | ( x ) ) , x ∈ M , (3)

and ∑ i = 1 ∞ ( ‖ g i − h i ‖ k g i ) = sup k x ∈ M ∑ i = 1 ∞ ( | g i − h i | g i ( x ) ) . Recall that two metrics g i , h i are said to be quasi-isometric if there exist C 1 , C 2 > 0 such that

C 1 g ( x ) ≤ ∑ i = 1 ∞ h i ( x ) ≤ C 2 g ( x ) , for all x ∈ M (4)

in the sense of positive definite quadratic forms. We shall write g i ~ h i for quasi-isometric metrics g i and h i . If g and h are quasi-isometric, then (4) implies that for all p , q ≥ 0 , there exist A P , q B p , q > 0 such that for every tensor field T on M of bidegree ( p , q ) we have

A p , q | T | ( x ) ≤ ∑ i = 1 ∞ | T | h i ( x ) ≤ B p , q | T | g ( x ) , x ∈ M (5)

Lemma 1. Let β be of moderate decay. Then there exist a constants C > 0 and c > 0 such that,

β ( x ) ≥ C e − c x , x ∈ [ 1 , ∞ ) (6)

Lemma 2. Let g , h ∈ C M be quasi-isometric. For every k ≥ 0 , there exists a polynomial P k ( X 1 , ⋯ , X k ) depending on the quasi-isometry constants, with nonnegative coefficients and vanishing constant term, such that

| g − h | k h ( x ) ≤ P k ( | g − h | g ( x ) , | ∇ g − ∇ h | g ( x ) , ⋯ , | ( ∇ g ) k − 1 ( ∇ g − ∇ h ) | g ( x ) ) , x ∈ M

Proof. From (4) follows that | g − h | h ( x ) ≤ C 3 | g − h | h ( x ) and

| ∇ g − ∇ h | h ( x ) ≤ C 4 | ∇ g − ∇ h | g ( x ) , x ∈ M . (7)

This is as important as the first two terms in (3) and deals with the question for k = 0 , 1 . Now we shall proceed by induction. Let k ≥ 2 and suppose that the lemma holds for l ≤ k − 1 . For each, p ≤ 0 we have

( ∇ h ) p ( ∇ h − ∇ g ) = ∇ g ( ∇ h ) p − 1 ( ∇ h − ∇ g ) + ( ∇ h − ∇ g ) ( ∇ h ) p − 1 ( ∇ h − ∇ g ) (8)

Let p ≤ k 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

( ∇ g ) p ( ∇ h ) l ( ∇ h − ∇ g ) = ( ∇ g ) ( p + 1 ) ( ∇ h ) ( l − 1 ) ( ∇ h − ∇ g ) + ( ∇ g ) p ( ∇ h − ∇ g ) ( ∇ h ) ( l − 1 ) ( ∇ h − ∇ g ) .

Applying the Leibniz rule, we get

| ( ∇ g ) p ( ∇ h − ∇ g ) ( ∇ h ) ( l − 1 ) ( ∇ h − ∇ g ) | g ( x ) ≤ C ∑ i = 0 p | ( ( ∇ g ) i ( ∇ h − ∇ g ) ) | g ( x ) ⋅ | ( ( ∇ g ) ( p − i ) ( ∇ h ) ( l − 1 ) ( ∇ h − ∇ g ) ) | g ( x )

for some C > 0 and all x ∈ M . Inserting (8) and iterating these formulas reduces everything to the induction hypothesis.

Lemma 3. Let β be a function of moderate decay. Then for all x , y , q ∈ M , we have

C β β ( 1 + d ( x , y ) ) ≤ β ( 1 + d ( x , q ) ) β ( 1 + d ( y , q ) ) ≤ 1 C β β ( 1 + d ( x , y ) ) (9)

Moreover, for every q ′ ∈ M there exists a constant C > 0 , depending only on q and q ′ such that

C − 1 β ( 1 + d ( x , q ′ ) ) ≤ β ( 1 + d ( x , q ) ) ≤ β ( 1 + d ( x , q ′ ) ) .

Lemma 4. There exists a constant C > 0 depend only on K such that

i ˜ ( x ) ≥ C i ˜ ( p ) n e − ( n − 1 ) K d ( x , p ) (10)

for all x , p ∈ M .

Lemma 5. For r ≤ i ˜ ( x 0 ) , 2 π n 2 Γ ( n 2 ) ∫ 0 r ( sin t K K ) ( n − 1 ) d t ≤ V O L ( B r ( x 0 ) ) ≤ 2 π n 2 Γ ( n 2 ) ∫ 0 r ( sin h K K ) ( n − 1 ) d t

We note that the inequality on the right-hand side holds for all r ∈ R . In particular V o l ( B r ( x 0 ) ) = 0 ( e ( n − 1 ) K r ) as r → ∞ .

It is also important to know the maximal possible decay of the injectivity radius.

Lemma 6. k ε ( M , g ; s ) finite for all s > ε . Moreover, there exist constants C , c > 0 , which depend only on K, such that for s > 2 π K + ε , we have k ε ( M , g ; s ) ≤ C e c s .

Lemma 7. Let k ≥ 1 be even. Assume that M has bounded curvature of order k. Let k > 0 be such that sup x ∈ M ∑ l = 0 2 k | ∇ l R ( x ) | ≤ k , there exist constants r 0 = r 0 ( k ) > 0 and C = C ( k ) > 0 such that for all x 0 ∈ M and r i ≤ min { r 0 , r ˜ ( x 0 ) } one has ∑ i = 1 ∞ ‖ u i ‖ W 2 K ( B r i ( x 0 ) ) ≤ C ∑ i = 1 ∞ ‖ u i ‖ H 2 k ( B r i ( x 0 ) ) for all u i ∈ C 0 ∞ ( B r i ( x 0 ) ) .

Lemma 8. Let k ∈ ℕ be even. Suppose that ( M , g i ) has bounded curvature of order 2k Let β : M → ℝ + be a function of moderate decay. Then there exists a canonical bounded inclusions H β τ ~ − 2 k n k ( M ) → W β k ( M ) and H β k ( M ) → W β τ ~ 2 k n k ( M )

Proof. By Theorem (2.6) in [

∀ x ∈ M : { x i | x ∈ ∑ i = 1 ∞ B τ ˜ ( x i ) ( x i ) } ≤ C (11)

Let φ ∈ C ∞ ( ℝ ) go be such that φ = 1 on [ 0 , 1 ] and φ = 0 on [ 2 , ∞ ) for x ∈ M and 1 ≤ j ≤ k , we define

∑ j = 1 k φ j , x ( y ) = { ∑ j = 1 k φ ( 2 j d ( x , y ) τ ˜ ( x ) ) , y ∈ B τ ˜ ( x ) ( x ) ; 0 , otherwise .

then ∑ j = 1 k φ j , x ∈ C 0 ∞ ( M ) . Let f ∈ H k ( M ) . Using Lemma 6, it follows that φ j , x f ∈ H k ( B τ ˜ x ( x ) ) . Then by Lemma 7, we get φ j , x f ∈ W k ( B τ ˜ x ( x ) ) and by the Leibniz rule there is C > 0 such that

| ∑ j = 1 k ∇ j ( φ k , x f ) | g ( y ) ≤ C ∑ p = 0 j | ∇ p φ k , x | g ( y ) ⋅ | ∇ j − p f | g ( y ) , y ∈ M .

By estimating the supremum-norm of the derivatives of φ k , x and using Lemma 7, we get

‖ φ k , x f ‖ W k ≤ C ‖ f ‖ W k ( B τ 2 k − 1 ( x ) ( x ) ) + C ′ ∑ p = 1 k ( k p ) ( x ) ‖ φ k − 1 , x f ‖ W k − p ≤ C ‖ f ‖ H k ( B τ ˜ 2 k − 1 ( x ) ( x ) ) + C ″ ∑ p = 1 k ( k p ) τ ˜ − p ( x ) ‖ φ k − 1 , x f ‖ H k − p

By induction, this yields

‖ φ k , x i f ‖ W k ≤ C τ ˜ − K ( x i ) ‖ f ‖ H k ( B τ ˜ 2 k − 1 ( x ) ( x ) ) + C ′ ∑ p = 1 k ( k p ) τ ˜ − P ( x ) ‖ φ k − 1 , x f ‖ W k − p (12)

Let f ∈ H β k . By Lemma 7, (11) and (12) we get

‖ f ‖ W β k ≤ C ∑ i = 1 ∞ β 1 2 ( x i ) ‖ φ k , x i f ‖ W k τ ˜ k ( x i ) ≤ C ∑ i = 1 ∞ β 1 2 ( x i ) ‖ φ k , x i f ‖ H k ≤ C ∑ i = 1 ∞ β 1 2 ( x i ) τ ˜ − k ( x i ) ‖ f ‖ H k ( β τ ( x i ) ( x ) )

By (10) there exists C 1 > 0 such that τ ˜ ( x i ) − k τ ˜ ( x ) k n ≤ C 1 for all i ∈ ℕ and x ∈ B τ ( x i ) ( x i ) . This implies ∑ i = 1 ∞ β 1 2 ( x i ) τ ˜ − k ( x i ) ‖ f ‖ H k ( B τ ( x i ) ( x i ) ) ≤ C 2 ‖ f ‖ H τ ˜ − 2 k n β k . Assume that ( M , g ) is complete. Then Δ : C 0 ∞ ( M ) → L 2 ( M ) is essentially self-ad joint and function f ( Δ ) can be defined by the spectral theorem for unbounded self-ad joint operators by f ( Δ ) = ∫ 0 ∞ f ( λ ) d E λ , where d E λ is

the projection spectral measure associate with Δ . Let f ∈ L 1 ( ℝ ) be even and let f ˜ ( λ ) = ∫ − ∞ ∞ f ( x ) cos ( λ x ) d x , then f ( Δ ) can also be defined by

f ( Δ ) = 1 2 π ∫ − ∞ ∞ f ˜ ( λ ) cos ( λ Δ ) d λ (13)

This representation has been used in [

Theorem 1. Assume that ( M , g ) has bounded curvature. Let β be a function of moderate decay. Then cos ( λ Δ ) extends to a bounded operator in L β 2 ( M ) for all s ∈ ℝ and there exist C , c > 0 , such that

‖ cos ( s Δ ) ‖ L β 2 , L β 2 ≤ C e c | s | , s ∈ ℝ . (14)

Moreover cos ( s Δ ) : L β 2 ( M ) → L β 2 ( M ) is strongly continuous in S.

Proof. Let s > 0 Choose a sequence { X k } k = 1 ∞ ⊂ M which minimizes. κ ( M , g i ; s ) . For k ∈ ℕ let P k denote the multiplication by the characteristic function of B s ( x k ) \ ∪ i = 0 k − 1 B s ( x i ) . Then each P k is an orthogonal projection in L 2 ( M ) and L β 2 ( M ) respectively. Moreover the projections satisfy P k P k ′ = 0 for k ≠ k ′ and ∑ k = 1 ∞ P k = 1 where the series is strongly convergent. Obviously the image of P k consists of functions with support in B s ( x k ) . Now recall that cos ( τ Δ ) has unit propagation speed [

sup p cos ( s Δ ) P k f ⊂ B 2 s ( x k ) and sup p cos ( s Δ ) ( ( 1 − χ B 3 s ( x k ) ) f ) ⊂ M − B 2 s ( x k ) Hence

‖ cos ( s Δ ) f ‖ β 2 = ∑ k = 1 ∞ 〈 cos ( s Δ ) P k f , cos ( s Δ ) f 〉 β = ∑ k = 1 ∞ 〈 cos ( s Δ ) P k f , cos ( s Δ ) 〉 (15)

Now observe that the norm of ( s Δ ) as an operation in L 2 ( M ) is bounded by 1. This implies

| 〈 cos ( s Δ ) P k f , cos ( s Δ ) ( χ B 3 s ( x k ) f ) 〉 | ≤ sup y ∈ B 3 s ( x k ) β ( y ) ‖ P k f ‖ L 2 ⋅ ‖ χ B 3 s ( x k ) f ‖ L 2

To estimate the right-hand side, we write sup y ∈ B 3 s ( x k ) β ( y ) ‖ P k f ‖ L 2 2 ≤ C β − 1 1 β ( 1 + 4 s ) ‖ P k f ‖ L β 2 2

Since the support of P k f is contained in B s ( x k ) we can use (9) to estimate the right-hand side. This gives sup y ∈ B 3 s ( x k ) β ( y ) ‖ P k f ‖ L 2 2 ≤ C β − 1 1 β ( 1 + 4 s ) ‖ P k f ‖ L β 2 2 . A similar inequality holds with respect to ‖ χ B 3 s ( x k ) f ‖ L 2 putting the estimations together, we get

| 〈 cos ( s Δ ) P k f , cos ( s Δ ) ( χ B 3 s ( x k ) f ) 〉 | ≤ C β − 1 1 β ( 1 + 6 S ) ‖ P k f ‖ L β 2 ‖ χ B 3 s ( x k ) f ‖ L β 2

Now recall that by Lemma 6, we have κ ( M , g ; s ) < ∞ . Hence together with (14) and (15) we obtain

‖ cos ( s Δ ) f ‖ L β 2 2 ≤ C β − 1 1 β ( 1 + 6 s ) ‖ f ‖ β 2 ∑ k = 1 ∞ ‖ χ B 3 s ( x k ) f ‖ L β 2 ≤ C β − 1 1 β ( 1 + 6 s ) κ ( M , g , s ) 1 2 ‖ f ‖ L β 2 2

Recall that by (1) we have β ( x ) ≤ C ( 1 + d ( x , p ) ) − 1 , x ∈ M . Therefore, L 2 ( M ) ⊂ L β 2 ( M ) , and L 2 ( M ) is a dense subspace of L β 2 ( M ) . This implies that cos ( s Δ ) extends to a bounded operator in L β 2 ( M ) . Moreover by (7) and Lemma 6, it follows that there exist constants C , c > 0 such that

‖ cos ( s Δ ) ‖ L β 2 , L β 2 2 ≤ C e c s , s ∈ [ 0 , ∞ ) . Since cos ( − s Δ ) = cos ( s Δ ) this extends to all s ∈ ℝ such that holds. The strong continuity is a consequence of the local bound of the norm and the strong continuity on the dense subspace cos ( − s Δ ) L 2 ( M ) ⊆ L β 2 ( M ) . Using Theorem 1, we can study f ( Δ ) as an operator in L β 2 ( M ) given c ≥ 0 , let F ′ ( c ) = { f ∈ L 1 ( ℝ ) : ∫ − ∞ ∞ | f ˜ ( λ ) | e c | λ | d λ < ∞ } .

Lemma 9. Let β a function of moderate decay. If λ and λ ¯ satisfy conditions (b) of Corollary 4.3 in [

H β 2 ( M ) = ( Δ − λ ) − 1 ( L β 2 ( M ) ) .

Proof. First: note that C 0 ∞ ( M ) is dense in L β 2 ( M ) . Indeed C 0 ∞ ( M ) is dense in L 2 ( M ) and L 2 ( M ) is dense in L β 2 ( M ) . Let f = ∑ i = 1 ∞ ( Δ − λ ) − 1 g i , g i ∈ L β 2 ( M ) . Then there exists a sequence { φ i } i ∈ ℕ ⊂ C 0 ∞ ( M ) which converges to ∑ i = 1 ∞ g i in L β 2 ( M ) and ( Δ − λ ) − 1 φ i converges to f in L 2 ( M ) . Let φ ∈ L 0 ∞ ( M ) . Then

〈 f , Δ φ 〉 = lim i → ∞ 〈 ( Δ − λ ) − 1 , Δ φ 〉 = lim i → ∞ 〈 φ i + λ ( Δ − λ ) − 1 φ i , φ 〉 = 〈 g + λ f , φ 〉 .

Thus Δ f = ∑ i = 1 ∞ ( g i + λ f ) ∈ L β 2 ( M ) and hence f ∈ L β 2 ( M ) now suppose that f ∈ L β 2 ( M ) and set g = ( Δ − λ ) f . Then g ∈ L β 2 ( M ) and we need to show that f = ∑ i = 1 ∞ ( Δ − λ ) − 1 g i . Let φ ∈ C 0 ∞ ( M ) . By definition of ∑ i = 1 ∞ ( Δ − λ ) − 1 g i , there exists a sequence { g } i ∈ ℕ ⊂ L 2 ( M ) such that ( Δ − λ ) − 1 g i converges to ( Δ − λ ) − 1 g in L β 2 ( M ) as i → ∞ . Using this fact, we get

〈 ∑ i = 1 ∞ ( Δ − λ ) − 1 g i , ∑ i = 1 ∞ φ i 〉 = 〈 g i , ( Δ − λ ¯ ) − 1 ∑ i = 1 ∞ φ i 〉 = 〈 ( Δ − λ ) f , ( Δ − λ ¯ ) − 1 φ 〉 . (16)

Now, observe that ( Δ − λ ) − 1 ∑ i = 1 ∞ φ i belongs to H 2 ( M ) . By Lemma (3.1) in [

〈 ( Δ − λ ) f , ( Δ − λ ¯ ) − 1 φ 〉 = lim i → ∞ 〈 ( Δ − λ ) f , ∑ i = 1 ∞ φ i 〉 = 〈 f , ( Δ − λ ¯ ) ∑ i = 1 ∞ φ i 〉 = 〈 f , φ 〉 .

Together with (16) this implies that ( Δ − λ ) − 1 g i .

Lemma 10. Let β be of moderate decay. Assume that g i ~ β k h i then the Sobolev spaces W ξ k ( M ; g i ) and W ξ k ( M ; h i ) are equivalent.

Proof. First note that by Lemma 1.7 in [

| ( ∇ g i ) a ( ∇ h i ) b f | h i ( x ) ≤ C l ∑ i = 0 ( a + b ) | ( ∇ g ) i f | ( x ) ( x ) , x ∈ M . (17)

Let l = 1 . Since on functions the connections equal, (17) follows from quasi-isometry of g and h i . Next suppose that (17) holds for 1 ≤ l < k . To establish (17) for l + 1 , we proceed by induction with respect to a. Let a , b ∈ ℕ 0 with a + b = l + 1 . We may assume that a < l + 1 . Using

∑ i = 1 ∞ ( ∇ g i ) a ( ∇ h i ) b f = ∑ i = 1 ∞ ( ∇ g i ) a ( ∇ h i − ∇ g i ) ( ∇ h i ) ( b − 1 ) f + ( ∇ g i ) ( a + 1 ) ( ∇ h i ) ( b − 1 ) f ,

and g ~ β k h , it follows that (17) holds for l + 1 . Especially, putting a = 0 we get

∑ i = 1 ∞ | ( ∇ h i ) l f | h i ( x ) ≤ C l ∑ i = 0 l | ( ∇ g ) i f | g i ( x ) , x ∈ M , l ≤ k . (18)

Suppose that f ∈ C ∞ ( M ) ∩ W ξ k ( M ; g i ) then (18) implies that f ∈ C ∞ ( M ) ∩ W ξ k ( M ; h i ) and ‖ f ‖ W ξ k ( M ; h i ) ≤ C ‖ f ‖ W ξ k ( M ; g i ) .

By Lemma (3.1) in [

Next we compare the Sobolev spaces H ξ 2 k ( M ; g i ) and H ξ 2 k ( M ; h i ) . Let Δ g i denote the Laplace operator with respect to the metric g. Recall, that ∑ i = 1 ∞ Δ g i = ∑ i = 1 ∞ ( ∇ g i ) * ∇ g i , and that the formal ad joint ( ∇ g i ) * of ( ∇ g i ) is given by ( ∇ g ) * = − T r ( g − 1 ∇ g i ) . Where ∑ i = 1 ∞ Δ g i = ∑ i = 1 ∞ ( ∇ g i ) * ∇ g i is the isomorphism induced by the metric and T r : T * M ⊗ T M → ℝ denotes ∑ i = 1 ∞ Δ g i = ∑ i = 1 ∞ ( ∇ g i ) * ∇ g i contraction. Since contraction commutes with covariant differentiation and ∇ g i g i − 1 = 0 , we get the well-known formula Δ = − T r ( g − 1 ∇ 2 ) . This can be iterated. For ω 1 ⊗ ⋯ ⊗ ω k ∈ ( T * M ) ⊗ k define g j − 1 ( ω 1 ⊗ ⋯ ⊗ ω k ) : ω 1 ⊗ ⋯ ⊗ ω j − 1 ⊗ g − 1 ( ω j ) ⊗ ω j + 1 ⊗ ⋯ ⊗ ω k , and let T r i . j ( g j − 1 ) denote, ( g j − 1 ) followed by the contraction of the ith and jth component using. That contraction commutes with covariant differentiation and ∇ g i g i − 1 = 0 , we get

Δ g k = ( − 1 ) k T r 1 , 2 ( g 2 − 1 ) ∘ ⋯ ∘ T r 2 k − 1 , 2 k ( g 2 k − 1 ) ( ∇ g ) 2 k . (19)

In more traditional notation this mean Δ g i k f = ( − 1 ) k ∑ i 1 , ⋯ , i k f ; i 1 i 1 i 2 i 2 ⋯ i k i k . For short notation we will write T r ( ( g − 1 ) ⊗ k ) : = T r 1 , 2 ( g 2 − 1 ) ∘ ⋯ ∘ T r 2 k − 1 , 2 k ( g 2 k − 1 ) .

Lemma 11. Assume that g i ~ β 2 k β . Then for each l , 0 ≤ l ≤ 2 k and j , 0 ≤ j ≤ 2 l , there exist section ξ j l g , ξ j l h ∈ C ∞ ( H o m ( T * M ) ⊗ j , ℝ ) such that ∑ l = 0 2 k ( Δ g l − Δ h l ) = ∑ j = 0 2 l ξ j l g ∘ ( ∇ g ) j = ∑ j = 0 2 l ξ j l h ∘ ( ∇ h ) j and there exists C < 0 such that for 0 ≤ p ≤ l , ∑ j = 0 2 l ( | ( ∇ g ) p ξ j l g | ( x ) ) ≤ C β ( x ) , ∑ j = 0 2 l ( | ( ∇ h ) p ξ j l h | h ( x ) ) ≤ C β ( x ) , x ∈ M .

Lemma 12. Assume that β is a function of moderate decay and there exist real numbers a , b such that

(i) b ≥ 1 , and a + b = 2 ,

(ii) β b 3 ∈ L 1 ( M ) ,

(iii) β a 3 τ ˜ − n ( n + 2 ) ∈ L ∞ ( M ) .

Let M β be the operator of multiplication by β . Then the operator all M τ − 2 n M β Δ p e − t Δ is a trace-class operator for β ∈ ℕ and t in a compact interval, the trace-class norm is bounded.

The main verification results are the following corollaries and lemma.

Corollary 1. Let K , λ > 0 be given. There exists r 0 = r 0 ( K , λ ) > 0 and C = C ( λ ) > 0 such that for all r i ≤ r 0 , p ∈ ε l l m ( r i , K , λ ) and x 0 ∈ B r i . ∑ i = 1 n ‖ u i ‖ W m ( B r i ) ≤ C ∑ i = 1 n ( ‖ P u i ‖ L 2 ( B r i ) + ‖ u i ‖ L 2 ( B r i ) ) for all ∑ i = 1 n u i ∈ C 0 ∞ ( ∑ i = 1 n B r i ) .

Proof. Let 1 ≥ r i > 0 and let P ∈ ε l l m ( r i , K , λ ) . Put P 0 = ∑ | α | = m a α ( 0 ) D α . By Lemma 17.1.2 in [

∑ i = 1 n ‖ u i ‖ W m ( B r i ) ≤ C ∑ i = 1 n ( ‖ P 0 u i ‖ L 2 ( B r i ) + ‖ u i ‖ L 2 ( B r i ) ) . (20)

Now ∑ i = 1 n p u i = ∑ i = 1 n P 0 u i + ∑ i = 1 n ( P − P 0 ) u i . Thus ∑ i = 1 n ‖ u i ‖ W m ( B r i ) ≤ C ∑ i = 1 n ( ‖ P u i ‖ L 2 ( B r i ) + ‖ ( P − P 0 ) u i ‖ L 2 ( B r i ) + ‖ u i ‖ L 2 ( B r i ) ) . Next observe that

∑ i = 1 n ( P − P 0 ) u i = ∑ i = 1 n ∑ | α | = m ( a α ( x ) − a α ( 0 ) ) D α u i + ∑ i = 1 n ∑ | α | < m a α ( x ) D α u i

Hence by lemma 17.1.2 in [

∑ i = 1 n ‖ ( P − P 0 ) u i ‖ L 2 ( B r i ) ≤ ∑ i = 1 n r i ∑ | α | = m ‖ a α ‖ C 1 ( B r i ) ‖ u i ‖ W m ( B r i ) + ∑ i = 1 n ∑ | α | < m ‖ a α ‖ C O ( B r i ) ‖ u i ‖ W m − 1 ( B r i ) ≤ K ∑ i = 1 n ( r i ‖ u i ‖ W m ( B r i ) + ‖ u i ‖ W m − 1 ( B r i ) ) (21)

By the Poincare inequality there exists C 2 > 0 which is independent of ∑ i = 1 n r i ≤ 1 such that for all ∑ i = 1 n u i ∈ C 0 ∞ ∑ i = 1 n ( B r i ) : ∑ i = 1 n ‖ u i ‖ W m − 1 ( B r i ) ≤ ∑ i = 1 n r i C 2 ‖ u i ‖ W m ( B r i ) . Using this inequality, it's follows from (21) that ∑ i = 1 n ‖ ( P − P 0 ) u i ‖ L 2 ( B r i ) ≤ ∑ i = 1 n r i C ( K ) ‖ u i ‖ W m ( B r i ) . Together with (20) we get

∑ i = 1 n ( 1 − r i C C ( K ) ) ‖ u i ‖ W m ( B r i ) ≤ C ∑ i = 1 n ( ‖ P u i ‖ L 2 ( B r i ) + ‖ u i ‖ L 2 ( B r i ) )

Set r 0 = min { 1 , 1 2 C C ( K ) } then it follows that for all ∑ i = 1 n r i ≤ r 0 and

∑ i = 1 n u i ∈ C 0 ∞ ∑ i = 1 n ( B r i ) : ∑ i = 1 n ‖ u i ‖ W m ( B r i ) ≤ 2 C ∑ i = 1 n ( ‖ P u i ‖ L 2 ( B r i ) + ‖ u i ‖ L 2 ( B r i ) ) .

Corollary 2. Assume ( M , g i ) has bounded curvature and let β be functions of moderate decay. Then there exists a constant C = C ( M , g i , β ) such that for all functions f i ∈ F ′ ( c ) , the operator f i ( Δ ) extends to abounded operator in L β 2 ( M ) . Moreover, there exists a constant C 1 = C 1 ( M , g i , β ) > 0

such that ∑ i = 1 n ‖ f i ( Δ ) ‖ L β 2 , L β 2 ≤ C 1 ∑ i = 1 n ‖ f ^ ‖ L e c | . | 1 for all f i as above. If κ ( M , g i ; s ) is at most sub-exponentially increasing, then c ( M , g i ; β ) > 0 can be chosen arbitrarily.

Proof. By Theorem 1, there exist constants C , c > 0 , depending on ( M , g i , β ) such that ‖ cos ( Δ ) ‖ L β 2 , L β 2 ≤ C e c | . | , for all s ∈ ℝ . Let φ ∈ L 2 ( M ) using (15), it follows that ∑ j = 1 n ‖ f j ( Δ ) φ ‖ L β 2 ≤ C 2 π ∑ i = 1 n ‖ f ^ i ‖ L e c | . | 1 . Since L 2 ( M ) = L β 2 ( M ) , it follows from (2) that f ( Δ ) extends to a bounded operator in L β 2 ( M ) . The last statement is obvious.

Corollary 3. Let β be a function of moderate decay. Assume that there exist real numbers a , b such that:

(i) a + b = 2 ,

(ii) β b ∈ L 1 ( M ) ,

(iii) β a t ˜ − 1 2 n ( n + 1 ) ∈ L ∞ ( M ) .

Let M β the operator of multiplication by β . Then for every p ∈ ℕ 0 the operator M β ( ∑ i = 1 n Δ g i p ) e − t Δ g i is Hilbert-Schmidt. For e − t ( ∑ i = 1 n g i ) in a compact interval in ℝ + the Hilbert-Schmidt norm is bounded.

Proof. We have M β Δ p e − t Δ = ( M β e − 1 2 Δ ) ( Δ p e − 1 2 Δ ) . Note that the operator norm of Δ P e − 1 2 Δ is bounded on compact subsets of ℝ + . Hence we assume that p = 0 . Lemma 11, (i) implies that e − t Δ I ∈ L β b 2 ( M ) . Let e − t Δ ( x , y ) be the kernel e − t Δ then 〈 I , e − t Δ 〉 L 2 = ∫ M ∫ M ∏ i = 1 n β b ( x ) e − t Δ g i ( x , y ) d y d x . The integral converges since e − t Δ ( x , y ) ≥ 0 we get

∫ M ∫ M ∏ i = 1 n | β ( x ) e − t Δ g i | 2 ( x , y ) d y d x = ∫ M ∫ M ∏ i = 1 n β 2 ( x ) ( e − t Δ g i ( x , y ) ) 2 d y d x ≤ sup z , w ∈ M | ∏ i = 1 n β a ( z ) e − t Δ g i ( z , w ) | ∫ M ∫ M ∏ i = 1 n β b ( x ) e − t Δ g j ( x , y ) d y d x ≤ C sup z ∈ M | β a ( z ) t ˜ − n ( n + 1 ) 2 ( z ) | ∫ M β b ( x ) ( e − t Δ ( 1 ) ) ( x ) d x ≤ C 1 ‖ e − t Δ ( 1 ) ‖ L β b 2 .

This proves the corollary.

Lemma 13. Let β be a function of moderate decay, satisfying the conditions of Lemma 11. Let g i , h i be two complete metrics on M such that g i ~ β 2 h i . Let Δ g i and Δ h i be the Laplacians of g i and h i , respectively. Then ∑ i = 1 ∞ ( Δ g i − Δ h i ) e − t Δ g i and ∑ i = 1 ∞ e − t Δ g i ( Δ g i − Δ h i ) are trace class operators, and the trace norm is uniformly bounded for τ in a compact subset of ( 0 , ∞ ) .

Proof. We decompose e − τ Δ g i as e − τ Δ g i = ∑ i = 1 ∞ ( e − t Δ g i M β − 1 3 ) ⋅ ( M β 1 3 e − t 2 Δ g i ) . By Lemma 11, the second factor is a Hilbert-Schmidt operator and it suffices to show that ( Δ g i − Δ h i ) e − t Δ g M β − 1 3 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

∑ i = 1 ∞ ( ‖ ( Δ g i − Δ h i ) e − t Δ g M β − 1 3 ‖ 2 2 ) ≤ C ∑ i = 0 2 ∫ M ∫ M | ( ∇ g ) i e − t Δ g i ( x , y ) β − 1 3 ( y ) | g i 2 β 2 ( x ) d x d y = C ∑ i = 0 ∞ ∫ M ‖ e t Δ g i ( . , y ) β − 1 3 ( y ) ‖ W β 2 2 2 d y ≤ C 1 ∑ i = 0 ∞ ∫ M ‖ e t Δ g i ( . , y ) β 1 3 ( y ) ‖ H β t 2 − 4 n 2 2 d y ≤ C 2 ∑ q = 0 1 ∫ M ‖ β ( . ) t ˜ − 2 n ( . ) Δ g q e − t Δ g ( . , y ) β − 1 3 ( y ) ‖ 2 2 d y = C 2 ∑ q = 0 1 ∫ M ‖ M β M i ˜ − 2 n Δ g i q e − t Δ g M β − 1 3 ‖ 2 2 d y .

By Lemma 13, the right-hand side is finite and bounded for t in a compact interval of ℝ + prove that ∑ i = 1 ∞ e − t Δ g i ( Δ g i − Δ h i ) is a trace class operator, it suffices to establish it for its adjoint ∑ i = 1 ∞ ( Δ g i − ( Δ h i ) * g i ) e − t Δ g i with respect to t. By (19) and (18) we have ∑ i = 1 ∞ Δ g i ( Δ h i ) * g = ∑ i = 1 ∞ ( ( ξ 01 g i ) * g + ( ∇ ) * g ∘ ( ξ 11 g i ) * g + [ ( ∇ g i ) * g ] 2 ∘ ( ξ 21 g i ) * g ) using (14) and (16), it follows that there exists η j ∈ C ∞ ( H o m ( ( T * M ) * j ℝ ) ) such that ∑ i = 1 ∞ ( Δ g i − ( Δ h i ) * g i ) = ∑ i = 1 ∞ ( η 0 + η 1 ∘ ∇ g i + η 2 ∘ ( ∇ g i ) 2 ) and these sections satisfy

∑ i = 1 ∞ | η j | g i ( x ) ≤ C β ( x ) , 0 ≤ j ≤ 2 , x ∈ M . (22)

By principle we have

∑ i = 1 ∞ ( e − t Δ g i − e − t Δ h i ) = ∑ i = 1 ∞ ( ∫ 0 t e − s Δ g i ( Δ h i − Δ g i ) e − ( t − s ) Δ h i d s ) = ∑ i = 1 ∞ ( ∫ 0 t 2 e − s Δ g i ( Δ h i − Δ g i ) e − ( t − s ) Δ h i d s + ∫ t 2 t e − s Δ g i ( Δ h i − Δ g i ) e − ( t − s ) Δ h i d s ) (23)

Using (22) and (23) we can proceed as above and prove that ∑ i = 1 ∞ ( Δ g i − ( Δ h i ) * g i ) e − t Δ g i is a trace class operator.

The authors declare no conflicts of interest regarding the publication of this paper.

Youssif, M.Y. and Dalam, E.E.E. (2020) Verifications of the Scattering Theory on Manifolds. Advances in Pure Mathematics, 10, 645-657. https://doi.org/10.4236/apm.2020.1011040