Asymptotic Behaviour of Resonance Eigenvalues of the Schr\"odinger operator with a Matrix Potential

We will discuss the asymptotic behaviour of the eigenvalues of Schr\"{o}dinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$ matrix with $m\geq 2$, $d\geq 2$ , when the eigenvalues belong to the resonance domain, roughly speaking they lie near planes of diffraction. \textbf{Keywords:} Schr\"{o}dinger operator, Neumann condition, Resonance eigenvalue, Perturbation theory. \textbf{AMS Subject Classifications:} 47F05, 35P15

Since fu (x)g 2 +0 2 is a complete system in L 2 (F ), for any q(x) in L 2 (F ) we have In our study, it is convenient to use the equivalent decomposition (see [9]) where q = 1 (F ) (q(x); u (x)) for the sake of simplicity. That is, the decomposition (3) and (4) are equivalent for any d 2: Thus, according to (4), each matrix element v ij (x) 2 L 2 (F ) of the matrix V (x) can be written in its Fourier series expansion v ij (x) = X F v ij (x)dx i; j = 1; 2; : : : ; m.
We assume that l > (d+20)(d 1) 2 + d + 3 and the Fourier coe¢ cients v ij of v ij (x) satisfy X 2 2 jv ij j 2 (1 + j j 2l ) < 1; 488 SEDEF KARAKILIÇ, SETENAY AKDUM AN, AND DIDEM COŞKAN the condition (6) Here O( p ) is a function in L 2 (F ) with norm of order p . Furthermore, by (6), we have for all i; j = 1; 2; : : : ; m. Notice that, if a function q(x) is su¢ ciently smooth q(x) 2 W l 2 (F ) and the support of rq(x) = @q @x1 ; @q @x2 ; : : : ; @q @x d is contained in the interior of the domain F , then q(x) satis…es condition (6) (See [7]). There is also another class of functions which is periodic with respect to a lattice = f(m 1 a 1 ; m 2 a 2 ; : : : ; m d a d ) : m k 2 Z; k = 1; 2; : : : ; dg and thus it also satis…es condition (6). As in [17]- [22], we divide R d into two domains: Resonance and Non-resonance domains. In order to de…ne these domains, let us introduce the following sets: Let 0 < < 1 d+20 , k = 3 k , k = 1; 2; : : : ; d 1 and ! where b 6 = 0, i 6 = 0, i = 1; 2; : : : ; k and the intersection in E k is taken over 1 ; 2 ; : : : ; k which are linearly independent vectors and the length of i is not greater than the length of the other vector in T i R: The set U ( 1 ; p) is said to be a non-resonance domain, and the eigenvalue j j 2 is called a non-resonance eigenvalue if 2 U ( 1 ; p): The domains V b ( 1 ), for b 2 (p ) are called resonance domains and the eigenvalue j j 2 is a resonance eigenvalue if 2 V b ( 1 ).
As noted in [20]- [21], the domain V b ( 1 ) n E 2 , called a single resonance domain, has asymptotically full measure on V b ( 1 ); that is, hold. Since 0 < < 1 d+20 , the conditions in (9) hold. In most cases, it is important to know the asymptotic behavior of the eigenvalues of the Schrödinger operator L(V ): In this paper, [3] and [8], we construct the asymptotic formulas in the high energy region for eigenvalues of the operator L(V ): In [3], we obtain the asymptotic formulas of arbitrary order for the eigenvalue of L(V ) corresponding to the non-resonance eigenvalues j j 2 of L(0) in arbitrary dimension d 2: In [8], we constructed the high energy asymptotics of arbitrary order for the eigenvalue of L(V ) corresponding to resonance eigenvalue j j 2 when belongs to the special single resonance domains V ( 1 ) n E 2 ; where is from fe 1 ; e 2 ; : : : ; e d g and e 1 = a1 ; 0; : : : ; 0 ; : : : ; e d = 0; : : : ; a d , d 2.
In this paper, we study the case for which j j 2 is a resonance eigenvalue. More precisely, in Theorem (1) and (2) of Section(2), we assume that 2 ( ) n E k+1 , k = 1; 2; : : : ; d 1 and = 2 V e k ( 1 ) for k = 1; 2; : : : ; d and prove that the corresponding eigenvalue of L(V ) is close to the sum of the eigenvalue of the matrix V 0 and the eigenvalue of the matrix C = C( ; 1 ; : : : ; k ) (See (14)).
In Section(3), this time we assume that 2 V ( 1 ) n E 2 ; 2 2 nfe 1 ; e 2 ; : : : ; e d g, that is, is in a single resonance domain and we prove the main result Theorem (7) which gives a connection between the eigenvalues of L(V ) corresponding to a single resonance domain and the eigenvalues of the Sturm-Liouville operators.
Note that, the case = e i ; i = 1; 2; : : : ; d, was considered in [8], by a di¤erent but simpler method and better formulas were obtained.
then by (17) and (18), we have Now, we prove that if (18) holds then for any j = 1; 2; : : : ; m. Here we remark that 0 6 = 0. If it were the case, then we would have from h 0 = 2 B k ( ; p 1 ) that h = 2 B k ( ; p 1 ) which is a contradiction. So, to prove (20), we argue as Theorem 2.2.2 (a) of [22]: Since N satis…es the inequality (18), by (19) . Using this, in the equation (13) instead of we write h 0 to get Substituting this equation (21) into the right hand side of (20), we obtain X In this manner, iterating p 1 times, we get X Taking norm of both sides of the last equality, using (19), the relation (8) and the fact that p 1 k+1 p 1 2 > p , we obtain which implies (20). Therefore, the equation (16) becomes Since h 0 2 B k ( ; p 1 ), using the notation h = h 0, the decomposition (22) can be written as Isolating the terms where h h = 0 in (23), we get Writing the equation (24) for all j = 1; 2; : : : ; m and for any = 1; 2; : : : ; b k , , we get the system of equations where I is an m m identity matrix, V h h is given by (15), is an m 1 vector and A(N; h ) is the m 1 vector for any = 1; 2; : : : ; b k . Letting N; = N jh j 2 , we have where I is an mb k mb k identity matrix, C is given by (14), A(N; h 1 ; h 2 ; : : and the right side of the system (28) is the mb k 1 vector whose norm is Theorem 1. Let j j 2 be a resonance eigenvalue of the operator L(0), that is, ) n E k+1 , k = 1; 2; : : : ; d 1 where j j , and N an eigenvalue of the operator L(V ) for which (18) holds and its corresponding eigenfunction N satis…es j< ;j ; N >j> c 4 c : Then there exists an eigenvalue s ( ), 1 s mb k of the matrix C such that Proof. Since (18) is satis…ed, (28) holds. Then multiplying both sides of the equation (28) by [ N I C] 1 , then taking norm of both sides and by (30), we get jA(N; h 1 ; h 2 ; : : : Using the fact that is one of h 1 ; h 2 ; : : : ; h (See de…nition of B k ( ; p 1 )) and hence by (31) and (32), we obtain Theorem 2. Let j j 2 be a resonance eigenvalue of the operator L(0), that is, 2 Multiplying the equation (28) by s , since C is symmetric (see (14) and (15)), we get  is the standard basis of R mb k . Now, we use the following notation which, together with Parseval's relation, imply Now we estimate the …rst summation in the expression (42): The assumption j s j j 2 j < 3 8 1 of the theorem and j s jh j 2 j < 1 8 1 imply that jj j 2 jh j 2 j < 1 2 1 . So by the well-known formula for j N j j 2 j 1 2 2 1 , and jj j 2 jh j 2 j < 1 2 2 1 , using (39), we have (45) To calculate the order of each term in (44), we use Bessel's inequality and the orthogonality of h ;i . So we have (46) for r = 0; 1; 2; : : : ; k. Now let K be the number of h satisfying j s jh j 2 j < 1 8 1 , then the order of the last summation in (46) is: and we can always choose k in O( 2(k+1) 1 ) such that which together with the estimations (44), , (45) and (46) imply Therefore, from the decomposition (42) we have Since the number of indexes N satisfying j N j j 2 j < 1 2 2 1 is less then d 1 , we have which implies together with the relation (41) that jA(N; h 1 ; h 2 ; : : : It follows from the equation (35) and the estimation (48) that ; that is, (36) holds.