THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT SCHR ODINGER OPERATOR WITH ENERGY DEPENDENT POTENTIAL Hideo Nakazaa Department of Mathematics, Tokyo Metropolitan University. 1.9 1.Introduction In this paper e shall sho the principle of limiting absorption for the quadratic operator pencil (1.1) L() =1 ib in the N{dimensional euclidean space ith N 6=. Here 1 is the N{dimensional Laplacian, C and B denotes the multiplication operator by a real-valued function b(x) on. If e consider the solutions of the form (x; t) =u(x)e it in (1:1), then u(x) satises L()u(x) =(1 ib(x) )u(x) =: Throughout the paper e require the folloing condition: (1.) There exist constants b and (; 1] such that <b < p and jb(x)j b (1 + r) 1 for any x ; here r = jxj: The operator L() is derived from the ave equation (1.3) tt (x; t) 1(x; t)+b(x) t (x; t) = here x, t, t = @ @t, tt = @ @t and b(x) C 1 ( ). If b(x), b(x) t represents a friction of viscous type. As is knon in Mochizuki [14], under a more general assumption than (1:) ith b(x), the solutions of (1:3) are asymptotically equal for t! +1 to those of free ave equations (1.4) tt (x; t) 1 (x; t) =: More precisely, he proved Assume N 6=. Ifb(x) satises b(x) b (1+r) 1 for some >, then the folloing statements hold: (1) The Mller Weve operators exist: 9 W+ = lim t!+1 U(t)U (t); 9 W = lim t!+1 U (t)u(t); 8 and8 here U (t) 9tR U(t) represent a one{parameter group of unitary operators and semi{group 9t of contraction operators, respectively (lim means the strong limit in uniform operator topology). 1
() They both are not identically vanishing contraction operators in H _ H 1 ( ) L ( ). (3) U (t) and U(t) are intertined by W 6, i.e., W + U (t) =U(t)W + ; W U(t) =U (t)w for any t : (4) The scattering operator, dened by S = W W +, commutes ith U (t): SU (t) =U (t)s for any t R: If e consider the solutions of the form (x; t) =u(x)e it in (1:3), then u(x) satises (1.5) L()u(x) =(1 ib(x) )u(x) =: The principle of limiting absorption states that there exists to Hilbert spaces such that L ;(1+)= ( ) and L ;(1+)= ( ) L ;(1+)= ( ) L ( ) L ;(1+)= ( ) (the denition of these spaces is given in section 3), and each of R ( 6 i) 1 L(6i) 1 is continuously extended to = as an operator from L ;(1+)= ( )tol ;(1+)= ( ). More precisely, Main Theorem. Assume =8 N 6= and (1:). Let K + + = + i C nfg 9 I =(a1 ;a );(; 1) ith <a 1 <a < 1. For + i K + +,andf L;(1+)= ( )(< 1), e dene u(; ) =R ( + i) 1 f: Then u(; ) L ;(1+)= ( ) converges in L ;(1+)= ( ) as #. Moreover this convergence is uniform ith respect to I. If e denote this limit by u(), then u() is a solution of L()u(x) =f(x) and is uniformly continuous ith respect to I in L ;(1+)= ( ). Remark. In this theorem, e can replace K + + by K +, K + or K : K + =8 = i C nfg I =(a1 ;a ); (; 1) 9 ; K + =8 = + i C nfg I =(a ; a 1 ); (; 1) 9 ; K =8 = i C nfg I =(a ; a 1 ); (; 1) 9 : The principle of limiting absorption has been studied for the Schrodinger equation (1.6) 1+V (x) 1 u(x) =f(x) in : After the pioneering ork of Edus [], Agmon [1], Ikebe{Saito [4] and Mochizuki [11] extended it to a ider class of real valued potentials. The spectral theory is also developed for complex potentials by Kato [6], Mochizuki [9], [1] and Saito [14], [15], and Saito extended the principle there to complex potentials. The methods developed in these orks are not directly applied to our problem since ib(x) depends on the energy. In order to overcome the diculty e suppose the smallness condition (1:). Then Mochizuki's method [1] directly applies to obtain the key a-priori estimate ( Theorem 3.4 (3.5) ). From this follos the uniqueness of solutions for L()u(x) =f(x) (Im R ) hich plays an important role in the proof of Main Theorem (Corollary 3.5 (1)).
. Spectral Structure for L() In this section, for the sake of simplicity, e assume b(x). We dene the linear operator H by D(H )=H ( ), H = 1. As is ell knon, H is a selfadjoint operator and it holds that (H )=C n [; 1), (H )= ess (H )= c (H )=[; 1), r (H )= p (H )=;, here (H ), (H ), ess (H ), c (H ), r (H )and p (H ) denote the resolvent set, spectrum, essential spectrum, continuous spectrum, residual spectrum, and point spectrum, respectively. For C nfg, e dene the linear operator L() byl() =H ib ; D(L()) = H ( ). This is a non-selfadjoint but closed operator. The adjoint operator L 3 () ofl() isdendbyl 3 () = H + ib, D(L 3 ()) = H ( ). The resolvent set, the spectrum, the essential spectrum, the continuous spectrum, the residual spectrum and the point spectrum of L() are dened as follos ( cf. Markus [7] pp.56, Saito [14] pp.46 ). (L()) =n C Ker(L()) = fg; R(L()) = L ( ); (L()) 1 L L ( ) 1 o; (L()) = C n (L()) ; ess (L()) =n (L()) 9 ff n gd(l()) s.t kf n k L =1; L()f n! S (n!1); f n! (n!1)o; (here! S and! denote the strong and eak convergence in L respectively); c (L()) =n (L()) Ker(L()) = fg; R (L()) = L ( ); (L()) 1 6 L L ( ) 1 o; r (L()) =n (L()) Ker(L()) = fg; R (L()) 6= L ( )o p (L()) 8 8 = (L()) Ker(L()) 6= fg 9 ; here Ker(L()) = f D (L()) L()f = 9, R (L()) is the range of the operator L(), R (L()) denotes the closure of R (L()) in L ( ), and L 1 L ( ) is the set of the bounded linear operator on L ( ). For (L()), e put R( )=(L()) 1 and call it resolvent ofl(). We can dene the spectrum sets for L 3 () similarly as above. As in the case of the usual denition, e see (.1) r (L()), p (L 3 ()); 6 p (L()): From this e obtain the absence of the residual spectrum of L(). In these denitions, noting that ib is H { and (H ib){compact, e can follo the same line of proof as Schechter [16] Chapter 3 to sho the stability theorem of essential spectrum. Moreover, if e assume N 6= and the folloing condition: There exist positive constants b and such that <b <1 (.) and b(x) b (1 + r) 1 for any x ; then the absence of positive eigenvalue of L() is obtaind by the modications of the proof of (3:5) of Theorem 3.4 belo. Therefore, e obtain Theorem.1. Assume that N 6= and (:). Then the folloing assertions hold: (L()) = ess (L()) = c (L()) = [; 1); p (L()) = r (L()) = ;; (L()) = C n [; 1): ; 3
3. Some Inequalities-Resolvent Estimate In this section e shall sho some inequalities for L -solutions of the Schrodinger equation of the form (3.1) 1u ib(x)u u = f(x) in ; here = + i ((6= ); R) C nfg and f(x) L ;(1+)= ( ), here L ; (G) = ( f(x)kfk ;G = If G =, e shall omit the second symbol G. We deneto operators D 6 and D 6 r [14], [15] ): G (1 + jxj) jf(x)j dx 1 < 1 ) : as follos ( cf. Ikebe{Saito [4], Mochizuki [11], [1], [13], Saito D 6 u = ru + N 1 u x r r 7 iu x r Im R 1 ; D 6 r u = x r 1D6 u = u r 1 + N 1 u 7 iu Im R : r Proposition 3.1. If u is the solution of (3:1), then the folloing inequality holds: dx + jj (1 + r) 1 jb(x)j (3.) 1 RN (1 + r) u r + N 1 r u 5 (1 + r) 1 jd 6 r uj dx + jfiujdx Im R 1 : juj dx Lemma 3.. Let ' = '(r) be areal-valued C 1 {function. If u is a solution of (3:1), then the folloing identity holds: (3.3) 6 + n6'+ ' r R N RN b(x)' =Re From this, e obtain jd 6 uj + ' r uj1o r ' jd 6 uj jd 6 r! u N 1 r + r u jd 6 r uj + jj juj ' juj 6 8 r r (N 1)(N 3) f'd 6 r udx Im R 1 : dx dx (N 1)(N 3) 4r 'juj dx Proposition 3.3. Assume that N 6=. If u is a solution of (3:1), then the folloing inequality holds: (3.4) (1 + r) 1+ jd 6 uj dx 1 jb(x)j(1 + r) jd 6 uj + jj juj + u N 1 r + R r N By Proposition 3.1 and 3.3, e have (1 + r) jfd 6 r ujdx Im R 1 : 4! u dx
Theorem 3.4. Assume that N 6= and (1:). Ifu is a solution of (3:1) then there exist positive constants C 1 and C independent of such that the folloing inequalities hold: 1 Im R ; (3.5) jjkuk (1+)= C 1 kfk (1+)= (3.6) kd 6 uk (1+)= C kfk (1+)= Im R 1 : From these, e nd Corollary 3.5. Assume that N 6= and (1:). (1) There exists a unique solution u of (3:1) in L ;(1+)= ( ) for Im R. Im R 1 ; () There exists a positive constant C 3 independent of such that the folloing inequality holds: (3.7) jj kuk (1+)=;E R C 3 (1 + R) kfk (1+)= here E R = fx jjxj >Rg for R>. From Theorem3.4, Corollary 3.5 and the elliptic estimate for (3:1), e can sho Main Theorem.. 4. The Principle of Limiting Absorption for i 1 1 B First of all, note that (1:3) ith initial data (x; ) = 1, t (x; ) = is equivalent to (4.1) i t i 1 1 B t = () ; t = 1 : () t (e f;eg) H = We dene the Hilbert space H H _ 1 ( ) L ( )( H _ 1 ( ) being the Beppo-Levi space ) ith inner product for e f = f1 f ;eg = g1 H. g If b(x) then the solution of (4:1) is given by here W = i 1 [1] pp.384). 1 B t rf1 1 rg 1 + f 1 g 1 dx = e iw t 1 8 iw is the generator of the contraction semi-group e on H (cf. Mochizuki t9t Theorem 4.1. Assume that N 6=, (1:) and f e = f1 f The limiting absorption principle for W can be stated as follos: L ;(1+)= ( )(H). Ifeu =eu 6i = u1;6i u ;6i (4.) Weu ( 6 i)eu = e f is a solution of the equation: H 1 _H 1 ( ) \ L ;(1+)= ( ) 5
( R nfg, >), then there exists the limit lim #eu 6i = fu 6 in H H 1;(1+)= ( ) L ;(1+)= ( )(H) and fu 6 solves the equation here H 1;(1+)= ( )= f(x) RN Wfu 6 fu 6 = e f; (1 + r) 1 jrf(x)j + jf(x)j 1 dx < 1 : References 1. S. Agmon, Spectral properties of Schrodinger operators and scattering theory, Ann. Scuola. Norm. Sup. pisa(4). (1975), 151{18.. D. M. Edus, The principle of limiting absorption, Math. Sb. 57 (99) (196), 13{44 = Amer. Math. Soc. Transl. () 47 (1965), 157{191. 3. T. Ikebe, A spectral theory for reduced ave equation ith a complex refractive index, Publ. RIMS Kyoto Univ. 8 (197/73), 579{66. 4. T. Ikebe and Y. Saito, Limiting absorption method and absolute continuity for the Schrodinger operators, J. Math. Kyoto. Univ. 1 (197), 513{54. 5. N. Iasaki, On the principle of limiting amplitude, Publ. RIMS Kyoto Univ. 3 (1968), 373{39. 6. T. Kato, Wave operators and similality for some non-selfadjoint operators, Math. Ann. 16 (1966), 58{79. 7. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, vol.71, A.M.S. Soc. Providence, Rhode Island, 1988. 8. S. Mizohata and K. Mochizuki, On the principle of limiting amplitude for dissipative ave equations, J. Math. Kyoto. Univ 6 (1966), 19{17. 9. K. Mochizuki, On the large perturbation by a class of non-selfadjoint operators, J. Math. Soc. Japan 19 (1967), 13{158. 1., Eigenfunction expansions associated ith the Schrodinger operator ith a complex potential and the scattering theory, Publ. RIMS Kyoto Univ 4 (1968), 419{466. 11., Spectral and Scattering Theory for Second Order Elliptic Dierential Operators in an Exterior Domain, Lecture Notes Univ Utah. Winter and Spring, 197. 1., Scattering theory for ave equations ith dissipative terms, Publ. RIMS Kyoto Univ 1 (1976), 383{39. 13., Scatteing Theory for Wave Equations, Kinokuniya, Tokyo, 1984. (japanese) 14. Y. Saito, The principle of limiting absorption for the non-selfadjoint Schrodinger operator in (N 6= ), Publ. RIMS Kyoto Univ 9 (1974), 397{48. 15., The principle of limiting absorption for the non-selfadjoint Schrodinger operator in R, Osaka. J. Math 11 (1974), 95{36. 16. M. Schechter, Operator Methods in Quantum Mechanics, North Holland, Ne York, 1981. 6