Asymptotics of Resonant States

Consider a Schriidinger operator h = -V’ + V(x). x E R-’ with a compactly supported potential. ‘Its resolvent kernel G(x. J’, k) is a meromorphic function of k = Eli’, where E is the energy parameter. The poles of G in Im k ( 0 (or the second sheet of the Riemannian surface of fi) are called resonances. The corresponding solutions of the homogeneous equation hy = k2ty, I,U r-’ exp(ikr) exponentially growing as r = Ix + co. are called the resonant states. It is easy to prove that if ty exp(-ikr) + 0 as r + co then I,U E L2(lR3) and v = O(r-’ exp(-ikr)) as r 4 00. In particular. if V(x) is real-valued potential and k = -ib, b > 0, then rt,u exp(-ikr) -+ 0, r + co, implies that either k = ib corresponds to the bound state with the energy E = -b2 or w E 0. We prove this and a more general result for an operator equation. This generalization includes complex-valued potentials. Our result for the Schrodinger operator is of the dichotomy type obtained by T. Kato [ 1, Theorem 31. His result roughly speaking says that if Au+q(x)u=O, Ixl>R,, xER3, Re(exp(i0) q(x)} < 0. / 191 < 742, then either u(x) grows as Ix] -+ co or it decays. In the problem of resonant states we obtained actually a precise order of the growth. In [5 ] and [6] one can find some other results on dichotomy for ordinary differential equations and for the Schriidinger equation.