David W. Pravica. Department of ...... J. V. Ralston, Trapped rays in spherically symmetric media and poles of the scattering matrix, Comm. Pure Appl. Math.
Journal of Differential Equations DE3134 journal of differential equations 129, 262289 (1996) article no. 0118
Shape Resonances: A Comparison between Parabolic and Flat Bottom Potentials David W. Pravica Department of Mathematics, East Carolina University, Greenville, North Carolina 27858 Received July 28, 1995; accepted December 17, 1995
Sharp exponential bounds are obtained for resonance states and complex eigenvalues z for the Schrodinger equation H *#(&d 2dx 2 +*V) =z on [0, ). In a compact region V(x) is either a square-well or a parabolic potential. As x , V(x)tV +V M x &+ for some +>0. The three cases V 0, V(c)=V M #sup(V)>0;
(H1)
(H2) _= 1 , ; 0 and V (}) which is analytic on C; 0 =[c+= 1 +\e i; | \>= 1, | ;| ; 0 ], (a)
V (x)=V(x) for x # (c+2= 1 , ),
(b) j=1, 2, 3,
|(}&d ) j V ( j )(})(V (})&V(d ))| 0 so that \} # C; 0 , |Im V (})| C 1 |Re V (})&V | +C 2 ; U(x) # L , and supp(U )/[b, c+2= 1 ].
(H3)
Remark 1.1. A potential which satisfies (H1) and V(x)=V M(cx) + +V for +>0 in the exterior region x c.
(H2)
is
Asymptotic energy levels for the P and F potentials are, for n=1, 2, 3, ... (P) (F)
a=b, E Pn #2 - * n - V"(a), n #n&14, * * a0 and B # [0, 1) are appropriately chosen. The maximum distance * * * B&12 ). is then d * n, * #\ n, *, B( ). It is easily shown that |d n, * &\ n (*)| =O(* Main Theorem (NTH). Consider potentials V(x), U(x) which satisfy (H1)(H3). Suppose _d>c+2= 1 so that V(x) also satisfies the following conditions, (H4)
(a) V | (b, d ) >0, (b) V | J d 0 and V | J c >V ,
V$ | (c+2= 1 , ) 0 there is an s.r. value and an eigenvalue within E 0 of the bottom of the essential spectrum of H *. The techniques used here must be modified considerably in order to follow the transition from an s.r. value into an eigenvalue, as * increases. A study of the + 2 cases is given in [10]. The paper is organized as follows. The next section is a review of definitions and important results. The basic estimate is established which ensures that (H *(%)&`) &1 is a meromorphic function of ` for some % in a complex neighborhood of 0. Section 3 begins by demonstrating convergence of projections for appropriate eigenstates. It also contains a proof of Main Theorem (NTH). The section following demonstrates the modifications required to handle the TH case. We also discuss the method of [13] further. The PTH case is discussed in Section 5. In the final section we apply our results to the study of waves in cylindrically symmetric media, and obtain an extension of the work of Marcuse [8].
2. PRELIMINARIES An s.r. value for a self-adjoint operator H is defined to be a (complex) eigenvalue for an associated non-self adjoint operator H(%), % # C. One obtains H(%) from H using a group of operators U% which are unitary for % # R. If U% becomes non-unitary for Im(%){0, then the spectrum of can differ from _(H). In physical applications, where H H(%)#U% HU &1 % corresponds to a Hamiltonian operator, one often has the decomposition
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DAVID W. PRAVICA
_(H)=_ ess(H) _ _ pp(H). Then in general _(H(%))=_ ess(H(%)) _ _ pp(H) _ _ sr(H) where the essential spectrum of H(%) moves in some continuous manner with % to reveal the complex eigenvalues _ sr(H). Note that the pure point spectrum of H does not change with %. Once revealed, the same is true for the s.r. values. Our analysis begins by defining the vector field on J a , v(x)=
{
(x&d 1 )(1&exp[&_* &$ 1(F 1 &*V(x))]), 0,
x>d 1 x
