Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system

Abstract

Consider the 2 \Theta 2 first order system due to Zakharov-Shabat, LY := i ` 1 0 0 \Gamma1 ' Y 0 + ` 0 / 1 / 2 0 ' Y = Y with / 1 ; / 2 being complex valued functions of period one in the weighted Sobolev space H w j H w C : Denote by spec(/ 1 ; / 2 ) the set of periodic eigenvalues of L(/ 1 ; / 2 ) with respect to the interval [0; 2] and by spec Dir (/ 1 ; / 2 ) the set of Dirichlet eigenvalues of L(/ 1 ; / 2 ) when considered on the interval [0; 1]. It is well known that spec(/ 1 ; / 2 ) and spec Dir (/ 1 ; / 2 ) are discrete. Theorem Assume that w is a weight such that, for some ffi ? 0, w \Gammaffi (k) = (1 + jkj) \Gammaffi w(k) is a weight as well. Then for any bounded subset B of 1-periodic elements in H w \ThetaH w there exist N 1 and M 1 so that for any jkj N , and (/ 1 ; / 2 ) 2 B , the set spec(/ 1 ; / 2 )"f 2 1 C j j \Gamma kj ! =2g contains exactly one isolated pair of eigenvalues f + k ; \Gamma k g and spec Dir (/ 1 ; / 2 ) " f 2 C j ..