Generic nonselfadjoint Zakharov–Shabat operators

Authors: T Kappeler P Lohrmann P Topalov

Publish Date: 2014/01/09

Volume: 359, Issue: 1-2, Pages: 427-470

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Abstract

In this paper we develop tools to study within a family of nonselfadjoint operators Lvarphi depending on a parameter varphi in a real Hilbert space those with partially simple spectrum As a case study we consider the Zakharov–Shabat operators Lvarphi appearing in the Lax pair of the focusing NLS on the circle In particular the main result implies that the set of potentials varphi of Sobolev class HN Nge 0 so that all non real eigenvalues of Lvarphi are simple is path connected and denseFor any varepsilon 0 sufficiently small there exists an open neighborhood V subseteq L2 c of psi such that for any varphi in V chi cdot varphi has exactly m roots z 1varphi ldots z mvarphi listed with their multiplicities in the open disk Dvarepsilon equiv D varepsilon z psi = lambda in mathbb Clambda z psi varepsilon and no roots on the boundary partial Dvarepsilon of Dvarepsilon By the analyticity of chi cdot psi there exists varepsilon 0 so that chi cdot psi does not vanish on overlineDvarepsilon setminus z psi By the analyticity of chi it then follows that there exists a neighborhood V of psi in L2 c so that for any varphi in V chi cdot varphi does not vanish in a small tubular neighborhood of partial Dvarepsilon in mathbb C One then concludes by the argument principle that for any varphi in V chi cdot varphi has precisely m zeros in Dvarepsilon when counted with their multiplicities

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