Generic non-selfadjoint Zakharov–Shabat operators

Abstract

In this paper we develop tools to study families of non-selfadjoint operators L(ϕ), ϕ ∈ P , characterized by the property that the spectrum of L(ϕ) is (partially) simple. As a case study we consider the Zakharov-Shabat operators L(ϕ) appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials ϕ of Sobolev class H N , N ≥ 0, so that all small eigenvalues of L(ϕ) are simple, is path connected and dense.

Arnold-Liouville theorem for integrable PDEs: a case study of the focusing NLS equation

“…Next we argue as in the proof of Corollary 3.3 in [14] to construct a continuous path : γ : [0, 1] → U tn ∩ iL 2 r so that γ(0) = ψ (0) , γ(1) = ψ (1) , and for any potential ϕ ∈ γ [0, 1] the operator L(ϕ) has only simple (and hence non-real) periodic eigenvalues in the disk B R ′ . In view of the compactness of γ we can find a connected open neighborhood…”

“…This part of the phase space contains the set of potentials ϕ ∈ iH N r which have the property that all multiple eigenvalues are real with geometric multiplicity two whereas all simple eigenvalues are non-real and appear in complex conjugated pairs. In [14], it is shown that this set is open and path connected. We remark that in order to prove that the actions and the angles, first defined in an open neighborhood of the potential ψ ∈ iH N r , analytically extend to an open neighborhood of Iso o (ψ), we make use of the assumption that all periodic eigenvalues of L(ψ) are simple.…”

“…In fact, by [14,Lemma 2.3], the algebraic multiplicity of a periodic eigenvalue coincides with its multiplicity as a root of χ p (•, ϕ). We say that two complex…”

On normalized differentials on hyperelliptic curves of infinite genus

“…• . Then L 2 • is open, dense, and contains the zero potential ( [10]). In the case of the defocusing NLS equation, L(ϕ) is self-adjoint.…”

Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications

“…">IntroductionDirac operators attracted considerable attention in recent years, in particular in the context of non-self-adjoint spectral theory [4,5,6,9,12,24,25], nonlinear Schrödinger equations e.g. [3,21] or as an effective model for graphene [1,7,15,20]. In this paper we analyze eigenvalues emerging from the thresholds of the essential spectrum of the one-dimensional Dirac operator in L 2 (R) perturbed by a general matrix-valued and non-symmetric potential V preserving the essential spectrum.Our main results include the existence and asymptotics of weakly coupled eigenvalues for the one dimensional Dirac operator (Theorem 2.2) and Lieb-Thirring type inequalities (Theorem 2.4) in the massive as well as the massless case.…”