Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system (Q2730794)

To the nonlinear Schrödinger equation may be assigned (or represented and analyzed in terms of) a Lax pair of operators of Zakharov-Shabat (in essence: perturbed Dirac operator) type. This context and related specific range of applications limit, often, the assumptions and analysis, typically, to the selfadjoint case. The authors deal with a generalized nonselfadjoint two-by-two complex Zakharov-Shabat linear differential operator, with (in the Dirac-equation language) ``potential'' term assumed to lie within a certain (suitably weighted) Sobolev space. NEWLINENEWLINENEWLINEThe spectrum is still known to be discrete (for both the periodic and Dirichlet boundary conditions). Suitable asymptotic expansions of the eigenvalues are available in the selfadjoint case but not in its present generalization. The authors employ a nonstandard method (called a Lyapunov-Schmidt type decomposition in the paper) and offer two basic theorems on the asymptotic distribution of the (complex) eigenvalues. NEWLINENEWLINENEWLINEThe text is inspired by the recent similar analysis of linear Schrödinger operators. This puts this paper in a perspective of a natural methodical development since, in the later context, the method is well known under many other names (e.g., as a Loewdin projection operator method in perturbation theory and in its applications in quantum chemistry or as Feshbach's effective operator method in quantum mechanics and nuclear physics, etc). Its present application is innovative and leads to a surprisingly compact picture of the generalized case, with the key idea lying in a successful elimination of all ``irrelevant'' Fourier components of the eigenfunctions so that we are left with the two-by-two ``effective'' algebraic eigenvalue problem.