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Description

Some PDEs and ODEs admit a Lax pair (a pair of linear operators) to be completely solve the equation. One of these operators defines a spectral problem. For some equations (as for the Korteweg-deVries, KdV, equation) this operator is self-adjoint and, consequently, its discrete spectrum is real. However, for some other equations, this operator is non-selfadjoint, such as for the nonlinear Schrödinger (NLS) equation or the Ablowitz-Ladik equation. In 2001, M. Klaus and J. K. Shaw found symmetries and conditions on the potentials for the Zakharov-Shabat system (spectral problem for the NLS equation) for the eigenvalues to lie on the imaginary axis. In this talk, I will show which would be an equivalent to the Klaus-Shaw theorem for the Ablowitz-Ladik lattice. This is a work in progress. This is a joint work with P. Shipman (Colorado State University) and S. Shipman (Louisiana State University).