Spectrally meromorphic operators and non-linear systems

Moutard transform approach to generalized analytic functions with contour poles

“…2. M 2 is the simple Moutard transform for equations ( 16), (17) with coefficient ũ = M 1 u, given by:…”

“…Theorem 2 Let M 1 be a simple Moutard transform defined as in ( 18)- (20), where ω f 1 ,f + 1 = 0. Let M 2 be the simple Moutard transform for equations ( 16), (17) with coefficient ũ = M 1 u, given by ( 25)- (27), where ψ = M 1 ψ, ψ+ = M 1 ψ + , f2 = f , f + 2 = f + , where f and f + are defined in (30). Then: 1.…”

“…Proceeding from the aforementioned inverse scattering motivation, we consider the generalized analytic function equation (1) with contour poles for the case when u is of the form as in (6), and when this equation has sufficiently many (more or less as in the regular case) local solutions ψ of the form as in (6). Adopting the terminology of [17] we say that in this case equation ( 1) is of meromorphic class. Note also that equation (1) may be of meromorphic class only if principal terms of u near pole contours satisfy solvability conditions; for simple contour poles these conditions were found in [15].…”

“…Adopting the terminology of [17] we say that in this case equation (1) is of meromorphic class. Note also that equation (1) may be of meromorphic class only if principal terms of u near pole contours satisfy solvability conditions; for simple contour poles these conditions were found in [15].…”

On the s-meromorphic OD operators

“…For the formally self-adjoint case we proved recently for all higher order algebrogeometric rank one operators [4] that product of 2 eigenfunctions f g never has nontrivial Laurent terms of the order (x − x j ) −1 at any singular point x j .…”