Singular soliton operators and indefinite metrics

Abstract

Abstract. We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebro-geometrical potentials (i.e. "singular finite-gap" and "singular solitons") satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functions with singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2 . Here g is the genus of spectral curve.

Moutard transform approach to generalized analytic functions with contour poles

“…In particular,f andf + defined in (30) satisfy equations (16) and (17), respectively, with the coefficientũ given by (20).…”

“…Let M 2 be the simple Moutard transform for equations (16), (17) with coefficientũ = M 1 u, given by (25)- (27) (30). Then:…”

“…In particular, in [13] we give examples of coefficients u when equation (1) is of meromorphic class and can be efficiently studied using Moutard-type transforms. In addition, we were stimulated by results of [1], [4], [16], [25], [29], [27] on efficient applications of Darboux-Crum and Moutard-type transforms to studies of some important linear ODE's and PDS's with singular coefficients.…”

Complex Exceptional Orthogonal Polynomials and Quasi-invariance

“…The appearance of negative norms for singular potentials was first emphasised by Grinevich and Novikov [14].…”

“…In certain classes such operators were explicitly described in terms of Wronskians in [4,7,10,15]. Grinevich and Novikov studied the spectral properties of these and more general singular finite-gap operators and emphasised the important link with the theory of Pontrjagin spaces (see [14] and references therein). Our paper can be considered as dealing with the implications of all these results for the theory of exceptional orthogonal polynomials.…”