BK-factorization of bivariate LPDOs is outlined. As an example, class of factorized hyperbolic operators is constructed as well as general solution in quadratures for operators of this class. Coupling of BK-factorization with Darboux-Laplace transformations is described which provided additional factorization conditions (demand of generalized commutativity) in the case when BK-factorization doe...

Introduction. During the last 3 years the present author made a series of works [1, 2, 3, 4, 5, 8, 6, 7, 9] dedicated to the study of the unusual spectral properties of low-dimensional continuous and discrete (difference) Schrodinger Operators. Some of these works were done in collaboration with A.Veselov, I.Taimanov and I.Dynnikov. First, let me briefly describe the list of problems discussed ...

We show that the quantum field theoretical formulation of the τ function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic vertex operators are Lévy and adjoint Lévy transformations and iii) the diagonal s...

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases o...

An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric int...

appear (the uj run over an orthonormal basis of automorphic Laplaceeigenforms), so our formula (Theorem 1) is a duality between such integrals and certain geodesic integrals of u. New integral transformations are involved depending on the Laplace-eigenvalue of u. We invert these integral transformations in Section 5, Theorem 2. We develop the formula for finite volume Fuchsian groups, so (as in...

The aim of this article is to present and reformulate systematically what is known about surfaces in the projective 3-space, in view of transformations of surfaces, and to complement with some new results. Special emphasis will be laid on line congruences and Laplace transformations. A line congruence can be regarded as a transformation connecting one focal surface with the other focal surface....