Related papers: Faddeev eigenfunctions for two-dimensional Schrodi…
By using the Moutard transformation of two-dimensional Schroedinger operators we derive a procedure for constructing explicit examples of such operators with rational fast decaying potentials and degenerate $L_2$-kernels (this construction…
By the Moutard transformation method we construct two-dimensional Schrodinger operators with real smooth potential decaying at infinity and with a multiple positive eigenvalue. These potentials are rational functions of spatial variables…
In the framework of the Moutard transformation formalism we find multi-point delta-type potentials of two-dimensional Schrodinger operators and their isospectral deformations on the zero energy level. In particular, these potentials are…
The discrete spectra of certain two-dimensional Schrodinger operators are numerically calculated. These operators have interesting spectral properties, i.e. their kernels are multi-dimensional and the deformations of potentials via the…
We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the…
We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for point (delta-type) potentials in two dimensions. In particular, we obtain the first explicit examples of such eigenfunctions with…
Using Moutard transformations we show how explicit examples of two-dimensional Schroedinger operators with fast decaying potential and multidimensional $L_2$-kernel may be constructed
In this paper we study two-dimensional discrete operators whose eigenfunctions at zero energy level are given by rational functions on spectral curves. We extend discrete operators to difference operators and show that two-dimensional…
The existence of potentials for relativistic Schrodinger operators allowing eigenvalues embedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger…
We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for multipoint potentials in two and three dimensions. For single point potentials in 3D such formulas were obtained in an old unpublished…
We develop the d-bar -approach to inverse scattering at zero energy in dimensions d>=3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction…
Different cases of sequences of the Laplace Transformations for the 2D Schrodinger operator in the periodic magnetic field and electric potential are considered. They lead to the exactly solvable operators with nonstandard spectral…
The nonlocal Darboux transformation of the two - dimensional stationary Schr\"odinger equation is considered and its relation to the Moutard transformation is established. It is shown that a special case of the nonlocal Darboux…
We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the…
The generalized Moutard transformation of the stationary axially symmetric Schr\"odinger equation is considered. It is shown that a superposition of two Moutard transformations can provide new potentials for the eigenvalue problem. Examples…
In this paper we study a connection between finite-gap on one energy level two-dimensional Schrodinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators.…
The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All…
We continue to develop the method for creation and annihilation of contour singularities in the $\bar\partial$--spectral data for the two-dimensional Schr\"odinger equation at fixed energy. Our method is based on the Moutard-type transforms…
This work studies the direct and inverse fixed energy scattering problem for two-dimensional Schroedinger equation with rather general nonlinear index of refraction. In particular, using the Born approximation we prove that all…
We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…