Transmission eigenvalues for multipoint scatterers

We study the transmission eigenvalues for the multipoint scatterers of the Bethe-Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions $d=2$ and $d=3$. We show that for these scatterers: 1) each positive energy $E$ is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex $E$ is an interior transmission eigenvalue of infinite multiplicity. The case of dimension $d=1$ is also discussed.

Creation and annihilation of point-potentials using Moutard-type transform in spectral variable

We continue to develop the method for creation and annihilation of contour singularities in the $barpartial$--spectral data for the two-dimensional Schrodinger equation at fixed energy. Our method is based on the Moutard-type transforms for generalized analytic functions. In this note we show that this approach successfully works for point potentials.

Moutard transform for the conductivity equation

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 Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrodinger equation at zero energy are also shown.

Multipoint scatterers with zero-energy bound states

We study multipoint scatterers with zero-energy bound states in three dimensions. We present examples of such scatterers with multiple zero eigenvalue or with strong multipole localization of zero-energy bound states.

Moutard transform approach to generalized analytic functions with contour poles

We continue studies of Moutard-type transforms for the generalized analytic functions started in arXiv:1510.08764, arXiv:1512.00343. In particular, we show that generalized analytic functions with the simplest contour poles can be Moutard transformed to the regular ones, at least, locally. In addition, the later Moutard-type transforms are locally invertible.

Generalized analytic functions, Moutard-type transforms and holomorphic maps

We continue the studies of Moutard-type transform for generalized analytic functions started in our previous paper: arXiv:1510.08764. In particular, we suggest an interpretation of generalized analytic functions as spinor fields and show that in the framework of this approach Moutard-type transforms for the aforementioned functions commute with holomorphic changes of variables.

An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse Scattering Transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, we have recently esatablished that, in the nonlocal part of its evolutionary form $v_{t}= v_{x}v_{y}-partial^{-1}_{x},partial_{y},[v_{y}+v^2_{x}]$, the formal integral $partial^{-1}_{x}$ corresponding to the solutions of the Cauchy problem constructed by such an IST is the asymmetric integral $-int_x^{infty}dx$. In this paper we show that this results could be guessed in a simple way using a, to the best of our knowledge, novel integral geometry lemma. Such a lemma establishes that it is possible to express the integral of a fairly general and smooth function $f(X,Y)$ over a parabola of the $(X,Y)$ plane in terms of the integrals of $f(X,Y)$ over all straight lines non intersecting the parabola. A similar result, in which the parabola is replaced by the circle, is already known in the literature and finds applications in tomography. Indeed, in a two-dimensional linear tomographic problem with a convex opaque obstacle, only the integrals along the straight lines non-intersecting the obstacle are known, and in the class of potentials $f(X,Y)$ with polynomial decay we do not have unique solvability of the inverse problem anymore. Therefore, for the problem with an obstacle, it is natural not to try to reconstruct the complete potential, but only some integral characteristics like the integral over the boundary of the obstacle. Due to the above two lemmas, this can be done, at the moment, for opaque bodies having as boundary a parabola and a circle (an ellipse).

Moutard transform for the generalized analytic functions

We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.

On the s-meromorphic OD operators

We consider linear spectral-meromorphic (s-meromorphic) OD operators at the real axis such that all local solutions to the eigenvalue problems are meromorphic for all $lambda$. By definition, rank one algebro-geometrical operator $L$ admit an OD operator $A$ such that $[L,A]=0$ and rank of this commuting pair is equal to one. All of them are s-meromorphic. In particular, second order ``singular soliton operators satisfy to this condition. Operator $L^+$ formally adjoint to s-meromorphic operator $L$ is also s-meromorphic. For singular eigenfunctions of operators $L,L^+$ following scalar product $<f,g>=int_R fbar{g}dx$ is well-defined such that $<Lf,g>=<f,L^+g>$ avoiding isolated singular points. For the case $L=L^+$ this formula defines indefinite inner product on the spaces of singular functions $f,gin F_L$ associated with operator $L$. They are $C^{infty}$ outside of singularities and have isolated singularities of the same type as eigenfunctions $Lf=lambda f$. Every s-meromorphic operator can be approximated by algebro-geometric rank one operators in any finite interval

Faddeev eigenfunctions for multipoint potentials

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 work of L.D. Faddeev. For single point potentials in 2D such formulas were given recently by the authors in arXiv:1110.3157 .