Related papers: Solvability Conditions for Some non Fredholm Opera…
We derive necessary and sufficient conditions for a one-dimensional periodic Schr\"odinger (i.e., Hill) operator H=-d^2/dx^2+V in L^2(R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the…
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time…
In this note we present an algorithm to generate new Schr\" odinger type equations explicitly solvable in terms of orthogonal polynomials or associated special functions.
We characterize lower semi-Fredholm and Fredholm of weighted composition operators on $C(K)$ in the case when the corresponding map is an open surjection of the compact space $K$ onto itself. The obtained criterions involve the notion of…
We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave…
Operator-type estimates of homogenization are obtained for elliptic operators of arbitrary even order equal or greater than two. Operators under consideration are non-selfadjoint with lower-order terms.
We turn back to the well known problem of interpretation of the Schrodinger operator with the pseudopotential being the first derivative of the Dirac function. We show that the problem in its conventional formulation contains hidden…
There is studied problem on existence of solutions to non-homogeneous differential equation of higher even order. Similar problem arises while studying soliton and soliton-like solutions to partial differential equations of integrable type.…
Based on a method that produces the solutions to the Schrodinger equations of partner potentials, we give two conditionally exactly solvable partner potentials of exponential type defined on the half line. These potentials are…
Motivated by the dynamics of defects in planar pattern-forming systems, we study Fredholm properties of elliptic operators with singular coefficients in weighted Sobolev spaces. In particular, we consider a family of doubly weighted spaces…
We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex…
Motivated by the interest in non-relativistic quantum mechanics for determining exact solutions to the Schrodinger equation we give two potentials that are conditionally exactly solvable. The two potentials are partner potentials and we…
We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(\Omega; d^n x)$, $n\geq 2$, where $\Omega$ is an open set with a compact, nonempty boundary…
We report branches of explicit expressions for nonlinear modes in parity-time (PT) symmetric potentials of several types. For the single-well and double-well potentials the found solutions are two-parametric and appear to be stable even…
The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations \cite{Evans}. In this article, we demonstrate a generalized…
In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schr\"odinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.
There is studied problem on solvability of linear non-homogeneous differential equation of higher even order. There is proved the theorem on necessary and sufficient conditions on existence of solutions to the equation in the Schwartz…
We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the…
After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which…
The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be…