Related papers: A theorem for the normalization of continuous spec…
Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially low-regular stationary solutions. This work provides a comprehensive theory of asymptotic stability of bounded, discontinuous or…
This is a survey paper based on previous results of the author. In the paper, we define and discuss the generalizations of linear partial differential equations to multidimensional variational problems. We consider two examples of such…
We review an exact WKB resolution method for the stationary 1D Schr\"odinger equation with a general polynomial potential. This contribution covers already published material: we supply a commented summary here, stressing a few aspects…
An explicit formula for the wave operators associated with Schroedinger operators on the discrete half-line is deduced from their stationary expressions. The formula enables us to understand the wave operators as one dimensional…
In this work we show that results of Rayleigh-Schr\"{o}dinger perturbation theory can be easily obtained using the recently proposed supersymmetric expansion algorithm. Our formalism avoids the sums over intermediate states and yield…
We present an analysis of the stability spectrum for all stationary periodic solutions to the sine-Gordon equation. An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about…
We solve the one-dimensional Schr\"odinger equation for the bound states of two potential models with a rich structure as shown by their "spectral phase diagram". These potentials do not belong to the well-known class of exactly solvable…
We present new approaches for solving constrained multicomponent nonlinear Schr\"odinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose…
We propose a method to establish the rapid stabilization of the bilinear Schr\"odinger control system and its linearized system, and the finite time stabilization of the linearized system using the Grammian operators. The analysis of the…
In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schr\"odinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the…
For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…
We present a new kind of normalization theorem: linearization theorem for skew products. The normal form is a skew product again, with the fiber maps linear. It appears, that even in the smooth case, the conjugacy is only H\"older…
This paper communicates recent results in theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schr\"odinger operators. In particular, we propose a formula for…
A class of generalized Schr\"{o}dinger problems in bounded domain is studied. A complete overview of the set of solutions is provided, depending on the values assumed by parameters involved in the problem. In order to obtain the results, we…
Static soliton bound states in nonlinear systems are investigated analytically and numerically in the framework of the parametrically driven, damped nonlinear Schr\"odinger equation. We find that the ordinary differential equations, which…
We prove the existence of normalized ground state solutions for the biharmonic Schr\"odinger equation with combined nonlinearities and show that all ground states correspond to the local minima of the associated energy functional restricted…
We generalize the Lewis-Riesenfeld technique of solving the time-dependent Schrodinger equation to cases where the invariant has continuous eigenvalues. An explicit formula for a generalized Lewis-Riesenfeld phase is derived in terms of the…
The objective of this paper is to report on recent progress on Strichartz estimates for the Schr\"odinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in…
We investigate the stability of ground states to a nonlinear focusing Schr\"odinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability…
We consider the Cauchy problem for the nonlinear Schroedinger eqiation with initial data close to a sum of N decoupled solitons. Under some suitable assumptions on the spectral structure of the one soliton linearizations we prove that for…