Quasi-Feynman Formulas providing Solutions of Multidimensional Schrödinger Equations with Unbounded Potential

Mathematical Notes -     

Abstract Theory of Pointwise Decay with Applications to Wave and Schrödinger Equations

We prove pointwise in time decay estimates via an abstract conjugate operator method. This is then applied to a large class of dispersive equations.     

Solitary Waves for Linearly Coupled Nonlinear Schrödinger Equations with Inhomogeneous Coefficients

Motivated by the study of matter waves in Bose–Einstein condensates and coupled nonlinear optical systems, we study a system of two coupled nonlinear Schrödinger equations with inhomogeneous parameters, including a linear coupling. For that system, we prove the existence of two different kinds of homoclinic solutions to the origin describing solitary waves of physical relevance. We use a Krasnoselskii fixed point theorem together with a suitable compactness criterion.     

Fractional Schrödinger Equations with Logarithmic and Critical Nonlinearities

Acta Mathematica Sinica, English Series - In this paper, we study a class of the fractional Schrödinger equations involving logarithmic and critical nonlinearities. By using the Nehari...     

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.

     

The Cauchy Problem for Coupled Nonlinear Schrödinger Equations with Linear Damping: Local and Global Existence and Blowup of Solutions

The authors study, by applying and extending the methods developed by Cazenave(2003), Dias and Figueira(2014), Dias et al.(2014), Glassey(1994–1997), Kato(1987), Ohta and Todorova(2009) and Tsutsumi(1984), the Cauchy problem for a damped coupled system of nonlinear Schrdinger equations and they obtain new results on the local and global existence of H~1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.     

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure on a Brelot space (X,) with Green function G, it is well known that -harmonic functions (i.e., in the classical case, finely continuous versions of solutions to u–u=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, ) admitting a Green function T (G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric :=(G ^–1+^* G ^–1)/2 yields -Hölder continuity for quotients of -harmonic functions as well.     

Schrödinger Operator with Morse Potential and Zeros of the Riemann Zeta Function

Mathematical Notes -     

Multi-bump Bound States of Schrödinger Equations with a Critical Frequency

We consider existence and qualitative properties of standing wave solutions $\Psi(x,t) = e^{-iEt/h}u(x)We consider existence and qualitative properties of standing wave solutions to the nonlinear Schr?dinger equation with E being a critical frequency in the sense that inf . We verify that if the zero set of W − E has several isolated points x _i () near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k, , a standing wave solution which concentrates simultaneously on , where is any given subset of . This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).Supported by the Alexander von Humboldt foundation and NSFC(No:10571069).     

Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger Equations

We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schrödinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of “irregularity” for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well–posedness theory of the equations. We develop two different approaches to the analysis of the effects of the modulation. A first approach is based on novel estimates for the regularizing effect of the modulated dispersion on the non-linear term using the theory of controlled paths. A second approach is an extension of a Strichartz estimated first obtained by Debussche and Tsutsumi in the case of the Brownian modulation for the quintic NLS.     

Rotation Numbers of Linear Schrödinger Equations with Almost Periodic Potentials and Phase Transmissions

In this paper we study the linear Schrödinger equation with an almost periodic potential and phase transmission. Based on the extended unique ergodic theorem by Johnson and Moser, we will show for such an equation the existence of the rotation number. This extends the work of Johnson and Moser (in Commun Math Phys 84:403–438, 1982; Erratum Commun Math Phys 90:317–318, 1983) where no phase transmission is considered. The continuous dependence of rotation numbers on potentials and transmissions will be proved.     

Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions

We study spectral properties of Hamiltonians H _X,β,q with δ′-point interactions on a discrete set ${X = \{x_k\}_{k=1}^\infty \subset (0, +\infty)}$ . Using the form approach, we establish analogs of some classical results on operators H _q =  ?d²/dx ² + q with locally integrable potentials ${q \in L^1_{\rm loc}[0, +\infty)}$ . In particular, we establish the analogues of the Glazman–Povzner–Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with δ-interactions, spectral properties of operators H _X,β,q are closely connected with those of ${{\rm H}_{X,q}^N = \oplus_{k}{\rm H}_{q,k}^N}$ , where ${{\rm H}_{q,k}^N}$ is the Neumann realization of ?d²/dx ² + q in L ²(x _k-1,x _k ).     

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schrödinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schrödinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1+2)-dimensional cubic Schrödinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.     

The Cauchy Problem for Defocusing Nonlinear Schrödinger Equations with Non-Vanishing Initial Data at Infinity

For rather general nonlinearities, we prove that defocusing nonlinear Schrödinger equations in ? ⁿ (n ≤ 4), with non-vanishing initial data at infinity u ₀, are globally well-posed in u ₀ + H ¹. The same result holds in an exterior domain in ? ⁿ , n = 2, 3.     

On the Negative Spectrum of One-Dimensional Schrödinger Operators with Point Interactions

We investigate negative spectra of one-dimensional (1D) Schrödinger operators with δ- and δ′-interactions on a discrete set in the framework of a new approach. Namely, using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of Albeverio and Nizhnik (Lett Math Phys 65:27–35, 2003; Methods Funct Anal Topol 9(4):273–286, 2003). For instance, we propose an algorithm for determining the number of negative squares of the operator with δ-interactions. We also show that the number of negative squares of the operator with δ′-interactions equals the number of negative strengths.     

Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

Potential Analysis - We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential...     

On a Family of Solvable Multiparticle Schrödinger Equations with Pair Potential

Mathematical Notes -     

Negative powers of a singular Schrödinger operator and convergence of spectral decompositions

We compare theL ₂( ^N )-norms of negative powers of various Laplace and Schrödinger operators possessing a singular potential whose singularities lie on some manifolds. We write out sufficient conditions for uniform convergence and localization of spectral decompositions of functions from the Liouville class.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 428–436, March, 1996.The author wishes to express deep gratitude to Prof. Sh. A. Alimov for his attention to this work.