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.     

Localization and Schrödinger Perturbations of Kernels

We study iterations of integral kernels satisfying a transience-type condition and we prove exponential estimates analogous to Gronwall’s inequality. As a consequence we obtain estimates of Schrödinger perturbations of integral kernels, including Markovian semigroups.     

Self-adjointness of Schrödinger operator and Wiener integral

In this paper we prove theorems of self-adjointness of the operatorH=–+V and its powersH ^p . The proof is based on the analysis of Wiener's integrals.     

Eigenvalues of Schrödinger operators and Dirichlet-to-Neumann maps

Eigenvalues and eigenspaces of selfadjoint Schrödinger operators on are expressed in terms of Dirichlet-to-Neumann maps corresponding to Schrödinger operators on the upper and lower half space. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bäcklund Transformations for the One-Dimensional Schrödinger Equation

Journal of Applied and Industrial Mathematics - We study the system of equations which bases on the one-dimensional Schrödinger equation and connects the potential, amplitude, and phase...     

Quantum diffusion of the random Schrödinger evolution in the scaling limit

We consider random Schrödinger equations on R ^d for d???3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and \(\psi_t\) the solution with initial data \(\psi_0\). The space and time variables scale as \(x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)}\). We prove that, in the limit λ?→?0, the expectation of the Wigner distribution of \(\psi_t\) converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data.The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the non-ladder diagrams is smaller than their “naive size” by an extra λ ^c factor per non-(anti)ladder vertex for some c?>?0. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

Sigma Functions and Lie Algebras of Schrödinger Operators

Functional Analysis and Its Applications - In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g &gt; 0$$ , a...     

On Combinatorics of Schrödinger Perturbations

We give a tight upper bound for Schrödinger-type perturbations of integral kernels.     

Self-similar solutions of Schrödinger flows

In this note, we showed the existence of equivariant self-similar solutions with finite local energy for the Schrödinger flow from \({\mathbb{C}^n}\) into \({\mathbb{C}P^n}\) (n ≥ 2).     

Spherical Perturbations of Schrödinger Equations

We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights and on , for which the a priori inequality holds with constant independent of . Here is the Laplace-Beltrami operator on the sphere . Due to the relation between and the tangential component of the gradient, , we obtain some "Morawetz-type" estimates for on . As a consequence we establish some new estimates for the free Schr?dinger propagator , which may be viewed as certain refinements of the -(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem for certain time dependent first order spherical perturbations of the dimensional Schr?dinger equation.     

Inverse theory of Schrödinger matrices

In this note we discuss the inverse spectral theory for Schrödinger matrices, in particular a conjecture of Gesztesy-Simon [1] on the number of distinct iso-spectral Schrödinger matrices. We consider 3 × 3 matrices and obtain counter examples to their conjecture.     

Schrödinger equation and oscillatory Hilbert transforms of second degree

Let $h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $ ( $(i = \sqrt { - 1;} t,x$ -real variables). It is proved that in the rectangle $D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}$ , the function h satisfies the followingfunctional inequality: $\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle. Variation and complexity of solutions of Schrödinger equation are discussed.     

Mixed Norm Estimates of Schrödinger Waves and Their Applications

In this paper we establish mixed norm estimates of interactive Schrödinger waves and apply them to study smoothing properties and global well-posedness of the nonlinear Schrödinger equations with mass critical nonlinearity.     

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.     

Gauge Optimization and Spectral Properties of Magnetic Schrödinger Operators

We establish new necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with positive scalar potentials. We also derive two-sided estimates for the bottoms of the spectrum and essential spectrum. The main idea is to optimize the gauges of the magnetic field on cubes, thus reducing the quadratic form on the cubes to ones without magnetic field (but with appropriately adjusted scalar potentials).     

Decay estimates and Strichartz estimates of fourth-order Schrödinger operator

We study time decay estimates of the fourth-order Schrödinger operator $H={(?\mathrm{\Delta })}^2+V(x)$ in ${\mathbb{R}}^d$ for $d=3$ and $d\ge 5$. We analyze the low energy and high energy behaviour of resolvent $R(H;z)$, and then derive the Jensen–Kato dispersion decay estimate and local decay estimate for $e^{?itH}{P}_{ac}$ under suitable spectrum assumptions of H. Based on Jensen–Kato type decay estimate and local decay estimate, we obtain the $L^1\to L^{\infty }$ estimate of $e^{?itH}{P}_{ac}$ in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of $e^{?itH}{P}_{ac}$ for $d\ge 5$. Furthermore, using the local decay estimate and the Georgescu–Larenas–Soffer conjugate operator method, we prove the Jensen–Kato type decay estimates for some functions of H.