Related papers: On the Schrodinger maximal function in higher dime…
A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in…
We establish some weighted $L^2$ estimates for the Fourier extension operator in $\mathbb{R}^2$ and discuss several applications to $L^p$ problems. These include estimates for the maximal Schr\"odinger operator and the maximal extension…
We consider certain Littlewood-Paley square functions on $\Bbb R^2$ and prove sharp estimates for them, from which we can deduce $L^p$ boundedness of maximal functions defined by Fourier multipliers of Bochner-Riesz type on $\Bbb R^2$. This…
A new family of solvable potentials related to the Schroedinger-Riccati equation has been investigated. This one-dimensional potential family depends on parameters and is restricted to the real interval. It is shown that this potential…
The Schroedinger equation with one and two dimensional potentials are solved in the frame work of the sl(2) Lie algebra. Eigenfunctions of the Schroedinger equation for various asymmetric double-well potentials have been determined and the…
We prove a sharpened version of the Strichartz inequality for radial solutions of the Schr\"odinger equation in $\mathbb{R}^2\times \mathbb{R}$. We establish an improved upper bound for functions that nearly extremize the inequality, with a…
New exactly solvable quantum models are obtained with the help of the supersymmetric extencion of the nonstationary Schr/"odinger equation.
In this paper, we introduce some new ideas to study Schrodinger equations in RN with power-type nonlinearities.
This article considers the classification of matrix superpotentials that corresponds to exactly solvable systems of Schrodinger equations. Superpotentials of the following form are considered: $W_k = kQ + P + \frac1kR$, where $k$ ---…
We have derived some new results for the Mellin transform formulas, as well as for the Gauss hypergeometric function. Also, we have found the connection between the Legendre functions of the second kind. Some of the results obtained we used…
Monotone linear relations play important roles in variational inequality problems and quadratic optimizations. In this paper, we give explicit maximally monotone linear subspace extensions of a monotone linear relation in finite dimensional…
We present new sharp assertions concerning multipliers in various spaces of harmonic functions in the unit ball of $R^n$
The general solution of the one-dimensional stationary Schroedinger equation in the form of a formal power series is considered. Its efficiency for numerical analysis of initial value and boundary value problems is discussed.
This paper proves Strichartz estimates for the Schrodinger Equation with a potential term and white noise dispersion in dimension $1$. We also explore dispersive estimates using previous results in the field.
We discuss the explicit construction of the Schroedinger equations admitting a representation through some family of general polynomials. Almost all solvable quantum potentials are shown to be generated by this approach. Some generalization…
This article is aimed at presenting the Schr\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\"odinger problem. We also give a survey of the related literature.…
It is shown that the Schrodinger symmetry algebra of a free particle in d spatial dimensions can be embedded into a representation of the higher-spin algebra. The latter spans an infinite dimensional algebra of higher-order symmetry…
We give a survey on recent developments on nonlinear Schr\"odinger equations with dissipative structure based on the authors' recent works.
Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances…
In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin's maximum principle from optimal control theory for the Loewner equation in several complex…