Related papers: Note on Strichartz inequalities for the wave equat…
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential \[iu_{t} +\Delta u-c|x|^{-2}u=\lambda|x|^{-b} |u|^{\sigma } u,\; u(0)=u_{0} \in…
We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schr\"{o}dinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a…
We study the operator symmetric modulus (|Z|+|Z^*|)/2 for matrices Z. Several triangle type inequalities are given.
The one-particle three-dimensional Dirac equation with spherical symmetry is solved for the Hulthen potential. The s-wave relativistic energy spectrum and two-component spinor wavefunctions are obtained analytically. Conforming to the…
The potential concept that is successful in classical electrodynamics should also be applicable to the nonlinear electromagnetic forces acting on matter. The obvious method of determining these potentials should be provided by Helmholtz's…
In this paper, we study the decoherence of a wave described by the solution to a Schroedinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze…
We prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work…
For the one-dimensional case, we establish the long-time asymptotics of solution to Cauchy problem and prove existence of modified wave operators. In particular, we show that the part of the wave travels ballistically if the potential is…
We study a semilinear wave inequality with double damping on a complete noncompact Riemannian manifold. The considered problem involves a potential function $V$ depending on the space variable in front of the power nonlinearity and an…
The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the…
With using the algebraic approach Lie symmetries of Schr\"odinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, the admissible…
In this work we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically.…
Some inequalities and reverses of classic H\"{o}lder and Minkowski types are obtained for scalar Birkhoff weak integrable functions with respect to a non-additive measure.
We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in $\mathbb{R}^{3}$: \[ \partial_{tt}u-\Delta u+\sum_{j=1}^{m}V_{j}\left(x-\vec{v}_{j}t\right)u=0. \]…
The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schr\"odinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend…
Sharp extensions of Pitt's inequality and bounds for Stein-Weiss fractional integrals are obtained that incorporate gradient forms and vector-valued operators. Such results include Hardy-Rellich inequalities.
It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be…
In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schr\"{o}dinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe…
Strichartz estimates for a time-decaying harmonic oscillator were proven with some assumptions of coefficients for the time-decaying harmonic potentials. The main results of this paper are to remove these assumptions and to enable us to…
We prove Strichartz-type estimates for Schroedinger's equation with time-dependent potentials. The time derivative of the potentials need not be integrable, so the total variation of the potentials may be infinite.