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Register a new userAchieving energy permutation of modes in the Schrodinger equation with moving Dirac potentials
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In this work, we study the Schrodinger equation $ipartial_tpsi=-Deltapsi+eta(t)sum_{j=1}^Jdelta_{x=a_j(t)}psi$ on $L^2((0,1),C)$ where $eta:[0,T]longrightarrow R^+$ and $a_j:[0,T]longrightarrow (0,1)$, $j=1,...,J$. We show how to permute the energy associated to different eigenmodes of the Schrodinger equation via suitable choice of the functions $eta$ and $a_j$. To the purpose, we mime the control processes introduced in [17] for a very similar equation where the Dirac potential is replaced by a smooth approximation supported in a neighborhood of $x=a(t)$. We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.
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In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrodinger equation (NLS) with partial harmonic potential begin{equation*}tag{NLS} ipartial_t u + left(Delta_{mathbb{R}^3 }-x^2 right) u = |u|^2 u, quad u|_{t=0} = u_0. end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrodinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies.
New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.
We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
In this article, we study the decay of the solutions of Schrodinger equations in the exterior of an obstacle. The main situations we are interested in are the general case (no non-trapping assumptions) or some weakly trapping situations
We study the direct and inverse scattering problem for the one-dimensional Schrodinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and prescribed decay to their asymptotics. These results are important for solving the Korteweg-de Vries equation via the inverse scattering transform.
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