arXiv subjects

Takaaki Nakamura

Publications and source records attributed to Takaaki Nakamura.

4 recordsLinked to original sources

arXiv2025

An explicitly solvable NLS model with discontinuous standing waves

We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.

math.AP↗

arXiv2020

Discontinuous Ground States for NLSE on $\mathbb{R}$ with a Fülöp-Tsutsui $δ$ interaction

We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schrödinger equation with a focusing power nonlinearity and a defect located at the origin. In this paper a ground state is defined as a global minimizer of the action functional on the Nehari manifold and the defect considered is a Fülöp-Tsutsui $δ$ type, namely a $δ$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results from the Grillakis-Shatah-Strauss theory.

math.AP↗

arXiv2019

Generating nonequilibrium stationary state from ground state condensate through an almost-adiabatic cycle

It is shown that the ground state of weakly interacting Bose particles in a quasi one-dimensional box trap can be converted into an excited stationary state by an adiabatic cyclic operation that involves a quench of interaction strength: A sharp impurity potential is applied and its strength is varied during the cycle, which induces a nonequilibrium stationary state exhibiting the inversion of population. This process is robust in the sense that the resultant stationary state is almost independent of the details of the cycle, such as the position of the impurity as long as the cycle is far enough from critical regions. The case of the failure of the population inversion due to the strong interparticle interactions is also examined.

cond-mat.quant-gas↗

arXiv2017

Spectral properties of nonlinear Schrödinger equation on a ring

The stationary states of nonlinear Schr{ö}dinger equation on a ring with a defect is numerically analyzed. Unconventional connection conditions are imposed on the point defect, and it is shown that the system displays energy level crossings and level shifts and associated quantum holonomies in the space of system parameters, just as in the corresponding linear system. In the space of nonlinearity parameter, on the other hand, the degeneracy occurs on a line, excluding the possibility of any anholonomies. In contrast to the linear case, existence of exotic phenomena such as disappearance of energy level and foam-like structure are confirmed.

quant-ph↗