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Critical nonlinear Schrödinger equations with and without harmonic potential. (English) Zbl 1029.35208

Summary: Bose-Einstein condensation is usually modeled by a nonlinear Schrödinger equation with harmonic potential \[ i\hbar \partial_t \psi^\hbar+ {\hbar^2\over 2m}\Delta \psi^\hbar= {m\over 2}\omega^2 x^2\psi^\hbar +{4\pi \hbar^2a\over m}|\psi^\hbar |^2\psi^\hbar, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^n, \] where \(\omega>0\) and \(a\) is the scattering length, whose sign differs according to the chemical element considered.
We use a change of variables that turns the critical nonlinear Schrödinger equation into the critical nonlinear Schrödinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on \(L^2\), and bijective on some time intervals. Using the known results for the critical nonlinear Schrödinger equation, this provides information for the properties of Bose-Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B33 Critical exponents in context of PDEs
35B35 Stability in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
82D99 Applications of statistical mechanics to specific types of physical systems

References:

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