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Confinement in the framework of Schrödinger equations: a revision of the classical "particle in a box" example and further free boundary results.

Díaz, I (Universidad Complutense de Madrid)
Thursday 22 May 2014, 14:00-15:00

Seminar Room 2, Newton Institute Gatehouse


One of the main modifications to the Classical Mechanics introduced by Quantum Mechanics is the impossibility to localize the state (position and velocity) in the dynamics of a particle (Heisenberg Principle). This fact is connected with the study of the support of solutions of the associated Schrödinger equation. Some of the more popular simplifications for the linear Schrödinger equation (attributed by him, in 1935, to George Gamow [1904-1968] and repeated in any text book in Quantum Mechanics) deals with the case of the stationary eigenvalue problem associated to several discontinuous potentials $V(x)$, which, among other things allows to illustrate the tunneling effect. Nevertheless, surprisingly enough, it seems that it was nor observed before in the literature (lecture by this author at Tours, 2012) that the confinement argument used in the case of "the infinite well potential" ($V(x)=V_{0}$ if $x\in \lbrack 0,L]$ and $ V(x)=+\infty $ if $x\notin \lbrack 0,L]$) leads to a serious mathematical mistake: the usual "popular" solution does not satisfy the global Schrödinger equation in $\mathbb{R}$ since a Dirac delta is generated at each point of the boundary of the box. A first goal of the lecture is to present a different confinement argument which requires the study of bifurcation diagrams associated to problems of the type \begin{equation*} \left\{ \begin{array}{lr} -\dfrac{d^{2}u}{dx^{2}}+V(x)u=\lambda u & \hbox{in }(0,L), \\ u=0 & \text{on }\partial (0,L), \end{array} \right. \end{equation*} $V(x)=\frac{V_{0}}{\left\vert u(x)\right\vert ^{1-m}}$, when $m\in \lbrack 0,1).$ By a suitable application of the "bifurcation from the infinity" method (Rabinowitz 1973) it is possible to show the existence of a numerable set of branches emanating (from the infinity) from the eigenvalue subspaces of the linear problem. Moreover, the exact multiplicity can be given by extending some previous joint results with J. Hernández, which leads to a complete description of each branch. In particular, in each branch, there exists a suitable value $\lambda ^{\ast }$ of the energy (the parameter $ \lambda $) such that if $\lambda \geq \lambda ^{\ast }$ then the solutions satisfy that $\dfrac{du}{dx}(0)=\dfrac{du}{dx}(L)=0$ and so the confinement argument does not develop any singularity on the boundary of the box. In a second part of the lecture, I will make mention (very briefly) to the question of the confinement when it is studied, by other type of methods (integral enery methods), for the Schrödinger equation with a singular nonlinear potential of the type \begin{equation*} i\frac{\partial u}{\partial t}+\Delta u=a|u|^{-(1-m)}u,\mbox{ in }(0,\infty )\times \mathbb{R}^{N}, \end{equation*} with $a\in C $ and $ 0<m<1 $ (a series of joint works with P. Bégout).


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