We study a non-relativistic charged quantum particle moving in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of a zero-range potential. In the electrostatic case the standing wave solutions take the form $\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies $-\Delta u+\alpha\varphi u-{\beta}\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by $-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.
Ground States of the Schrodinger–Maxwell system with Dirac mass: Existence and Asymptotics
COCLITE, Giuseppe Maria;
2010-01-01
Abstract
We study a non-relativistic charged quantum particle moving in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of a zero-range potential. In the electrostatic case the standing wave solutions take the form $\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies $-\Delta u+\alpha\varphi u-{\beta}\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by $-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.File in questo prodotto:
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