×

Approximation of a general class of quantum-mechanical one-dimensional point interactions by local self-adjoint interactions. (English) Zbl 0943.81010

Bainov, D. (ed.), Proceedings of the 9th international colloquium on differential equations, Plovdiv, Bulgaria, August 18-23, 1998. Utrecht: VSP. 393-400 (1999).
The paper deals with a general class of quantum-mechanical point interactions in one dimension. For systems with time-reversal symmetry, these is a three- parameter family of solutions characterized by the boundary condition at the position of the point interaction \[ \left(\begin{matrix} \psi' (\eta)\cr \psi(\eta) \end{matrix} \right) = \left( \begin{matrix} -\alpha & -\beta\cr -\delta & -\gamma \end{matrix} \right) \left(\begin{matrix} \psi' (-\eta)\cr \psi(-\eta) \end{matrix} \right), \tag{1} \] where \(\alpha, \beta, \gamma, \delta \in\mathbb{R}\) such that \(\alpha\gamma-\beta\delta = 1\) and \(\eta\) is a positive infinitesimal quantity. The authors show that all the boundary conditions in equation (1) are realized in the zero-range limit of local self-adjoint interactions.
For the entire collection see [Zbl 0914.00066].

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDFBibTeX XMLCite