A New Class of Point Interactions in One Dimension

✍ Scribed by P.R. Chernoff; R.J. Hughes

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 614 KB
• Volume: 111
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1993.1006

✦ Synopsis

We present a class of self-adjoint extensions of the symmetric operator (\left.-A \mid C_{0}^{\prime}\left(\mathbb{H}^{\prime} \backslash 0\right}\right)) which correspond formally to perturbations of the Laplacian by pseudopotentials involving (\boldsymbol{d}^{2}). These operators, which provide new examples of generalized point interactions in the sense of Seba, are defined by the boundary conditions (f\left(0^{+}\right)=e=f\left(0^{-}\right), r f\left(0^{+}\right)+f^{\prime}\left(0^{+}\right)=e^{z}\left[r f\left(0^{-}\right)+f^{\prime}(0)\right]), for (z \in \mathbb{C}), (r \in \mathbb{R}). We calculate their spectra, resolvents, and scattering matrices, and show that they can be realized as limits of Schrödinger operators with local short-range potentials. 1993 Academic Press, Inc.