On energy gaps in a new type of analytically solvable model in quantum mechanics
The authors study spectral properties of one-dimensional Schrödinger Hamiltonians of the form \[ (*)\quad H_{\{\beta_ j\}}=-d^ 2/dx^ 2+\sum_{j\in {\mathbb{Z}}} \beta_ jV_ j \] where \(V_ j\) is Dirac's delta function/derivative of Dirac's delta function at a point \(y_ j\in {\mathbb{R}}\) for all \(j\in {\mathbb{Z}}.\) In particular various generalizations of the Saxon-Hutner conjecture are proved: Let \(\{y_ j\}=a{\mathbb{Z}}\), and let \(H_{\beta}\) denote the operator (*) with \(\beta_ j=\beta\), \(j\in {\mathbb{Z}}\). Then it is e.g. proved that if (a,b)\(\subset \cap_{\beta \in \{\beta_ j\}} \rho (H_{\beta})\) then \((a,b)\subset \rho (H_{\{\beta_ j\}})\) where \(\rho\) (H) denotes the resolvent set of an operator H. Finally the authors analyze the case when the \(\beta_ j\) are random variables. Assume the \(\beta_ j's\) are independent identically distributed random variables with support in [0,\(\infty)\). Let \(\gamma\) denote the inf of the distribution of \(\beta_ j.\) Then \(\Sigma =\sigma (H_{\gamma})=\cup_{m\in {\mathbb{Z}}} [a_ m,b_ m]\) where \(\Sigma =\alpha (H_{\{\beta_ j\}})\) almost surely. For more extensive treatment of these operators we refer to the book of \textit{S. Albeverio}, \textit{F. Gesztesy}, R. Høegh-Krohn, \textit{H. Holden} [Solvable Models in Quantum Mechanics, Springer-Verlag (1988)].
spectral properties of one-dimensional Schrödinger Hamiltonians
independent identically distributed random variables
scientific article; zbMATH DE number 4096178
scientific article