Non-self-adjoint Schrodinger operators A which correspond to non-symmetric zero-range potentials are investigated. For a given A, the description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of A as a self-adjoint operator in a Krein space is studied, the problem of similarity of A to a self-adjoint operator in a Hilbert space is solved.

http://arxiv.org/abs/1309.5482
Mathematical Physics (math-ph); Spectral Theory (math.SP); Quantum Physics (quant-ph)

Let \(J\) and \(R\) be anti-commuting fundamental symmetries in a Hilbert space \(\mathfrak{H}\). The operators \(J\) and \(R\) can be interpreted as basis (generating) elements of the complex Clifford algebra \({\mathcal C}l_2(J,R):={span}\{I, J, R, iJR\}\). An arbitrary non-trivial fundamental symmetry from \({\mathcal C}l_2(J,R)\) is determined by the formula \(J_{\vec{\alpha}}=\alpha_{1}J+\alpha_{2}R+\alpha_{3}iJR\), where \({\vec{\alpha}}\in\mathbb{S}^2\).  Let \(S\) be a symmetric operator that commutes with \({\mathcal C}l_2(J,R)\). The purpose of this paper is to study the sets \(\Sigma_{{J_{\vec{\alpha}}}}\) (\(\forall{\vec{\alpha}}\in\mathbb{S}^2)\) of self-adjoint extensions of \(S\) in Krein spaces generated by fundamental symmetries \({{J_{\vec{\alpha}}}}\) (\({{J_{\vec{\alpha}}}}\)-self-adjoint extensions). We show that the sets \(\Sigma_{{J_{\vec{\alpha}}}}\) and \(\Sigma_{{J_{\vec{\beta}}}}\) are unitarily equivalent for different \({\vec{\alpha}}, {\vec{\beta}}\in\mathbb{S}^2\) and describe in detail the structure of operators \(A\in\Sigma_{{J_{\vec{\alpha}}}}\) with empty resolvent set.

http://arxiv.org/abs/1105.2969
Functional Analysis (math.FA); Mathematical Physics (math-ph)