It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions with PT-symmetric point interactions, the usual situation is that the C operator is unbounded. The fact that the C operator is unbounded is significant because, while there is a formal equivalence between a PT-symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is so because the mapping from one Hilbert space to the other is unbounded. This shows that PT-symmetric quantum theories are mathematically distinct from conventional Hermitian quantum theories.

http://arxiv.org/abs/1207.1176
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

The S-matrices corresponding to PT-symmetric \(\rho\)-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory.

http://arxiv.org/abs/1203.2110
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Generalized PT-symmetric operators acting an a Hilbert space $\mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scattering methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.

http://arxiv.org/abs/1202.1537
Mathematical Physics (math-ph); 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)