Unbounded C-symmetries and their nonuniqueness

TL;DR

This paper demonstrates that for PT-symmetric Hamiltonians with non-unique C operators, the C operator is unbounded, highlighting fundamental differences between PT-symmetric and conventional Hermitian quantum theories.

Contribution

It establishes the unboundedness of the C operator in most cases, except for specific finite-matrix or differential expression Hamiltonians, revealing the non-equivalence of the associated Hilbert spaces.

Findings

Non-uniqueness of C operator implies it is unbounded.

Unbounded C operators lead to inequivalent Hilbert spaces.

PT-symmetric quantum theories differ fundamentally from Hermitian theories.

Abstract

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.