Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model
Miloslav Znojil
Abstract
In quantum mechanics (formulated, say, in Schrödinger picture) only the knowledge of a complete set of observables $Λ_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $Θ$ such that $Λ_j^\dagger Θ=Θ\,Λ_j$, i.e., such that $Θ=Θ(Λ_j)$) unique. In many applications people simplify the model and consider just a single input observable (mostly an energy-representing Hamiltonian $Λ_1=H$) and pick up, out of all of the eligible metrics $Θ=Θ(H)$, just the simplest candidate (typically, in the case of the special self-adjoint input $H$ we virtually always work with trivial $Θ=I$). As long as this forces us to admit only the self-adjoint forms of any other input observable $Λ_j$, the scope of the theory is, without any truly meaningful phenomenological reason, restricted. In our present paper we describe a strictly non-numerical $N$ by $N$ matrix model in which such a restriction is replaced by another, phenomenologically non-equivalent restriction in which $Θ\neq I$ and in which the system reaches a collapse (i.e., a loss-of-bservability catastrophe) via unitary evolution.