Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model

In quantum mechanics (formulated, say, in Schr"{o}dinger picture) only the knowledge of a complete set of observables ${\mathrm{\Lambda }}_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $\mathrm{\Theta }$ such that ${\mathrm{\Lambda }}_j^{†}\mathrm{\Theta }=\mathrm{\Theta }{\mathrm{\Lambda }}_j$, i.e., such that $\mathrm{\Theta }=\mathrm{\Theta }({\mathrm{\Lambda }}_j)$) unique. In many applications people simplify the model and consider just a single input observable (mostly an energy-representing Hamiltonian ${\mathrm{\Lambda }}_1=H$) and pick up, out of all of the eligible metrics $\mathrm{\Theta }=\mathrm{\Theta }(H)$, just the simplest candidate (typically, in the case of the special self-adjoint input $H$ we virtually always work with trivial $\mathrm{\Theta }=I$). As long as this forces us to admit only the self-adjoint forms of any other input observable ${\mathrm{\Lambda }}_j$, 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 $\mathrm{\Theta }\mathrm{\ne }I$ and in which the system reaches a collapse (i.e., a loss-of-bservability catastrophe) via unitary evolution.