Quantum catastrophes II. Generic pattern of the fall into instability

Miloslav Znojil

The quantum-catastrophe (QC) benchmark Hamiltonians of paper I (M. Znojil, J. Phys. A: Math. Theor. 45 (2012) 444036) are reconsidered, with the infinitesimal QC distance \(\lambda\) replaced by the total time $\tau$ of the fall into the singularity. Our amended model becomes unique, describing the complete QC history as initiated by a Hermitian and diagonalized N-level oscillator Hamiltonian at \(\tau=0\). In the limit \(\tau \to 1\) the system finally collapses into the completely (i.e., N-times) degenerate QC state. The closed and compact Hilbert-space metrics are then calculated and displayed up to N=7. The phenomenon of the QC collapse is finally attributed to the manifest time-dependence of the Hilbert space and, in particular, to the emergence and to the growth of its anisotropy. A quantitative measure of such a time-dependent anisotropy is found in the spread of the N-plet of the eigenvalues of the metric. Unexpectedly, the model appears exactly solvable — at any multiplicity N, the N-plet of these eigenvalues is obtained in closed form.

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

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