| Summary: | A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability <i>alias</i> quantum phase transition is studied. A specific solvable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo mathvariant="normal">−</mo></mrow></semantics></math></inline-formula>state model is considered, characterized by a non-stationary non-Hermitian Hamiltonian. Our analysis uses quantum mechanics formulated in Schrödinger picture in which, in principle, only the knowledge of a complete set of observables (i.e., operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">Λ</mi><mi>j</mi></msub></semantics></math></inline-formula>) enables one to guarantee the uniqueness of the related physical Hilbert space (i.e., of its inner-product metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula>). Nevertheless, for the sake of simplicity, we only assume the knowledge of just a single input observable (viz., of the energy-representing Hamiltonian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>≡</mo><msub><mi mathvariant="normal">Λ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>). Then, out of all of the eligible and Hamiltonian-dependent “Hermitizing” inner-product metrics <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo>=</mo><mi mathvariant="normal">Θ</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula>, we pick up just the simplest possible candidate. Naturally, this slightly restricts the scope of the theory, but in our present model, such a restriction is more than compensated for by the possibility of an alternative, phenomenologically better motivated constraint by which the time-dependence of the metric is required to be smooth. This opens a new model-building freedom which, in fact, enables us to force the system to reach the collapse, i.e., a genuine quantum catastrophe as a result of the mere conventional, strictly unitary evolution.
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