Integrability, Supersymmetry and Coherent States pp 411-426 | Cite as

# On Some Aspects of Unitary Evolution Generated by Non-Hermitian Hamiltonians

## Abstract

The possibility of nontrivial quantum-catastrophic effects caused by the mere growth of the imaginary component of a non-Hermitian but \({\mathcal {P}\mathcal {T}}\)-symmetric *ad hoc* local-interaction potential *V* (*x*) is revealed and demonstrated. Via a replacement of coordinate \(x \in \mathbb {R}\) by a non-equidistant discrete lattice *x*_{n} with *n* = 0, 1, …, *N* + 1 the model is made exactly solvable at all *N*. By construction, the energy spectrum shrinks with the growth of the imaginary strength. The boundary of the unitarity of the model is reached in a certain strong non-Hermiticity limit. The loss-of-stability instant is identified with the Kato’s exceptional point of order *N* at which the model exhibits a complete *N*-state degeneracy. This phase-transition effect is accessible as a result of a unitary-evolution process in an amended physical Hilbert space.

## Keywords

Quantum systems Unitary evolution Three-Hilbert-space representation of states Non-Hermitian observables Quantum phase transitions Quantum catastrophes Exactly solvable model## References

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