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

A Unitary Way Towards Quantum Collapse
  • Miloslav ZnojilEmail author
Part of the CRM Series in Mathematical Physics book series (CRM)


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 xn 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.


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


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.NPI ASCRŘežCzech Republic

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