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.

Work supported by the GAČR Grant No. 16-22945S.

Appendix: The Metric as a Degree of Model-Building Freedom

The specification of quantum system ${\mathcal {S}}$ requires not only the knowledge of its Hamiltonian H ^(N)(z) [i.e., at any preselected dimension N and parameter z, the knowledge of matrix (20) in our case] but also a constructive access to the correct probabilistic interpretation of experiments. In other words, having solved the time-dependent Schrödinger equation (2) we still need to replace our manifestly unphysical working Hilbert space ${\mathcal {H}}^{{\mathrm{(auxiliary)}}}$ by the correct physical Hilbert space, i.e., we must modify the inner product accordingly [3].

The Abstract Theory Revisited

In the context of quantum theory of many-body systems it was Freeman Dyson [8] who conjectured that in some cases, an enormous simplification of the variational determination of the bound-state spectra could be achieved via a suitable non-unitary similarity transformation of the given realistic Hamiltonians

$$\displaystyle \begin{aligned} \mathfrak{h} \ \to \ H = \varOmega^{-1}\mathfrak{h} \varOmega\,,\ \ \ \ \varOmega^\dagger\varOmega \neq I. {} \end{aligned} $$

(25)

The trick proved particularly efficient in nuclear physics [9]. An amendment of the calculations has been achieved via a judicious choice of the operators Ω converting, e.g., the strongly correlated pairs of nucleons into weakly interacting effective bosons.

In spite of the initial success, the trial-and-error nature of the Dyson-inspired recipes and the fairly high formal mathematical costs of the replacement of the self-adjoint “realistic” operator $\mathfrak {h}=\mathfrak {h}^\dagger$ by its manifestly non-Hermitian, quasi-Hermitian [9] alternative

$$\displaystyle \begin{aligned} H = \varTheta^{-1} H^\dagger \varTheta \neq H^\dagger\,, \ \ \ \ \ \ \varTheta=\varOmega^\dagger\varOmega {} \end{aligned} $$

(26)

have been found, beyond the domain of nuclear physics, strongly discouraging (cf., e.g., [25]).

Undoubtedly, the idea itself is sound. In the context of abstract quantum theory its appeal has been rediscovered by Bender with Boettcher [4]. In effect, these authors just inverted the arrow in Eq. (25). They conjectured that one might start a model-building process directly from Eq. (26), i.e., directly from a suitable trial-and-error choice of a sufficiently simple non-Hermitian Hamiltonian with real spectrum. Their conjecture was illustrated by the family of perturbed imaginary cubic oscillator Hamiltonians

$$\displaystyle \begin{aligned} H_{\epsilon}\kern-1pt=\kern-1pt-\frac{d^2}{dx^2} \kern-1pt +\kern-1pt V_{\epsilon}(x) \neq H^\dagger_{\epsilon}\,, \ \ \ \ V_{\epsilon}(x) \kern-1pt=\kern-1pt {\mathrm{i}}x^3 ({\mathrm{i}}x)^\epsilon \,, \ \ \ \ x \kern-1pt\in\kern-1pt (-\infty,\infty), \ \ \ \ \epsilon \in (-1,1). {} \end{aligned} $$

(27)

Technical details may be found in reviews [2, 3, 9, 18] in which several formulations of the “inverted” stationary version of the quantum model-building strategy may be found.

The Unitarity of Evolution Reestablished

It is worth adding that strictly speaking, the latter strategies are not always equivalent (cf. also further comments in [22, 32]). For our present purposes we may distinguish between the older, more restrictive “quasi-Hermitian” formulation of quantum mechanics (QHQM) of Ref. [9], and the “${\mathcal {P}\mathcal {T}}$-symmetric” version of quantum mechanics (PTQM, [2]).

The key difference between the latter two pictures of quantum reality lies in the strictly required non-admissibility of the unbounded Hamiltonians in the QHQM framework of Ref. [9]. This requirement is by far not only formal, and it also makes the QHQM theory mathematically better understood. In contrast, the process of the rigorous mathematical foundation of the extended, phenomenologically more ambitious PTQM theory (admitting the unbounded Hamiltonians as sampled by Eq. (27)) is still unfinished (cf., e.g., the concise progress reports [23, 33]). Hence, also the toy models with the local but not real potentials are far from being widely accepted as fully understood and consistent at present (cf., e.g., [20, 21]).

One is forced to conclude that the ordinary differential (but, unfortunately, unbounded) benchmark model (27) of a ${\mathcal {P}\mathcal {T}}$-symmetric quantum system (where ${\mathcal {P}}$ means parity, while symbol ${\mathcal {T}}$ denotes the time reversal [4]) is far from satisfactory. At the same time, its strength may be seen in its methodical impact as well as in its simplicity and intuitive appeal. For all of these reasons one is forced to search for alternative ${\mathcal {P}\mathcal {T}}$-symmetric quantum models which share the merits while not suffering of the inconsistencies.

Needless to add that the unitarity of the quantum evolution can be reestablished for many non-Hermitian models with real spectra. One just has to return to the standard quantum theory in QHQM formulation. The details of the implementation of the idea may vary. Thus, Bender [2] works with an auxiliary nonlinear requirement $H {\mathcal {P}\mathcal {T}}={\mathcal {P}\mathcal {T}}H$ called “${\mathcal {P}\mathcal {T}}$-symmetry of H.” In a slightly more general setting Mostafazadeh [3] makes use of the same relation written in the equivalent form $H^\dagger {\mathcal {P}}={\mathcal {P}}H$, and he calls it “${\mathcal {P}}$-pseudo-Hermiticity of H.” Still, both of these authors respect the Stone theorem. This means that both of them introduce the correct physical Hilbert-space metric Θ and both of them use it in the postulate

$$\displaystyle \begin{aligned} H =\varTheta^{-1}H^\dagger \varTheta := H^\ddagger. {} \end{aligned} $$

(28)

of the so-called quasi-Hermiticity property of the acceptable Hamiltonians. Rewritten in the form

$$\displaystyle \begin{aligned} H^\dagger \varTheta =\varTheta\,H {} \end{aligned} $$

(29)

the equation can be interpreted as a linear-algebraic system which defines, for a given Hamiltonian matrix H with real spectrum, the N-parametric family of all of the eligible matrices of metric Θ. For the tridiagonal input matrices H, the solution is particularly straightforward because the algorithm can be given a recurrent form implying that the solutions exist at any input H [34].