Ad Hoc Physical Hilbert Spaces in Quantum Mechanics

Abstract

The overall principles of what is now widely known as PT-symmetric quantum mechanics are listed, explained and illustrated via a few examples. In particular, models based on an elementary local interaction V(x) are discussed as motivated by the naturally emergent possibility of an efficient regularization of an otherwise unacceptable presence of a strongly singular repulsive core in the origin. The emphasis is put on the constructive aspects of the models. Besides the overall outline of the formalism we show how the low-lying energies of bound states may be found in closed form in certain dynamical regimes. Finally, once these energies are found real we explain that in spite of a manifest non-Hermiticity of the Hamiltonian the time-evolution of the system becomes unitary in a properly amended physical Hilbert space.

Appendix A: Two-parametric family of regularized singular interactions: phenomenological aspects

One of the key merits of the bound-state Schrödinger equations of the ordinary differential form

$$ \left [-\frac{d^{2}}{dx^{2}}+V(x) \right ]\,\psi_{n}(x) = E_{n}\,\psi_{n}(x) \,, \ \ \ \ n = 0, 1, {\ldots} $$

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is that they combine a broad phenomenological applicability and methodical appeal with the formal friendliness of the linear differential equations of the second order. This has been re-emphasized in Ref. [16] where a judiciously chosen next-to-harmonic toy-model potential was studied in a specific strong-repulsion dynamical regime in which g ≫ 1. In a continuation and generalization of this analysis (cf. Section 4 here) one introduces a coordinate-dependent generalization of the couplings,

$$ V(x) = V_{\alpha,\beta,g}(x) = x^{2}\,\mu_{\alpha}(x) + \frac{g^{2}\,\nu_{\beta}(x)}{x^{6}}\,. $$

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Both of the additional non-constant functions of x possess the same one-parametric power-law forms of μ _α (x) = (i x)^α, α ≥ 0 and ν _β (x) = 1/(i x)^β, β ≥ 0.

This made the shape of the potential more flexible. Moreover, one may return to the original potential via an elementary limiting transition α → 0 and β → 0. Far from this limit, on the contrary, the shape of the function(s) μ _α (x) and ν _β (x) may be adapted more easily to phenomenological needs.

The list of formal reasons for our choice further incorporates also the quasi-solvable nature of similar forces (cf. the fifth item in Table Nr. 1 of Ref. [22]), i.e., the feature which was made popular in monograph [23]) or the tractability of at least some of the related eigenvalue problems using continued fractions [24] or a specific simplicity of the asymptotic estimates of wave functions [25].

It makes sense to add that the studies of non-Hermitian but real-spectra-exhibiting quantum models may be perceived as one of the most dynamical branches of development of quantum theory after 1998 (see, e.g., reviews [14, 15, 26]). One of the fairly productive subbranches of these efforts was devoted to the mathematical idea (which may be dated back to the early nineties [5, 27]) that the spectrum of bound states may be in fact controlled and modified by the mere ad hoc redefinition of the integration path of $x \in \mathcal {S} \subset \mathbb {C}$ (cf., e.g., [28]).

A consequent further extension of the latter mathematical idea (related closely to the presence of the strong singularities in V(x) but getting us already beyond the limits of our present considerations) may be based on the question of what happens when the localization of the underlying integration path $\mathcal {S}$ is allowed to leave the plain complex plane (endowed, possibly, with a cut oriented upwards). In this direction it has been proposed [29] that in the definition of the integration path $\mathcal {S}$ one may and should try to replace the (cut) complex plane $\mathbb {C}$ by a more general Riemann surface $\mathcal {R}$. In the latter scenario (cf. also [30, 31]) one treats the general Riemann surface $\mathcal {R}$ as composed, in usual manner, of a set of individual Riemann-sheet cut planes, $\mathcal {R}= \bigcup _{j} \mathcal {R}_{j}$ where $\mathcal {R}_{j} \sim \mathbb {C}$. Then the path $\mathcal {S}$ of integration may and should encircle the branch-point singularities of $\mathcal {R}$, giving rise to several alternative, non-equivalent quantum systems living on the respective “tobogganic” complex curves. Thus, every such a system is described not only by the ordinary differential equation but also by the topologically nontrivial tobogganic path $\mathcal {S}$ (connecting, in general, several individual Riemann sheets) and, in addition, by a suitable definition of inner product in the underlying sophisticated physical Hilbert space $\mathcal {H}^{(S)}$ (cf. [30] for a deeper discussion of the latter point in tobogganic context).