New Concept of Solvability in Quantum Mechanics

In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of Hamiltonian) requires a generalization allowing for a non-numerical form of metric. A few illustrative solvable quantum models of this type are presented.


Introduction
In our recent paper [1] it has been noticed that contrary to the current belief, an active use of an ad hoc variability of the inner products (i.e., in other words, of the freedom of choosing a nontrivial metric Θ = I in the 1 e-mail: znojil@ujf.cas.cz correct physical Hilbert space of quantum states H (S) where the superscript (S) stands for "standard") is not restricted to the so called non-Hermitian quantum mechanics and to its characteristic applications in nuclear physics [2] or in molecular physics [3] or in the relativistic quantum kinematical regime [4] or in the PT −symmetric quantum dynamical regime [5]. In our present paper we intend to develop this idea and to describe some of its consequences in some detail.
From the point of view of the recent history of quantum mechanics it was, certainly, fortunate that in some of the above-mentioned specific hidden-Hermiticity contexts people discovered the advantages of working with such an operator representation H of a given observable quantity (say, of the energy) which only proved Hermitian after a change of the inner product in the initially ill-chosen (i.e., by assumption, unphysical) Hilbert space H (F ) (the superscript (F ) might be read here as abbreviating "former", "first", "friendly" or, equally well, "false" [6]). Let us emphasize that the shared motivation of many of the above-cited papers speaking about non-Hermitian quantum mechanics resulted just from the observation that several phenomenologically interesting operators (say, Hamiltonians) H appear manifestly non-Hermitian in the "usual" textbook setting and that they only become Hermitian in some much less common representation of the Hilbert space of states.
The amendments of space were, naturally, mediated by the mere introduction of a non-trivial metric Θ = Θ (S) = I entering the upgraded, S−superscripted inner products, Such an inner-product modification changed, strictly speaking, the Hilbert space, H (F ) → H (S) . This had several independent reasons. Besides the formal necessity of re-installing the unitarity of the evolution law, the costs of the transition to the more complicated metric were found more than compensated by the gains due to the persuasive simplicity of Hamiltonian (cf. [2] or [5] in this respect). Moreover, for some quantum systems the transition F → S may prove motivated by physics itself. The most elementary illustration of such a fundamental reason can be found in our recent study [7] where a consistent simulation of the cosmological phenomenon of quantum Big Bang has been described. In the model the Hamiltonian remained self- For an entirely general quantum system characterized by two observables H and Q, Hermitian or not, a fully universal scenario may be found displayed in Fig. 1 an instructive sample of the time evolution may be chosen as generated by the Hamiltonian (i.e., quantum energy operator or matrix) of Ref. [8], Its eigenvalues E (2) ± = ± √ 1 − λ 2 are non-degenerate and real (i.e., in principle, observable) for λ inside interval (−1, 1). On the two-point domain boundary {−1, 1}, these energies degenerate. Subsequently, they complexify whenever |λ| > 1. In the current literature one calls the boundary points λ = ±1 "exceptional points" (EP, [9]). At these points the eigenvalues degenerate and our toy-model Hamiltonian ceases to be diagonalizable, becoming unitarily equivalent to a triangular Jordan-block matrix, At |λ| > 1, the diagonalizability gets restored but the eigenvalues cease to be real, E In the spirit of current textbooks, this leaves these purely imaginary complex conjugate energies unobservable.

Hidden Hermiticity: The set of all eligible metrics
Our matrix H (2) (λ) remains diagonalizable and crypto-Hermitian whenever −1 < λ = sin α < 1, i.e., for the auxiliary Hamiltonian-determining parameter α lying inside a well-defined physical domain D H such that α ∈ (−π/2, π/2). In such a setting, matrix H (2) (λ) becomes tractable as a Hamil-tonian of a hypothetical quantum system whenever it satisfies the abovementioned hidden Hermiticity condition The suitable candidates for the Hilbert-space metric are all easily found from the latter linear equation, All of their eigenvalues must be real and positive, This is satisfied for any positive σ = a + d > 0 and with any real δ = a − d such that Without loss of generality we may set σ = 2, put δ = cos α cos β and treat the second free parameter β ∈ (−π/2, π/2) as numbering the admissible metrics with eigenvalues Thus, all of the eligible physical Hilbert spaces are numbered by two parameters, H (S) = H (S) (α, β).

The second observable Q = Q ‡
What we now need is the specification of the domain D Q . For the general four-parametric real-matrix ansatz the assumption of observability implies that the eigenvalues must be both real and non-degenerate, Once we shift the origin and rescale the units we may set, without loss of generality, w = −z = −1. This simplifies the latter condition yielding our final untilded two-parametric ansatz At any fixed metric Θ (2) (physical) the crypto-Hermiticity constraint (3) imposed upon matrix (11) degenerates to the single relation The sum s = x + y may be now treated as the single free real variable which numbers the eligible second observables. The range of this variable should comply with the inequality in Eq. (11). After some straightforward additional calculations one proves that the physical values of our last free parameter remain unrestricted, s ∈ R, due to the validity of Eq. (12). We may conclude that our example is fully non-numerical. It also offers the simplest nontrivial explicit illustration of the generic pattern as displayed in 3 Hilbert spaces H (F ) of dimension N

Anharmonic Hamiltonians
During the developments of mathematics for quantum theory, one of the most natural paths of research started from the exactly solvable harmonicoscillator potential V (HO) (x) = ω 2 x 2 and from its power-law perturbations Perturbation expansions of energies proved available even at the "unusual", complex values of the coupling constants g / ∈ R + .
The particularly interesting mathematical results have been obtained at m = 3 and at m = 4. In physics and, in particular, in quantum field theory the climax of the story came with the letter [10] where, under suitable ad hoc boundary conditions and constraints upon g = g(m) (called, conveniently, PT −symmetry), the robust reality (i.e., in principle, observability) of the spectrum has been achieved at any real exponent m > 2 even for certain unusual, complex values of the coupling.
It has been long believed that the PT −symmetric Hamiltonians H = H(m) with real spectra are all consistent with the postulates of quantum theory, i.e., that these operators are crypto-Hermitian, i.e., Hermitian in the respective Hamiltonian-adapted Hilbert spaces H (S) (m) [5]. Due to the ill-behaved nature of the wave functions at high excitations, unfortunately, such a simple-minded physical interpretation of these models has been shown contradictory [11]. On these grounds one has to develop some more robust approaches to the theory for similar models in the nearest future.
In our present paper we shall avoid such a danger by recalling the original philosophy of Scholtz et al [2]. They simplified the mathematics by admitting, from the very beginning, that just the bounded-operator and/or discrete forms of the eligible anharmonic-type toy-model Hamiltonians H = H † should be considered.

Discrete Hamiltonians
For our present illustrative purposes we intend to recall, first of all, one of the most elementary versions of certain general, N−dimensional matrix analogues of the differential toy-model Hamiltonians, which were proposed in Refs. [8]. Referring to the details as described in that paper, let us merely This parametrization proved fortunate in the sense that it enabled us to replace the usual numerical analysis by a rigorous computer-assisted algebra.
In this sense, the model in question appeared to represent a sort of an exactly solvable model, precisely in the spirit of our present message.
The new parameter t ≥ 0 is auxiliary and redundant. It may be interpreted, say, as a measure of distance of the system from the boundary ∂D H of the domain of spectral reality. At very small t the local part of boundary ∂D H has been shown to have the most elementary form of two parallel hyperplanes in the J−dimensional space of parameters G n [12].
In the simplest nontrivial special case of N = 2 the present Hamiltonian  [14] and of the operators which are self-adjoint in the so called Krein spaces with indefinite metric [15] but also to the emergence of manageable non-Hermitian models in quantum field theory [16] or even in classical optics [17], etc.
After a restriction of attention to quantum theory, the key problem emerges in connection with the ambiguity of the assignment H → Θ(H) of the physical Hilbert space H (S) to a given generator H of time evolution. For many phenomenologically relevant Hamiltonians H it appeared almost prohibitively difficult to define and construct at least some of the eligible metrics Θ = Θ(H) in an at least approximate form (cf., e.g., Ref. [18] in this respect). Clearly, in methodical analyses the opportunity becomes wide open to finite-dimensional and solvable toy models. 4.1 Solvable quantum models with more than one observable Let us restrict the scope of this paper to the quantum systems which are described by a Hamiltonian H = H(λ) accompanied by a single other operator Q = Q(̺) representing a complementary measurable quantity like, e.g., angular momentum or coordinate. In general we shall assume that symbols λ and ̺ represent multiplets of coupling strengths or of any other parameters with an immediate phenomenological or purely mathematical significance.
We shall also solely work here with the finite-dimensional matrix versions of our operators of observables.
In such a framework it becomes much less difficult to analyze one of the most characteristic generic features of crypto-Hermitian models which lies in their "fragility", i.e., in their stability up to the point of a sudden collapse.
Mathematically, we saw that the change of the stability/instability status of the model is attributed to the presence of the exceptional-point horizons in the parametric space. In the context of phenomenology, people often speak about the phenomenon of quantum phase transition [17].
Let us now return to Fig. 1 where the set of the phase-transition points pertaining to the Hamiltonian H is depicted as a schematic circular boundary ∂D H of the left lower domain inside which the spectrum of H is assumed, for the sake of simplicity, non-degenerate and completely real. Similarly, the right lower disc or domain D Q is assigned to the second observable Q.

Quantum observability paradoxes
One of the most exciting features of all of the above-mentioned models may be seen in their ability of connecting the stable and unstable dynamical regimes, within the same formal framework, as a move out of the domain D though one of its boundaries. In this sense, the exact solvability of the N < ∞ toy models proves crucial since the knowledge of the boundary ∂D Θ remains practically inaccessible in the majority of their N = ∞ differential-operator alternatives [18].
In the current literature on the non-Hermitian representations of observables, people most often discuss just the systems with a single relevant observable H(λ) treated, most often, as the Hamiltonian. In such a next-to-trivial scenario it is sufficient to require that operator H remains diagonalizable and that it possesses a non-degenerate real spectrum. Once we add another observable Q into considerations, the latter conditions merely specify the interior of the leftmost domain D H of our diagram Fig. 1.
One may immediately conclude that the physical predictions provided by the Hamiltonian alone (and specifying the physical domain of stability as an overlap between D H and the remaining upper disc or domain D Θ ) remain heavily non-unique in general. According to Scholtz et al [2] it is virtually obligatory to take into account at least one other physical observable Q = Q(̺), therefore.
In opposite direction, even the use of a single additional observable Q without any free parameters may prove sufficient for an exhaustive elimination of all of the ambiguities in certain models [5]. One can conclude that the analysis of the consequences of the presence of the single additional operator Q = Q(̺) deserves a careful attention. At the same time, without the exact solvability of the models, some of their most important merits (like, e.g., the reliable control and insight in the processes of the phase transitions) might happen to be inadvertently lost. of our matrices H (2) (λ) in their three by three extension In a way discussed in more detail in our older paper [20], The shape of this line is shown in Fig. 2.
As a final result we obtain the formula for Thus, we may denote Θ = Θ (3) (a, f, m) and conclude that the metric is obtainable in closed form so that our extended, N = 3 quantum system remains also solvable.
If we also wish to determine the critical boundaries ∂D Θ of the related metric-positivity domain D Θ , the available Cardano's closed formulae for the corresponding three eigenvalues θ j yield just the correct answer in a practically useless form. Thus, we either have to recall the available though still rather complicated algebraic boundary-localization formulae of Ref. [20] or, alternatively, we may simplify the discussion by the brute-force numerical localization of a sufficiently large metric-supporting subdomain in the para-  The PT −symmetric and tridiagonal nine-by-nine-matrix Hamiltonian H (9) of Ref. [8] reads In the limit α → 0 it splits into a central one-dimensional submatrix with eigenvalue 0 and a pair of non-trivial four-by-four sub-Hamiltonians H (4) .
At α = 0 the special and easily seen feature of the latter operator (i.e.,

Boundary ∂D H
The t−independent level E 4 = 0 is a schematic substitute for a generic environment. Each of the two remaining subsystems remains coupled to this environment by the coupling or matrix element α. We shall choose its value as proportional to t via a not too large real coupling constant β, α = β t. The β = 1 results are sampled in Fig. 6. Inside the physical domain of t < 0, qualitatively the same pattern is still obtained even at the perceivably larger β = 2.73 (cf. Fig. 7). Once we are now getting very close to the critical value of β ≈ 2.738, the situation becomes unstable. In the unphysical domain of t > 0, for example, we can spot an anomalous partial de-complexification of the energies at certain positive values of parameter t.  At cca β ≈ 2.738 the two separate EP instants of the degeneracy and complexification/decomplexification of the energies fuse themselves. Subsequently, a qualitatively new pattern emerges. Its graphical sample is given in Fig. 8. First of all, the original multiple EP collapse gets decoupled. This implies that at β = 2.75 as used in the latter picture, the inner two levels degenerate and complexify at a certain small but safely negative t = t crit ≈ −0.004.
Due to the solvability of the model we may conclude that the boundary-curve ∂D H starts moving with parameter β.

Models with point interactions
Another interesting PT −symmetric single-particle differential-operator Hamiltonian H with the property H = H † in H (F ) has been proposed in Ref. [21].
The extreme simplicity of this model opened the way not only towards the elementary formula for the energy spectrum, but also towards the equally elementary construction of the complete family of the eligible metrics Θ (cf., e.g., Refs. [22] for the details).
The solvability as well as extreme simplicity of this model proved encouraging in several directions. In the present context, the mainstream developments may be seen in the study of its discrete descendants (cf. the next subsection). Nevertheless, before turning our attention to the resulting fam- composed of q edges e j , j = 0, 1, . . . , q − 1.
The idea still waits for its full understanding and consistent implementation. In particular, in Ref. [23] we showed that even for the least complicated equilateral q−pointed star graphs with q > 2 the spectrum of energies need not remain real anymore, even if one parallels, most closely, the q = 2 boundary conditions (16) and even if one does not attach any interaction to the central vertex. In our present notation this means that the domain D H of Fig. 1 becomes empty. In other words, the applicability of this and similar models remains restricted to classical physics and optics while a correct, widely acceptable quantum-system interpretation of the manifestly non-Hermitian q > 2 quantum graphs must still be found in the future. k+1,k = 1, k = 1, 2, . . . , N − 1. With this idea in mind we already studied, in Ref. [25], the most elementary model with

Discrete lattices
We succeeded in constructing the complete N−parametric family of the physics-determining solutions Θ of the compatibility constraint (3). In Ref. [26] we then extended these results to the more general, multiparametric boundary-condition-simulated perturbations etc. Thus, all of these models may be declared solvable in the presently proposed sense. At the same time, the question of the survival of feasibility of these exhaustive constructions of metrics Θ after transition to nontrivial discrete quantum graphs remains open [27].

Discussion
During transitions from classical to quantum theory one must often suppress various ambiguities -cf., e.g., the well known operator-ordering ambiguity of Hamiltonians which are, classically, defined as functions of momentum and position. Moreover, even after we specify a unique quantum Hamiltonian operator H, we may still encounter another, less known ambiguity which is well know, e.g., in nuclear physics [2]. The mathematical essence of this ambiguity lies in the freedom of our choice of a sophisticated conjugation T (S) which maps the standard physical vector space V (i.e., the space of ket vectors |ψ representing the admissible quantum states) onto the dual vector space V ′ of the linear functionals over V. In our present paper we discussed some of the less well known aspects of this ambiguity in more detail. Let us now add a few further comments on the current quantum-model building practice.
First of all, let us recollect that one often postulates a point-particle (or point-quasi-particle) nature and background of the generic quantum models.
Thus, in spite of the existence of at least nine alternative formulations of the abstract quantum mechanics as listed, by Styer et al, in their 2002 concise review paper [28], a hidden reference to the wave function ψ(x) which defines the probability density and which lives in some "friendly" Hilbert space (say, in H (F ) = L 2 (R d )) survives, more or less explicitly, in the large majority of our conceptual as well as methodical considerations.
A true paradox is that the simultaneous choice of the friendly Hilbert space H (F ) and of some equally friendly differential-operator generator H = △+V (x) of the time evolution encountered just a very rare critical opposition in the literature [29]. The overall paradigm only started changing when the nuclear physicists imagined that the costs of keeping the Hilbert space H (F ) (or, more explicitly, its inner product) unchanged may prove too high, say, during variational calculations [2]. Anyhow, the ultimate collapse of the old paradigm came shortly after the publication of the Bender's and Boettcher's letter [10] in which, for certain friendly ODE Hamiltonians H = △ + V (x) the traditional choice of space H (F ) = L 2 (R) has been found unnecessarily over-restrictive (the whole story may be found described in [5]).
The net result of the new developments may be summarized as an acceptability of a less restricted input dynamical information about the system. In other words, the use of the friendly space H (F ) in combination with a friendly Hamiltonian H = H † has been found a theoretician's luxury. The need of a less restrictive class of standard Hilbert spaces H (S) which would differ from their "false" predecessor H (F ) by a nontrivial inner-product metric Θ = I appeared necessary.
One need not even abandon the most common a priori selection of the friendly Hilbert space H (F ) of the ket vectors |ψ with their special Dirac's duals (i.e., roughly speaking, with the transposed and complex conjugate bra vectors ψ|) yielding the Dirac's inner product ψ 1 |ψ 2 = ψ 1 |ψ 2 (F ) . What is only new is that such a pre-selected, F −superscripted Hilbert space need not necessarily retain the usual probabilistic interpretation.
One acquires an enhanced freedom of working with a sufficiently friendly form of the input Hamiltonian H, checking solely the reality of its spectrum.
Thus, one is allowed to admit that H = H † in H (F ) . One must only introduce, on some independent initial heuristic grounds, the amended Hilbert space H (S) . For such a purpose it is sufficient to keep the same ket-vector space and just to endow it with some sufficiently general and Hamiltonian-adapted (i.e., Hamiltonian-Hermitizing) inner product (1) [2]. This is the very core of innovation. In the physical Hilbert space H (S) the unitarity of the evolution of the system must remain guaranteed, as usual, by the Hermiticity of our Hamiltonian in this space, i.e., by a hidden Hermiticity condition alias crypto-Hermititicity condition [6]. In the special case of finite matrices one speaks about the quasi-Hermiticity condition. Unfortunately, the latter name becomes ambiguous and potentially misleading whenever one starts contemplating certain sufficiently wild operators in general Hilbert spaces [14].
It is rarely emphasized (as we did in [1]) that the choice of the metric remains an inseparable part of our model-building duty even if our Hamiltonian happens to be Hermitian, incidentally, also in the unphysical initial Hilbert space H (F ) . Irrespectively of the Hermiticity or non-Hermiticity of H in auxiliary H (F ) , one must address the problem of the independence of the dynamical input information carried by the metric Θ. Only the simultaneous specification of the operator pair of H and Θ connected by constraint (20) defines physical predictions in consistent manner. In this sense, the concept of solvability must necessarily involve also the simplicity of Θ.