Hiddenly Hermitian quantum models: The concept of perturbations

08/08/2019 ∙ by Miloslav Znojil, et al. ∙ 0

In conventional Schrödinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, h→ H=Ω h Ω≠ H^† induces the change L→ K of the Hilbert space of states, reflected by the loss of the Hermiticity of H≠ H^†. In such a reformulation of the theory the introduction of an ad hoc inner-product metric reconverts K into the third, correct physical Hilbert space H, unitarily equivalent to L. The situation encountered, typically, in PT-symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

Abstract

In conventional Schrödinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian . A non-unitary preconditioning of the Hamiltonian, (encountered, e.g., in symmetric quantum mechanics) induces the change of space , reflected by the loss of the Hermiticity of . The introduction of an ad hoc inner-product metric is usually used to reconvert into a correct physical Hilbert space , unitarily equivalent to . The situation is shown more complicated after a perturbation of (of immediate practical interest, say, in Klein-Gordon quantum systems) because we have to modify the mapping, . The formulation and solution of the problem are presented. Some of the consequences (typically, for the analysis of stability) are discussed.

1 Introduction

Quasi-Hermitian [1], pseudo-Hermitian [2] or hiddenly Hermitian alias  crypto-Hermitian representations of unitary bound state systems working with Hamiltonian-dependent inner-product metrics offer one of the less widely used but, sometimes, remarkably efficient constructive tricks in quantum mechanics. In the past it found successful applications in condensed matter physics [3] or in the study of heavy nuclei [4]. Recently, the idea contributed to the correct interpretation of various parity-times-time-reversal-symmetric systems [5, 6] and of the Krein-space-unitary Hamiltonians [7, 8]

. Along these lines, last but not least, a long-missing completion of the first-quantized theory with non-negative probabilities was also achieved for the Klein-Gordon and Wheeler-DeWitt equations

[2, 9].

In the present paper we will adapt the formalism to cover also the unitary, hiddenly Hermitian quantum systems with perturbations. First, a concise review of the unperturbed case will be provided in section 2. In section 3 we will turn attention to the models in which the Hamiltonian remains quasi-Hermitian but changes with a parameter, . We will emphasize that the related physical inner-product metric acquires the dependence in two ways, viz, directly and indirectly. Indeed, besides the obvious dependence inherited directly from the Hamiltonian, , one must also take into account that at any fixed the assignment is ambiguous. This ambiguity may be characterized, formally, by some parameters (e.g., by in the illustrative example of paragraph 3.1). Thus, also these parameters may vary with in general, . In paragraph 3.2 we will explain the consequences.

In section 4

we emphasize that the resulting formal requirements of the consistence form a pattern which may be simplified. We will succeed in reducing the whole set of the formal and physical correspondences between the respective vector and Hilbert spaces, physical or auxiliary, to a comparatively transparent three-Hilbert-space (3HS) scheme (cf. paragraph

4.3).

In section 5 this will finally enable us to concentrate on the resolution of a few paradoxes related to the apparently uncontrollable choices of the inner products and norms. We will confirm that the theory remains compatible with the textbooks and that the description of the perturbed unitary quantum systems may still proceed in a mathematically fully consistent manner. We will point out, in particular, that the flexibility of the menu of the eligible physical Hilbert spaces of states still makes the 3HS formalism attractive, tractable and, for future applications, promising. This will be emphasized in the brief summary section 6.

2 Single quantum state in three representation spaces

In the quasi-Hermitian formalism of Refs. [1, 2, 5] one achieves a sophisticated amendment of the straightforward textbook recipe of the identification of any quantum bound state with a normalized ket-vector element of a unique  physical Hilbert space . The practical realization of the amendment may be split in two steps. In the first step we get rid of the (presumably, prohibitively complicated) initial Hilbert space via its replacement by a more suitable unitary equivalent partner space which is, by assumption, decisively user-friendlier. In nuclear physics, for example [4], people worked with the Fock space of fermions and with the unitarily equivalent space of effective bosons. This simplified the vector space of kets but, due to the unitarity requirement (with implementations attributed, in this context, to Primakoff [1]), the construction still led to a very complicated physical inner product in .

2.1 Simplification versus loss of Hermiticity

In the latter setting, a key to success was found by Dyson [3]. He revealed the possibility of a decisive simplification of the inner product in . Formally, one could say that Dyson introduced just a modified, user-friendliest Hilbert space of the same  kets. In reality, he proceeded in an opposite direction. Via a unitarity-violating map he introduced, first of all, the persuasively preferable space endowed with a conventional inner product . Only then he defined the ultimate physical space by means of the following redefinition of the inner product,

Among the shortcomings of the innovation one finds the fact that in the auxiliary, working Hilbert space the Hamiltonians become formally non-Hermitian, . Their necessary Hermitization (i.e., a reconstruction of metric ) need not be easy [10]. Still, the formalism can provide, in principle at least, all of the relevant experimental predictions as well as a consistent picture of physical reality, fully compatible with conventional textbooks [11].

In its earliest applications the recipe had the following three-step structure:

(1)

Typically, the fermionic, Pauli-principle-controlled Fock-space choice of and the simpler, bosonic-space choices of and together with the Hermiticity of in and with the unitary equivalence between the two alternative physical Hilbert spaces and formed a firm theoretical background of various interacting boson models in nuclear physics, etc [1].

2.2 Dynamics controlled by non-Hermitian Hamiltonians

The flowchart of Eq. (2) can be inverted:

(2)

In full detail the latter, inverted version of the 3HS pattern has been worked out in Ref. [1]. The authors formulated the general criteria “for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation” [1]. Recently, such an innovated formulation of quantum mechanics found a truly non-trivial fructification and application in relativistic quantum mechanics. It led to a consistent probabilistic interpretation of the first-quantized Klein-Gordon and Wheeler-DeWitt fields [9, 12] and of the first-quantized Proca fields [13, 14].

3 Perturbation-dependent representation spaces

For the crypto-Hermitian subclass of the unitary quantum models which are non-Hermitian in but self-adjoint in the two alternative physical Hilbert spaces and the spectra of the operators of observables are real. In an optimal strategy the calculations should be performed in and interpreted in or . In such a setting the non-Hermiticity of Hamiltonians in is, by assumption, more than compensated by their enhanced technical user-friendliness. In a rigorous formulation, the mathematical background of the 3HS theory includes also the very important condition of the boundedness of the observables (cf. p. 78 in [1]). Although such a constraint looks rather artificial and over-restrictive, its importance is well known to mathematicians [15]. Not so much to physicists: during the recent revitalization of interest in hidden Hermiticity in quantum physics (working, in , with the non-Hermitian but symmetric Hamiltonians [5, 6]) the constraint of boundedness of the Hamiltonian was not introduced at all. This inspired the recent criticism by mathematical physicists [16, 17] as well as a return to the bounded-operator benchmark models as sampled, in section 3 of [1], by the non-Hermitian generalization of the Lipkin-Meshkov-Glick model.

3.1 Example

An elementary but still sufficiently realistic by -matrix example was provided, in Ref. [18], by the discrete version of the free Klein-Gordon Hamiltonian . The first nontrivial special case of this model with already offers a highly schematic but exactly solvable two-level quantum system characterized by the real two-by-two-matrix Hamiltonian

with a real variable parameter . In spite of a manifest non-Hermiticity of at the bound-state energies are real and non-degenerate, . The property survives a transition to arbitrary matrix dimension (cf. also [19]). Every element of such an illustrative family is non-Hermitian but its eigenstates still admit the conventional probabilistic interpretation. It is the third space in which the eigenstates of the upper-case Hamiltonian acquire their standard physical probabilistic interpretation. In comparison, the role of the friendlier but manifestly unphysical Hilbert space (with non-amended inner product) is auxiliary and purely mathematical.

For our illustrative Eq. (3.1) where is the conventional two-dimensional real vector space it is easy to show that the Hamiltonian will satisfy the “hidden”, space-Hermiticity relation

(3)

if and only if we introduce the amended, “physical” Hilbert-space-metric in ,

This matrix contains another auxiliary real variable such that . Its variability reflects the generic ambiguity of the assignment via diagram (2). Moreover, also the possibility of a free variability of this parameter with the change of Hamiltonian must be kept in mind as an inseparable, characteristic part of the 3HS formulation of quantum theory.

3.2 Formulation of perturbation theory using five Hilbert spaces

Quantum system represented by two mutually isospectral Hamiltonians and remains the same. One may only encounter a number of technical complications. In flowchart (2) and, in particular, in its step (B), for example, every change of the parameter-dependent Hamiltonian with a real (i.e., e.g., with small of Eq. (3.1)) may cause (and, usually, causes, cf. Eq. (3.1)) an induced change of metric and, hence, of the relevant physical Hilbert space . Before the final step (C) one must, moreover, factorize the metric into its Dyson-mapping factors so that the step-(A) choice of a independent space would induce an anomalous and counterintuitive explicit dependence in the ultimate textbook space .

The apparent paradox has an elementary resolution: it is more natural to start from a independent textbook space with the geometry controlled by the dynamics-independent trivial metric . This offers the more acceptable and intuitive picture of physics. Thus, we will rather demand that , and we will transfer the explicit dependence to the friendly, unhatted induced Hilbert space . As a marginal benefit the latter convention will also enable us to work with the concept of the “sufficient smallness” of the norm of the operators of perturbation because such a concept is only well defined in  [20]. Needless to add that the related perturbed Hamitonians acting in , viz.,

must still be treated as “inaccessible” and “prohibitively complicated” (i.e., for example, strongly non-local [2]). Otherwise, the use of the whole 3HS machinery would not make any sense. At any given the hiddenly non-Hermitian Hamiltonians acting in and/or in should be then introduced using the three-step connection pattern (1).

Once we merge the respective and diagrams (1) and (2), we arrive at the following shaped, five-Hilbert-space (5HS) flowchart of the quasi-Hermitian extension of the textbook perturbation theory,

(4)

The occurrence of as many as five different Hilbert spaces is a not too surprising complication which reflects merely the absence of any additional assumptions about dynamics and/or about the ambiguities mentioned in paragraph 3.1 above.

4 Elimination of the redundant Hilbert spaces

One of our most fundamental methodical requirements is that the quantum system in question is not an open system and that its evolution is unitary. In the perturbation term must be self-adjoint, . Otherwise, we could not accept any given candidate for the Hamiltonian, irrespectively of its particular representation, as an operator of an experimentally realizable observable. The emergence of any  non-Hermiticity in would indicate that our system ceased to be isolated from its environment. Being exposed to an external force, weak or strong, all of its measurable features might suffer a decay or growth.

Naturally, the unitarity restriction and the Hermiticity assumption are widely accepted when one works in the conventional framework of textbook space . These requirements must be translated into the language of the sophisticated, dependent physical Hilbert space .

4.1 Projection from to

Diagram (4) makes an impression of our having the full knowledge of the unperturbed system, i.e., of its upper case Hamiltonian acting in , and of its lower-case Hamiltonian acting in . The rest of the flowchart may be then read as a direct analogue of Eq. (1). One chooses a perturbation term in , and one makes a choice of the Dyson map (i.e., of in (1), or of in (4)). Then one merely diagonalizes the resulting (and, presumably, simpler) upper-case Hamiltonian, i.e., in the present setting, the perturbed Hamiltonian .

The first, most visible weakness of such a naive interpretation of the recipe lies in the manifest dependence of the working Hilbert space . A remedy is easy. It consists in our pulling the construction down from the dependent space to its independent partner , and in our elimination of the redundant Hilbert space . The purpose can be served by an operator representing the link between the two auxiliary Hilbert spaces,

(5)

The perturbed Hamiltonian is defined in and reads

(6)

It can formally be pulled down to ,

(7)

Perturbation defined in should be treated as our initial information on dynamics. Our flowchart (4) should be, in this sense, inverted and re-read as a perturbation-theory analogue of the 3HS reconstruction recipe prescribed by diagram (2). The calculations have to be performed in the user-friendly Hilbert space while the standard probabilistic interpretation of the results must be added, via metric , in .

4.2 Projection from to

Briefly, the 3HS theory may be characterized, for unperturbed as well as perturbed systems, by a separate description of mathematics in and of physics in . Such a split of the two traditional roles of the single textbook Hilbert space may simplify the picture. A deeper analysis of the change may be based on the factorization of the perturbed Dyson’s map into its unperturbed part and a small correction,

(8)

This enables us to move from to and to factorize the metric

(9)

We may now parallel the reconstruction step (B) of the general 3HS recipe (2). As long as we have and , and as long as the metric depends on , the reconstruction does not yield the original physical Hilbert space of diagram (4) but another, reduced physical Hilbert space. This space would be a new component of the theory. It deserves to be denoted by a separate dedicated symbol, say, . Thus, we arrive at the following operational interpretation of the scheme.

Lemma 1

After a transfer of the theory from to via Eqs. (6) and (7), the crypto-Hermiticity constraint (3) imposed upon the Hamiltonian may be given the form

(10)

Given perturbation , this equation specifies the (empty or non-empty) class of admissible (and, in general, non-unique) inner-product metrics converting our auxiliary, working Hilbert space into a new, reduced physical Hilbert space . Diagrammatically, the latter result may be presented as a merely slightly modified 3HS scheme,

(11)

4.3 Realizable perturbations

We can summarize that the pull-down process did not lead to any conceptual problems, requiring merely our knowledge of an appropriate and realistic input form (7) of the Hamiltonian. Moreover, we also have to notice that our key mathematical assumption of the existence of the non-singular one-to-one correspondence between the textbook space and its unitarily non-equivalent but friendlier alternative (i.e., of the invertible dependent Dyson map (8)) is, in some sense, equivalent to the physical assumption of the stability of the perturbed quantum system in question.

Lemma 2

Whenever the spectrum of the perturbed Hamiltonian (which is defined and non-Hermitian in the reduced, independent Hilbert space ) is not real, the positive definite solution of Eq. (8) does not exist.

Proof. As long as definition (9) implies the positive definiteness of metric in at any not too large , relation (10) should be re-read as a hidden Hermiticity requirement, i.e., as a spectral reality condition imposed upon our operator in .

We see that even the generalized 3HS formalism offers an expected and consistent picture of correspondence between the loss of the reality of the spectrum (occurring at the Kato’s exceptional-point limit of [21]) and the loss of the unitary equivalence between the respective user-friendly and user-unfriendly physical Hilbert spaces and . In and only in this sense we may formally separate the class of perturbations into its “sufficiently small” and “inadmissibly large” elements. At the same time, due to the manifest dependence of , the formulation of some useful criteria of such a split would be much more difficult than in conventional quantum mechanics (a persuasive illustrative example may be found, e.g., in [22]).

5 Which perturbations are “small”?

5.1 Norm-bounded perturbations in

In the perturbation-theory spirit, the value of is assumed small. Obviously, at every not too large values of , the statement of Lemma 2 could have been also inverted: The reality of the non-degenerate perturbed spectrum admits the amendment of the inner product, rendering possible the transition to the dependent physical Hilbert space as well as the conventional unitary and stable probabilistic interpretation of the quantum system in question. Unfortunately, the inversion of Lemma 2 would just be of a little practical value in applications because in advance we only know the unperturbed spectrum. In non-Hermitian cases, most of the conventional criteria of the “sufficient smallness” of perturbations cease to be applicable.

Let us now explain the reasons reminding the readers that one of the most important applications of perturbation theory occurs in the context of the analysis of stability of quantum systems with respect to random perturbations. A representative sample of such an analysis may be found in Ref. [16] where the authors work with several manifestly non-Hermitian Hamiltonians with real spectra (i.e., in our present notation, with operators defined in ). The Hilbert space is declared physical because the authors only pay attention to the non-unitary, open quantum systems. This allows them to characterize the “size” of by its norm in . Along these lines they arrive at the mathematically entirely correct conclusion that the systems in question are deeply unstable with respect to “small” (i.e., norm-bounded) perturbations.

In contrast, our present attention is paid to the closed, unitary quantum systems in which the phenomenologically motivated requirements of the norm-boundedness and smallness can only be applied to the self-adjoint perturbation operator which is defined as acting in the textbook physical Hilbert space . In such a formulation of the test of stability the main technical challenge lies in the necessity of translation of the “smallness in the technically inaccessible space ” to the “smallness in one of the accessible physical Hilbert spaces” (i.e., say, in , or in ).

5.2 The change of space

Naturally, nobody  would ever use the complicated 3HS representation of unitary quantum systems if the conventional textbook space were not  practically inaccessible. Thus, we are only allowed to work with the space image

(12)

of the space perturbation, and with the inherited, hidden Hermiticity constraint

(13)

in

. The advantage is that the estimate of the size of

may still be based, sometimes, on the knowledge of the detailed structure of the unperturbed matric and/or Dyson mapping .

The connection of operator with the experimental information carried by the upper-case perturbation or Eq. (10) is far from straightforward, being mediated by Eq. (6). Thus, let us recall this relation in the form reduced to its action in ,

(14)

After a further minor re-arrangement we obtain the full-fledged upper-case space version of the correspondence between and ,

(15)

Such a relation confirms that . A routine simplification of this formula can yield the explicit definition of the given, dynamical-input operator in terms of (with the norm-boundedness still under our control), and vice versa. In the latter case, in particular, the reconstruction of from any given input operator ,

(16)

represents the last constructive step towards the following important conclusion concerning the stability non-violation alias  physical admissibility properties of the Hamiltonian.

Lemma 3

The insertion of operator of Eq. (16) in Eq. (13) yields the operator-equation criterion which tests the hidden Hermiticity of the perturbed Hamiltonian at a fixed component of the physical Hilbert-space metric .

In the scenario with constant input , formula (16) implies a manifest dependence of output (or of ). The same observation would also hold in the opposite direction, moving from and to . In both of these scenarios one may conclude that

(17)

This means that the difference between and (caused by the dependence of the metric) is, in general, model-dependent and non-perturbative, i.e., not necessarily small at small .

The latter observation is one of the main differences between the present approach to (general, non-Hermitian) perturbations and the conventional forms of the textbook perturbation theory.

5.3 The evaluation of corrections

The construction of and of its factorization [cf. Eq. (9)] is a key to our understanding of the physical contents of the theory. The quantum system is represented, in the correct physical Hilbert space , in a way which is equivalent to its alternative (though, presumably, technically inaccessible) conventional textbook representation in . The idea (if not the practical realization) of such an equivalence remains vital for interpretation purposes. In diagram (2) the connection has been emphasized by the inclusion of the “Hermitian-Hamiltonian-reconstruction” step (C). After the present, perturbation-theory-related upgrade of the correspondence between spaces the amended, ultimate 3HS flowchart will have the form (11) which is valid at any fixed , For a given perturbation we then can, in principle, recall Eq. (10) and evaluate the metric . Its factorization (9) then yields the Dyson-map component . Subsequently, we may use Eq. (17) and obtain . The norm of the latter operator (or, better, of its image ) could finally be tested for its smallness.

Needless to add that such a type of the direct test is hardly feasible even in Hermitian models. The reason is that even in the mathematically most ambitious applications of perturbation theory, the “smallness of perturbation” (i.e., the radius of convergence of the Rayleigh-Schrödinger perturbation expansions [11], given by the position of the nearest Kato’s “exceptional point” [21] in the complex plane of ) is in fact extremely difficult to determine. In practice one should be much more pragmatic and test the convergence a posteriori.

The key idea should be, first of all, a simplification of the constructions at small . In this spirit we can make the analyticity assumption and start from the Taylor-like series

(18)

Similarly, we may put

(19)

This leads to the series

(20)

in Eq. (9), with

(21)

etc. Our task is now to specify the advantages of the use of these power series in our constructive considerations based on the crypto-Hermiticity relation (10) of Lemma 1.

Naturally, in the dominant order the latter relation reads

It is assumed satisfied by the two unperturbed operators and which are “known”, i.e., in most applications, at our full disposal. This means that what is of the true interest is the next-order relation

It is to be interpreted as an equation by which one converts the known dynamical input (i.e., operator ) into the eligible (i.e., non-uniquely specified) output information (i.e., the first order correction in the perturbed physical Hilbert-space metric).

Naturally, one could proceed to the second order relation

insert the selected operator (obtained in the preceding step) and deduce the class of the admissible second-order metrics .

Along the same lines one could evaluate the higher-order corrections. Alternatively, one could also recall Eqs. (21) and redefine the task as the search for the separate components of the Dyson-map-defining expansion (19). It is worth noticing that in the first-order case the reconstruction of from the leading-order perturbation component has the form of equation

in which we easily recognize the above-derived relation (13). This means that the abbreviation just represents the above-introduced operator in the leading-order approximation of Eq. (17). Only the interpretation of the equation is different because there are no s, and we now only have to deduce an eligible leading-order correction from the leading-order input .

6 Summary: Self-consistent nature of perturbations

The main problem solved by the 3HS (re-)formulation of quantum mechanics occurs when the conventional Schrödinger equation for bound states in the conventional Hilbert space of states is found prohibitively user-unfriendly. The basic idea of the amendment (attributed, most often, to Dyson [3]) is that via a suitable non-unitary pre-conditioning (say, (6)), the Hermiticity constraint is relaxed and replaced by the quasi-Hermiticity property (3) of the amended Hamiltonian.

In our present paper we explained that whenever the unitary quantum system in question gets exposed to a self-adjoint perturbation (i.e., whenever it is required unitary even if we modify with ), the consistent transition to a quasi-Hermitian picture becomes a perceivably more complicated task.

A general five-Hilbert-space (re-)formulation of quantum mechanics was given and shown to serve the purpose, in principle at least. Several amendments of the scheme were then proposed. Subsequently, several consequences were discussed. It has been clarified, first of all, that in the hiddenly Hermitian quantum models the traditional notions of the norm or smallness of perturbations in cannot be translated, at a reasonable cost, in the new language. Only the most elementary a posteriori  tests of convergence remain applicable. This implies, in particular, that the questions of stability of a given system with respect to random perturbations (answered, in open systems, by the construction of the pseudospectra [16, 23]) do not seem to have any easy answer for the closed crypto-Hermitian quantum systems at present.

References

  • [1] F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Ann. Phys. (NY) 213, 74-101, 1992.
  • [2] A. Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. 7, 1191-1306 (2010).
  • [3] F. J. Dyson, Phys. Rev. 102, 1217-1230 (1956).
  • [4] D. Janssen, F. Dönau, S. Frauendorf and R. V. Jolos, Nucl. Phys. A 172, 145-165 (1971).
  • [5] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
  • [6] C. M. Bender, PT Symmetry in Quantum and Classical Physics (World Scientific, Singapore, 2018).
  • [7] H. Langer and Ch. Tretter, Czech. J. Phys. 54, 1113 - 1120 (2004).
  • [8] F. Bagarello, J.-P. Gazeau, F. Szafraniec and M. Znojil, Eds., Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects (Wiley, Hoboken, 2015).
  • [9] M. Znojil, Ann. Phys. (NY) 385, 162-179 (2017).
  • [10] D. Krejčiřík, V. Lotoreichik and M. Znojil, Proc. Roy. Soc. A: Math., Phys. & Eng. Sci. 474, 20180264 (2018).
  • [11] A. Messiah, Quantum Mechanics (North Holland, Amsterdam, 1961).
  • [12] A. Mostafazadeh, Class. Quantum Grav. 20, 155-171 (2003).
  • [13] V. Jakubský and J. Smejkal, Czech. J. Phys. 56, 985 (2006).
  • [14] F. Zamani and A. Mostafazadeh, J. Math. Phys. 50, 052302 (2009).
  • [15] J. Dieudonne, Proc. Int. Symp. Lin. Spaces (Pergamon, Oxford, 1961), pp 115-122.
  • [16] D. Krejčiřík, P. Siegl, M. Tater and J. Viola, J. Math. Phys. 56, 103513 (2015).
  • [17] P. Siegl and D. Krejčiřík, Phys. Rev. D 86, 121702(R) (2012).
  • [18] I. Semorádová, Acta Polytechnica 57, 462-466 (2017).
  • [19] M. Znojil, J. Phys. A: Math. Gen. 37, 9557-9571 (2004).
  • [20] M. Znojil and F. Růžička, J. Phys. Conf. Ser. 1194, 012120 (2019).
  • [21] T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).
  • [22] M. Znojil, Phys. Rev. A 97, 032114 (2018).
  • [23] L. N. Trefethen and M. Embree, Spectra and pseudospectra (Princeton University Press, Princeton, 2005).