Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

One of the key obstacles encountered during transition from classical to quantum mechanics is that the corresponding evolution equations become operator equations. For this reason the experimentally testable quantum-theoretical predictions become, in general, incomparably more difficult. As a consequence, our understanding of the quantum dynamics becomes only too often dependent on an analysis mediated by some thoroughly simplified models of the physical reality in which, typically, a given self-adjoint Hamiltonian can be easily diagonalized, $\mathfrak{h} \to \mathfrak{h}_\textrm{diagonal}$.

In 1956, Dyson [1] had to deal with a fairly complicated multi-fermionic Hamiltonian $\mathfrak{h}$ for which the convergence of the conventional numerical diagonalization algorithms happened to be prohibitively slow. Still, he managed to find a way out of the difficulty. His construction of the bound states became nicely convergent when he preconditioned his Hamiltonian,

$$\begin{equation} \mathfrak{h}\ \to \ H = \Omega^{-1}\,\mathfrak{h}\,\Omega\,. \end{equation} \tag{ 1 }$$

The essence of his convergence-acceleration recipe lied in a judicious guess of a sufficiently effective preconditioning (1) mediated by a suitable invertible mapping Ω. In the language of physics, this choice just reflected the role of the correlations in the many-body system in question. In this sense, the Dyson's simplification-oriented model-building strategy found a number of applications, first of all, in nuclear physics where the role of the short-range correlations is fairly well understood as well as sufficiently easily simulated [2].

The originality of the Dyson's innovation was that his mappings Ω were allowed non-unitary, $\Omega^\dagger\,\Omega \neq I$. The simplification (1) has been achieved, paradoxically, at an expense of the loss of the Hermiticity of the Hamiltonian. In the language of mathematics, this can be perceived as an unusual, non-unitary transition from a conventional Hilbert space (say, ${\cal L}$) to another, auxiliary but user-friendlier Hilbert space (say, ${\cal H}_\textrm{math}$). In the language of operators one moves from the conventional textbook representation of a realistic Hamiltonian which is self-adjoint in ${\cal L}$, $\mathfrak{h} = \mathfrak{h}^\dagger$, to its isospectral (and, presumably, significantly simpler) manifestly non-Hermitian avatar $H \neq H^\dagger$ in ${\cal H}_\textrm{math}$.

In 1992, Scholtz et al [3] proposed a different, albeit closely related model-building strategy. These authors assumed that we are given a non-Hermitian operator H (or rather a set of such operators) in advance. Under this assumption they described the way how this operator or operators could constitute a consistent quantum mechanical system. Thus, in our present notation they just considered an inverted correspondence (1),

$$\begin{equation} H\ \to \ \mathfrak{h} = \Omega\,H\,\Omega^{-1}\,. \end{equation} \tag{ 2 }$$

In such a deeply innovative approach one preselects a suitable tentative non-Hermitian candidate for the Hamiltonian $H \neq H^\dagger$ from the very beginning. Although the approach has recently been enriched by the development of mathematical techniques in which the feasibility of practical calculations has been enhanced (see, e.g. the more recent review [4]), its mathematical aspects are still full of open questions (see, e.g. monograph [5]).

In applications, naturally, an internal consistency of the theory based on reconstruction (2) must be guaranteed. Thus, the spectrum of H must be real: in this respect it often helps when H is chosen parity-time symmetric [6, 7]. Secondly, many rather unpleasant emerging mathematical obstacles (see, e.g. their descriptions in [8–11]) may be circumvented when the states of the system in question are represented in an M-dimensional Hilbert space ${\cal H}_\textrm{math}^{(M)}$ where M is arbitrarily large but finite [1, 3].

Under these conditions (see also [4] for more details) the implicit, hidden Hermiticity (or, in mathematics, quasi-Hermiticity [12]) of the operator H representing an input information about dynamics has to be made explicit. Once we abbreviate $\Omega^\dagger\,\Omega = \Theta$ (calling this product a physical Hilbert-space inner-product metric), the standard and conventional textbook self-adjointness requirement $\mathfrak{h} = \mathfrak{h}^\dagger$ becomes formally equivalent to the quasi-Hermiticity of H in ${\cal H}_\textrm{math}^{(M)}$,

$$\begin{equation} H^\dagger\,\Theta = \Theta\,H\,. \end{equation} \tag{ 3 }$$

This makes the reconstruction (2) of $\mathfrak{h}$ redundant. In the words of review [3] one manages to find a physical inner-product metric Θ compatible with equation (3) if it exists (i.e. just in certain parameter regimes).

For practical purposes the use of the quasi-Hermitian formulation of quantum mechanics makes sense only if equation (3) as well as the related bound-state Schródinger equation

$$\begin{equation} H^{}\,|\psi_n^{}\rangle = E_n^{}\,|\psi_n^{}\rangle\,,\ \ \ \ \ n = 1,2, \ldots, M \end{equation} \tag{ 4 }$$

remain sufficiently user-friendly and solvable. In fact, there exist not too many solvable models of such a type. One category of the technical obstacles emerges when H is a differential operator. Indeed, as long as these operators are, typically, unbounded, the abstract quantum theory of [3] (where all of the operators of observables have been assumed bounded) cannot be applied.

Even when both of our above-mentioned Hilbert spaces ${\cal L}$ and ${\cal H}_\textrm{math}^{(M)}$ are kept finite-dimensional, $M \lt \infty$, the literature offers just a few toy-matrix models H which remain exactly solvable, at an arbitrary number of states $M \lt \infty$, in the manner which combines the availability of a closed form of all of the solutions $|\psi_n\rangle$ and E_n of Schrödinger equation (4) with the equally important availability of a closed form of at least one of the solutions $\Theta = \Theta(H)$ of equation (3).

In these models (see, e.g. [13, 14] or [15], with further references) one still has to work with the tridiagonal forms of the Hamiltonians. In what follows we intend to propose the class of solvable models in which the Hamiltonians form even a sparse-matrix subset of the similar tridiagonal models. They will form a new exactly solvable family of unitary quasi-Hermitian quantum models. We will see that these models can be perceived as an illustration of the situation in which the unitary quantum model based on a manifestly non-Hermitian Hamiltonian $H\neq H^\dagger$ appears preferable and, not quite expectedly, technically simpler than its isospectral Hermitian-matrix alternative of conventional textbooks.

It is not too surprising that in the majority of the realistic applications of the bound-state Schrödinger equations using a self-adjoint phenomenological Hamiltonian $\mathfrak{h}$ people recall the variational argument and approximations and keep the dimension M of the conventional textbook Hilbert space ${\cal L}$ finite [1–3]. Then, there are also no conceptual problems with the linear-algebraic correspondence between ${\cal L}$ and ${\cal H}_\textrm{math}^{(M)}$ and/or between $\mathfrak{h}$ and H (cf equation (1)).

The situation is different when the Hamiltonians $\mathfrak{h}$ and/or H are differential operators with $M = \infty$. On positive side, the standard 'kinetic plus potential energy' structure of such a class of operators makes them intuitively acceptable on physical grounds: typically, this renders them eligible in the role of prototype models in quantum field theory [7]. For this reason, even on the level of quantum mechanics the dedicated literature abounds with the exactly solvable models [16] as well as with the quasi-exactly solvable models [17–20] of such a type.

On negative side, the recent progress in the analysis of the $\mathfrak{h}\ \leftrightarrow \ H$ correspondence led to several disappointing disproofs of its existence [5]. Pars pro toto it is sufficient to mention papers [9, 10] containing the mathematically rigorous disproofs of the existence of any self-adjoint partner $\mathfrak{h}$ for the most popular imaginary cubic oscillator Hamiltonian H of [6].

After all, the very explicit words of warning were already written in the older review [3]. The authors required there that any  eligible non-Hermitian operator representing an observable should be bounded. In other words, under the warmly recommended auxiliary assumption $M\lt\infty$ the mathematics becomes perceivably simpler. The problems which remain to be resolved are purely technical, emerging usually just at sufficiently large matrix dimensions $M \gg 1$ and requiring only a sufficiently reliable numerical software.

The prevailing nature of results is then purely numerical. The exactly solvable bound-state models are rare. Even the diagonalization of a next-to-diagonal (i.e. tridiagonal) matrix form of $\mathfrak{h}$ may be ill-conditioned and just badly convergent [21]. In this context the guiding mathematical idea of our present project was that one of the rarely emphasized consequences of the choice of a non-Hermitian model H with real spectrum is that its nontrivial (i.e. non-diagonal) matrix representation can be 'sparse tridiagonal'.

An exciting formal appeal of the latter idea appeared accompanied by the emerging possibility of its transfer to the phenomenology and physics of various lattice models [22]. Both of these observations led us directly to the introduction and study of the M by M 'zig-zag-matrix' (ZZM) Hamiltonians

$$\begin{equation} H = H^\textrm{(ZZM)}(\vec{a},\vec{c}) = \left[ \begin{array}{cccccc} a_1&0&0&0&\ldots& \\ c_1&a_2&c_2&0&\ldots& \\ 0&0&a_3&0&0&\ldots \\ 0&0&c_3&a_4&c_4&\ddots \\ \vdots&\vdots&0&0&a_5&\ddots \\ &&\vdots&\ddots&\ddots&\ddots \end{array} \right] \end{equation} \tag{ 5 }$$

in which just $2M-1$ real parameters do not vanish.

A compact outline of some of the purely mathematical properties of matrices (5) may be found postponed to appendix below. The bound-state spectrum of these matrices (i.e. of the Hamiltonians of our present interest) coincides with the subset of parameters $a_1, a_2, \ldots, a_M$ occupying the main diagonal (see lemma 5 in the appendix). This means that the unitarity of the evolution is guaranteed by the reality of the spectrum of energies, i.e. by the reality of these dynamical-input parameters.

For the purposes of applications we are just left with the necessity of the construction of the wave functions i.e. in the conventional Dirac's notation, of the column-vector solutions $|\psi_1^{}\rangle$, $|\psi_2^{}\rangle$, ... of our Schrödinger equation (4) corresponding to the respective bound-state energies $E = a_n$ with $n = 1,2,\ldots$. Surprisingly enough, these ket-vectors can be obtained in closed form. Incidentally, the construction is most straightforward when $M = \infty$ because in such a case we do not need to separate the description of the solutions at even and odd $M\lt\infty$.

Another useful trick used during the explicit systematic construction of the solutions of our Schrödinger equation (4) is that at any $M \leqslant \infty$ we can concatenate our ket-vector columns into a single M by M matrix, say

$$\begin{equation} \{|\psi_1^{}\rangle,|\psi_2^{}\rangle,\ldots,|\psi_M^{}\rangle \} = Q_\textrm{solution}\,. \end{equation} \tag{ 6 }$$

Indeed, precisely the study of this matrix-of-solutions leads to the following important result.

Lemma 1. In Schrödinger equation (4) with $M = \infty$ and with the ZZM Hamiltonian $H = H^\mathrm{(ZZM)}(\vec{a},\vec{c})$, the column-vector eigenstates $|\psi_n^{}\rangle$ corresponding to the energies $E = a_n$ with $n = 1,2,\ldots$ and arranged in matrix (6) acquire precisely the ZZM-matrix form defined in terms of suitable vectors of parameters $\vec{x} = \{x_1,x_2,\ldots\}$ and $\vec{y} = \{y_1,y_2,\ldots\}$,

$$\begin{equation} Q_\mathrm{solution} = H^\mathrm{(ZZM)}(\vec{x},\vec{y})\,. \end{equation} \tag{ 7 }$$

Under the auxiliary ad hoc  assumption that $c_j\neq 0$ at all odd j we may accept, say, the following normalization of the separate ket-vector columns of $Q_\mathrm{solution}$,

$$\begin{equation} x_2 = x_4 = \ldots = 1\,,\ \ \ \ y_1 = y_3 = \ldots = 1\,. \end{equation} \tag{ 8 }$$

Then, the closed-form solution of our infinite-dimensional matrix Schrödinger equation is given by formulae

$$\begin{equation} x_j = (a_j-a_{j+1})/c_j\,,\ \ \ \ \ j = \mathrm{odd} \end{equation} \tag{ 9 }$$

and

$$\begin{equation} y_k = -\frac{(a_{k+1}-a_{k+2})c_k}{(a_{k}-a_{k+1})c_{k+1}}\,,\ \ \ \ \ k = \mathrm{even}\,. \end{equation} \tag{ 10 }$$

Proof. Proof is based on the auxiliary lemmas of appendix reflecting the remarkable properties of the algebra of ZZMs. The formulae themselves follow directly from the insertion of the solution in Schrödinger equation.

In this lemma our assumption $M = \infty$ enabled us to avoid the discussion of the role of the truncation of the matrix at $M \lt \infty$. In the latter case, fortunately, it proves sufficient to set, formally, $a_{M+1} = a_{M+2} = \ldots = 0$ and $c_M = c_{M+1} = \ldots = 0$. Also the apparent $c_j \to 0$ singularities at j < M are just an artifact of our normalization (8). Whenever needed, these singularities may be removed easily because our choice of the normalization has been dictated by the simplicity of the proof rather than by the simplicity or optimality of the formulae (7) and (8). The amendment is offered by the following re-normalized and more compact result.

Lemma 2. In Schrödinger equation (4) with the ZZM Hamiltonian $H = H^\mathrm{(ZZM)}(\vec{a},\vec{c})$, the column-vector eigenstates corresponding to the bound-state energies $E = a_n$ with $n = 1,2,\ldots,M$ can be given a differently normalized 'tilded' form

$$\begin{equation} \{ |\widetilde{\psi_1^{}}\rangle,|\widetilde{\psi_2^{}}\rangle, \ldots,|\widetilde{\psi_M^{}}\rangle \} = H^\mathrm{(ZZM)}(\vec{p},\vec{q}) \end{equation} \tag{ 11 }$$

where we employ a different, unit-diagonal normalization $p_j = 1$ at all j, and where we obtain the more compact formula for the off-diagonal parameters forming the vector $\vec{q}$,

$$\begin{equation} q_k = -c_k/(a_k-a_{k+1})\,,\ \ \ \ \ k = 1,2,\ldots,M-1\,. \end{equation} \tag{ 12 }$$

Proof. As long as we just changed the normalization convention, there exists a diagonal matrix (say, ϱ) such that $H^\textrm{(ZZM)}(\vec{x},\vec{y})\,\varrho = H^\textrm{(ZZM)}(\vec{p},\vec{q})$.

It is well known [23] that whenever we replace the manifestly non-Hermitian Hamiltonian H in Schrödinger equation (4) by its conjugate $H^\dagger$, the knowledge of the 'ketket' solutions of the associated Schrödinger equation

$$\begin{equation} H^\dagger\,|\psi_n^{}\rangle\!\rangle = E_n^{}\,|\psi_n^{}\rangle\!\rangle\,, \ \ \ \ n = 1,2,\ldots,M \end{equation} \tag{ 13 }$$

enables us to define all of the admissible metrics $\Theta = \Theta(H)$ (i.e. all of the admissible solutions of equation (3)) by formula

$$\begin{equation} \Theta = \Theta(\kappa^2_1\,,\kappa^2_2\,, \ldots,\kappa^2_M) = \sum_{n = 1}^M\,|\psi_n^{}\rangle\!\rangle\,\kappa^2_n\,\langle\!\langle \psi_n^{}|\,. \end{equation} \tag{ 14 }$$

This is not  a spectral representation of Θ because in general (i.e. due to the non-Hermiticity of H) the overlaps $\langle\!\langle \psi_m^{}| \psi_n^{}\rangle\!\rangle$ need not vanish even when m ≠ n. Still, this formula shows that the general metric can vary with as many as M freely variable real and positive parameter $\kappa^2_n$.

In comparison with equation (4), the most important comment concerning equation (13) is that as long as our toy-model ZZM Hamiltonians H are real, we now have to deal with the transposed matrices,

$$\begin{equation} H^\dagger = H^\mathrm T = H^\mathrm{(TZZM)}(\vec{a},\vec{c}) = \left[ \begin{array}{cccccc} a_1&c_1&0&0&\ldots& \\0&a_2&0&0&\ldots& \\0&c_2&a_3&c_3&0&\ldots \\0&0&0&a_4&0&\ddots \\\vdots&\vdots&0&c_4&a_5&\ddots \\ &&\vdots&\ddots&\ddots&\ddots \end{array} \right]\,. \end{equation} \tag{ 15 }$$

The crucial consequence is that it is sufficient to replace the ZZM theory of appendix by its transposed-matrix TZZM alternative.

Lemma 3. In Schrödinger equation (13) with the transposed Hamiltonian $H^\mathrm T = H^\mathrm{(TZZM)}(\vec{a},\vec{c})$, the collection of the column-vector eigenstates $|\psi_n^{}\rangle\!\rangle$ corresponding to the bound-state energies $E = a_n$ with $n = 1,2,\ldots,M$ can be given the TZZM form,

$$\begin{equation} \{ |\psi_1^{}\rangle\!\rangle,|\psi_2^{}\rangle\!\rangle,\ldots,|\psi_M^{}\rangle\!\rangle \} = H^\mathrm {(TZZM)}(\vec{p},\vec{q})\,. \end{equation} \tag{ 16 }$$

The normalization $p_j = 1$ (at all j) leads to the closed-form result

$$\begin{equation} q_k = -c_k/(a_k-a_{k+1})\,,\ \ \ \ \ k = 1,2,\ldots,M-1\,. \end{equation} \tag{ 17 }$$

Proof. Proof is a TZZM analogue of the ZZM proof of lemma 2.

Formula (16) containing M − 1 characteristics (17) of the Hamiltonian may be inserted in the definition of all of the eligible metrics (14). The resulting M by M matrices Θ would be, by construction, invertible, Hermitian and positive definite. Due to the reality and tridiagonality of the factor (16) and of its transposition, all of the metrics will have a real and symmetric pentadiagonal-matrix form. The explicit evaluation of their matrix elements is straightforward and constitutes our present main mathematical result.

Theorem 4. Every metric Θ guaranteeing the quasi-Hermiticity (3) of our $(2M-1)$-parametric ZZM Hamiltonian (5) can be given the three-component form

$$\begin{equation} \Theta = \Theta^\mathrm {(diag)}+ \Theta^\mathrm {(tridiag)}+ \Theta^\mathrm {(pentadiag)}\,. \end{equation} \tag{ 18 }$$

Its first component is just the invertible, q_j -independent and positive-definite diagonal matrix,

$$\begin{equation} \Theta^\mathrm {(diag)} = H^\mathrm {(TZZM)}(\vec{\kappa^2},\vec{0})\,. \end{equation} \tag{ 19 }$$

The second component has the sparse tridiagonal-matrix form with vanishing main diagonal,

$$\begin{equation} \Theta^\mathrm {(tridiag)}_{nn} = 0\,,\ \ \ \ n = 1,2,\ldots,M\,. \end{equation} \tag{ 20 }$$

Its off-diagonal elements

$$\begin{equation} \Theta^\mathrm {(tridiag)}_{m,m+1} = \Theta^\mathrm {(tridiag)}_{m+1,m} = q_m\kappa^2_{m+1}\,, \ \ \ m = 1,3,\ldots\ (\leqslant M-1) \end{equation} \tag{ 21 }$$

and

$$\begin{equation} \Theta^\mathrm {(tridiag)}_{n,n+1} = \Theta^\mathrm {(tridiag)}_{n+1,n} = q_n \kappa^2_{n} \,,\ \ \ \ n = 2,4,\ldots\ (\leqslant M-1)\, \end{equation} \tag{ 22 }$$

are all linear in q_j s. The remaining, third component of the metric has the pentadiagonal sparse-matrix form

$$\begin{equation} \Theta^\mathrm {(pentadiag)} = \left[ \begin{array}{cccccccc} {q_1}^{2}\kappa^2_2&0&q_1\,\kappa^2_2\,q_2&0&\ldots&&& \\0&0&0&0&0&\ldots&& \\q_1\,\kappa^2_2\,q_2&0&{q_2}^{2}\kappa^2_2+{q_3}^{2}\kappa^2_4&0&q_3\,\kappa^2_4\,q_4&0&\ldots& \\0&0&0&0&0&0&0&\ldots \\\vdots&0&q_3\,\kappa^2_4\,q_4& &{q_4}^{2}\kappa^2_4+{q_5}^{2}\kappa^2_6 &0&q_5\,\kappa^2_6\,q_6&\ddots \\ &\vdots&0&0&0&0&0&\ddots \\ &&\vdots&0&q_5\,\kappa^2_6 \,q_6&0&{q_6}^{2}\kappa^2_6+{q_7}^{2}\kappa^2_8&\ddots \\ &&&\vdots&\ddots&\ddots&\ddots&\ddots \end{array} \right]\, \end{equation} \tag{ 23 }$$

with elements which are all quadratic in q_j s.

Proof. The result follows directly from formulae (14) and (16).

In the context of physics the latter result is truly remarkable because it implies that it really does make sense to work with the present non-Hermitian ZZM or TZZM representations H of the Hamiltonians. For the simplest dynamical scenarios with the finite and not too large numbers of the bound-state levels $M\lt\infty$ at least, one could feel tempted to employ the standard algorithms of linear algebra and to factorize the pentadiagonal-matrix metric $\Theta = \Omega^\dagger\Omega$ of theorem 4. In principle, this would yield the explicit Dyson map Ω and, finally, enable us to return to the quantum mechanics of textbooks in which the conventional self-adjoint representation $\mathfrak{h}$ of the Hamiltonian would be 'easily' reconstructed via equation (2). Nevertheless, a feasible realization of such an alternative, more traditional version of the present models would require an invention of new methods.

Indeed, the first mathematical obstacle would emerge when we imagine that the metrics Θ of equation (14) and of theorem 4 are ambiguous, M-parametric [3]. Secondly, we would have to deduce, from factorization $\Theta = \Omega^\dagger\,\Omega$, a suitable sample of the Dyson map Ω. Then, indeed, a new set of free parameters forming a unitary matrix ${\cal U}$ would have to be introduced and considered here due to the ambiguity of the factorization of the metric itself, $\Theta = \Omega^\dagger\,\Omega = \Omega^\dagger\,{\cal U}^\dagger\,{\cal U}\,\Omega$. Thus, certainly, the 'conventional' Hermitian matrix $\mathfrak{h}$ would be a non-sparse, user-unfriendly matrix in general. Hence, the non-Hermitian matrix H really seems to offer the most economical representation of the Hamiltonian. One could hardly find reasons for a tedious reconstruction of its partner(s) $\mathfrak{h}$ of conventional textbooks.

In the dedicated literature, not too many quasi-Hermitian quantum models have the 'exact and complete solvability' property of our present class of M-level bound-state systems using the real ZZM Hamiltonians (5) with $2M-1$ free parameters. Typically, the algebraically solvable models of such a type are based on the use of tridiagonal matrix forms of H (cf e.g. a sample of such a class of quasi-Hermitian models in [15]). In general, given a realistic non-Hermitian H, the metric Θ assigned to the model is usually just approximate and not too flexible, corresponding usually just to a fixed choice of the set of parameters $\kappa_n^2$ in (14).

In comparison, our present model is rather exceptional in keeping the whole set of the metric-determining parameters $\kappa_n^2$ freely variable. Moreover, our restriction of the class of the Hamiltonians to the mere sparse ZZMs of appendix proved fortunate: We discovered that the full sets of the eigenstates of H appeared to belong to the same (viz., ZZM) subclass of the highly sparse ZZMs. One would even like to say 'serendipitiously fortunate' because the same comment appeared to apply also to the TZZM subclass and to the transposed Hamiltonian $H^\dagger$ playing a key role in the construction of the complete  set of the eligible metrics $\Theta = \Theta(H)$.

In the context of physics one of the remarkable properties of the model is that its bound-state energy spectrum coincides, due to the ZZM sparsity of the Hamiltonian, with its main diagonal. As an input information about dynamics it can be, therefore, fixed in advance. This means that the remaining M − 1 freely variable off-diagonal matrix elements of H can be interpreted as playing an energy-complementing role of parameters responsible for the operator metric Θ, i.e. for the correct physical geometry of the Hilbert space. In this manner these parameters influence, directly and implicitly, the selection and form of the other possible observable features of the system [3].

A final complementary comment may be also added on the existence and structure of the exactly solvable quantum models of unitary systems occurring and widely used within the framework of the conventional Hermitian quantum mechanics in which the metric is kept trivial, $\Theta_\textrm {conventional} = I$. Indeed, once the operators of the observables (including the Hamiltonians) become required, in the conventional textbook spirit, self-adjoint, the first nontrivial matrix form of an observable (or of the Hamiltonian) has to be real and symmetric, i.e. fully tridiagonal, i.e. from the numerical-manipulation perspective, perceivably more complicated than our 'maximally sparse' ZZM models (5).

No new data were created or analysed in this study.

Miloslav Znojil: Writing—original draft.

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

By 'zig-zag matrices' (ZZMs) we will understand, in this paper, the real and tridiagonal M by M matrices of equation (5) which may be finite- or infinite-dimensional (i.e. $M \leqslant \infty$). Whenever needed, the transpositions of these matrices will be called, for the sake of definiteness, the 'transposed zig-zag matrices' (TZZMs). Both of these classes of matrices have a few truly remarkable properties.

Lemma 5. The spectrum of $H^\mathrm {(ZZM)}(\vec{a},\vec{c})$ coincides with the M-plet of parameters $\vec{a}$.

Proof. It is sufficient to recall the definition of the secular determinant $\det (H^\mathrm {(ZZM)}(\vec{a},\vec{c})-E)$.

Lemma 6. The standard matrix product of two ZZM factors retains the ZZM property,

$$\begin{equation} H^\mathrm {(ZZM)}(\vec{a},\vec{c}) \cdot H^\mathrm {(ZZM)}(\vec{b},\vec{d}) = H^\mathrm {(ZZM)}(\vec{u},\vec{v}) \end{equation} \tag{ A1 }$$

with

$$\begin{equation} u_j = a_jb_j\,,\ \ \ \ j = 1,2,\ldots,M \end{equation} \tag{ A2 }$$

$$\begin{equation} v_k = c_k b_k+ a_{k+1}d_k\,,\ \ \ \ k = \mathrm{odd}\,,\ \ k \leqslant M-1 \end{equation} \tag{ A3 }$$

$$\begin{equation} v_k = a_k d_k+c_k b_{k+1}\,,\ \ \ \ k = \mathrm{even}\,,\ \ k \leqslant M-1\,. \end{equation} \tag{ A4 }$$

Proof. Proof is obtained directly from the definition of the standard matrix product.

Corollary 7. Any positive integer power of ZZM retains the ZZM property.

Lemma 8. The negative integer powers of $H^\mathrm {(ZZM)}(\vec{a},\vec{c})$ retain the ZZM property, provided only that the zero does not belong to the spectrum, $a_j \neq 0$, $j = 1,2,\ldots, M$.

Proof. It is sufficient to set the product $H^\mathrm {(ZZM)}(\vec{u},\vec{v})$ in lemma 6 equal to the unit matrix. Then we may check that in this case (i.e. with all ${u}_j = 1$ and with all $v_j = 0$) one can define the inverted matrix $H^\mathrm {(ZZM)}(\vec{b},\vec{d}) = [H^\mathrm {(ZZM)}(\vec{a},\vec{c})]^{-1}$ via the following elementary formulae,

$$\begin{equation} b_j = 1/a_j\,,\ \ \ \ j = 1,2,\ldots,M\,,\ \ \ \ \ d_k = -c_k /(a_k a_{k+1})\,,\ \ \ \ k = 1,2,\ldots, M-1\,, \end{equation} \tag{ A5 }$$

i.e. by the formulae independent of the parity of k.

In addition to these observations it is also easy to check, by the explicit matrix multiplication, that

$$\begin{equation} H^\mathrm {(ZZM)}(\vec{a},\vec{0})\,[H^\mathrm {(ZZM)}(\vec{a},\vec{c})]^{-1}\,H^\mathrm {(ZZM)}(\vec{a},\vec{0}) = H^\mathrm {(ZZM)}(\vec{a},-\vec{c})\,. \end{equation} \tag{ A6 }$$

Moreover, depending on the specific needs in applications it is also not too difficult to deduce multiple other auxiliary formulae in which the fractions are eliminated, say, from the formulae for the higher negative powers of the matrices. For illustration let us only display here their first nontrivial sample,

$$\begin{equation} [H^\mathrm {(ZZM)}(\vec{a},\vec{0})]^2\, [H^\mathrm {(ZZM)}(\vec{a},\vec{c})]^{-2}\,[H^\mathrm {(ZZM)}(\vec{a},\vec{0})]^2 = H^\mathrm {(ZZM)}(\vec{\widetilde{a}},-\vec{\widetilde{c}})\, \end{equation} \tag{ A7 }$$

where $\widetilde{a}_j = a_j^2\,,\ \ \ j = 1, 2, \ldots, M$ and $\widetilde{c}_k = c_k\,(a_k+a_{k+1})\,,\ \ \ k = 1, 2, \ldots, M-1$.