Quasi-hermitian quantum mechanics and a new class of user-friendly matrix hamiltonians

In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.


Motivation.
The concept of a "user-friendly" quantum Hamiltonian is vague: In multiple realistic applications of Schrödinger equation people, typically, require that the user-friendly Hamiltonians should be just self-adjoint in Hilbert space L and, in calculations, "sufficiently easily" diagonalizable.
In practice, even the latter two elementary requirements may prove difficult to satisfy simultaneously.One of the very old (but still quite persuasive) examples of such a "user-unfriendly" self-adjoint (but difficult to diagonalize) many-body h = h † has been studied by Dyson [1].The point is that he was still able to re-classify his problem as "user-friendly".
What Dyson managed to find was a new strategy of solving Eq. ( 1) via a simulation of certain decisive, diagonalization-preventing multi-fermionic correlations.In technical terms, the success resulted from his judicious guess of a Hilbert-space-amending ansatz |ψ(t)≻ = Ω |ψ(t) , |ψ(t) ∈ H . ( In contrast to the common practice, the invertible mapping Ω was chosen non-unitary, i.e., such that Ω † Ω = I.As a consequence, the original Dyson's "unfriendly" Schrödinger equation (1) living in a fermionic Hilbert space L acquired an equivalent but decisively user-friendlier form Here, one only had to replace the conventional Hermiticity constraint h = h † as valid in L by the so called quasi-Hermiticity [2] alias Θ−quasi-Hermiticity [3] property of the new, isospectral Hamiltonian operator H acting in another Hilbert space H. Incidentally (or rather on purpose), the new Hilbert space H of the Dyson's multifermionic model was required to be the space of the so called "effective" or "interacting" bosons (cf.also the later successful use of the idea in nuclear physics [4]).This was, in some sense, the reason why the manifestly non-Hermitian H happened to become, not quite expectedly, "diagonalization-friendlier".
In 1992, the ideas behind the old Dyson's quasi-Hermitian version (3) of the interacting boson model have been reinterpreted in review [2].Its authors proposed to invert the flowchart and to build, systematically, the whole new family of user-friendly quantum models by preselecting a set of suitable non-Hermitian observables Λ j which all obeyed an analogue of Eq. ( 4), and which, naturally, included also the upper-case Hamiltonian as special case, Λ 0 = H.A few years later the idea has been further developed to its perfection and wide-spread appeal by Bender with Boettcher [5].Unfortunately, a weakness of the specific Bender's and Boettcher's ordinary differential Hamiltonians H (BB) appeared to be deep.After a rigorous mathematical analysis [6] it has been proved that up to a few remarkable exceptions [7], most of the popular toy-model operators H (BB) cannot be assigned, for certain subtle but persuasive reasons, any admissible Hilbert-space metric Θ (BB) .
Along the lines indicated in monograph [8] the only mathematically consequent way out of the crisis has been found in a return to the more restrictive version of the innovative quasi-Hermitian quantum mechanics of review [2] in which all of the eligible non-Hermitian operators of observables Λ j were required bounded in H.
These developments put in doubt the possibility of an internally consistent applicability of the quasi-Hermitian quantum theory to all of the truly user-friendly linear differential candidates for the Hamiltonians which happened to be unbounded.One of the consequences was that the attention of many quantum physicists had to be redirected back to the models with finite-dimensional Hamiltonian matrices (cf.their characteristic samples in [2,9,10]).
The major part of the currently developing non-Hermitian picture of dynamics had to be realized in finite-dimensional Hilbert spaces.One of the collateral practical loses was that even then, most of the restricted matrix or bounded-operator models admitted just a brute-force numerical analysis, the results of which often happened to be impractical, mainly due to the non-Hermitian matrix nature of the M by M Hamiltonians H (M ) .
Our present project of a search for the user-friendly quantum Hamiltonians H (M ) with any matrix dimension M ≤ ∞ was motivated precisely by the latter discouraging observations.Zig-zag matrix algebra.
Some encouraging preliminary results as well as an immediate inspiration of our present paper can be found in Ref. [11] (to be cited as paper P1 in what follows).One of us noticed there that there exist several parallels between the simplest "irreducible" special case of a non-diagonal self-adjoint (i.e., Hermitian) matrix h (M ) (i.e., between the tridiagonal matrix defined, in general, in terms of 3M −2 independent real parameters) and its simplest non-Hermitian analogues H (M ) having one of the two alternative sparse-tridiagonal forms called "zig-zag matrix" and "transposed zig-zag matrix", respectively.For the reasons as explained in [11] it makes sense to notice that the spectra of Hs in (7) coincide with the elements {a 1 , a 2 , . . ., a M } of the vector a (cf.Lemma Nr. 5 in P1).This enables one to impose the spectrum-reality constraint Re a j = a j = 0 and to treat these matrices as quasi-Hermitian candidates for quantum bound-state Hamiltonians compatible with Eq. ( 4).As a consequence, the analogy with the tridiagonal Hermitian matrices is even closer because every matrix H of Eq. ( 7) becomes also defined strictly in terms of 3M − 2 independent real parameters.
Among several other remarkable properties of the latter (i.e., ZZ or, analogously, TZ) matrices let us also mention that they form the two respective sets which are closed with respect to the multiplication (cf.Lemma Nr. 6 in paper P1).For our present purposes it makes also sense, as it did in P1, to simplify the discussion and to work just with the real and invertible ZZ or TZ matrices H with non-degenerate spectra, i.e., just with the 2M − 1 nontrivial real matrix elements such that Im c j = 0 and a j = a k at all j and k = j.

Quantum zig-zag-matrix Hamiltonians.
Besides the wealth of the remarkable algebraic characteristics of the ZZ and/or TZ sets (cf., e.g., the closed-form matrix-inversion rule of Lemma Nr. 8 in P1), the main emphasis has to be put upon their use, in the role of the user-friendly operators of observables, in quantum physics.In this setting it is necessary to emphasize the emergence of their two formal merits.The first one lies, after both of the alternative ZZ or TZ choices of the Hamiltonian, in the closed-form solvability of the time-independent Schrödinger equations (cf.Lemmas Nr. 1, 2 and 3 in P1).This means that not only the spectra but also the states become available in closed form.
The second formal merit is even more important.In P1 this merit was given the form of the main Theorem Nr. 4 stating that for any given ZZ or TZ H, all of the operators Θ compatible with Eq. (4) (i.e., solving this equation and making the matrix H quasi-Hermitian) become available in an explicitly specified pentadiagonal-matrix form.
The latter result is of paramount importance in applications.For two reasons.The first one is physical, implying the conventional probabilistic interpretation of the theory.This enables us to call the matrix H the Hamiltonian of a quantum system.The second, mathematical reason is almost trivial: Whenever one tries to follow the quasi-Hermitian model-building recipe of review [2] and whenever one considers a suitable non-Hermitian ZZ or TZ candidate H (M ) for the Hamiltonian, one has no problem with the necessary proof of the reality of the spectrum alias of the observability of all of the bound-state energies (i.e., one simply recalls that Im a j = 0 at all j).
This guarantees the existence of at least one matrix of metric Θ with which the quasi-Hermiticity constraint (4) becomes satisfied.In fact, the assignment of Θ to a preselected H (with real spectrum) is one of the main technical challenges encountered during any successful construction of a quasi-Hermitian physical quantum system [3].Hence, its closed-form feasibility as mentioned above is in fact one of the decisive merits of the ZZ and/or TZ matrix models of paper P1.
In the context of physics, the assignment H → Θ(H) has to be perceived as ambiguous.In this sense one can speak about another merit of the closed-form solvability of Eq. ( 4).Indeed, its closed-form solutions as prescribed by Theorem Nr. 4 of paper P1 form an M−parametric set, i.e., they represent an exhaustive and entirely general solution of the eligibleinner-product problem.Covering all of the existing complementary choices of the other observables Λ j .In this sense, relations (5) can be recalled as the additional input information about dynamics fixing the parameters and removing completely the ambiguity of Θ [2].
It is worth adding that the essence of the availability of the ZZ or TZ inner-product metric Θ lies in its above-mentioned factorization Θ = Ω † Ω.It is related to a slightly re-arranged formula taken from Eq. (3), i.e., to relation The subscripted zero indicates here that a "special" Hermitian partner matrix h 0 is chosen diagonal.
According to Lemma Nr. 3 of P1 we may then start, say, from the lefthand-side Hamiltonian H (M ) ZZ ( a, c) of Eq. ( 7), and we may re-write Eq. ( 8) in the form of equation Nr. (13) of P1, i.e., in its explicit Schrödinger-equation form of which the solutions are known in closed form, With the normalization p j = 1 (at all j) one gets (cf.Lemma Nr. 3 in P1).The climax of the story then comes with the formula alias definition in which all of the real parameters κ2 j = 0 are arbitrary (cf.[12]).It is possible to conclude that once we finally have the correct physical operator of the inner-product metric (12) at our disposal, we are prepared to build the consistent quantum-mechanical models along the more or less conventional lines and, moreover, in a strictly non-numerical manner.In these models the "obligatory" requirement of the self-adjointness of all of the eligible and relevant observables Λ j becomes merely generalized and replaced by the Θ−quasi-Hermiticity rule (5).

Generalized zig-zag matrix algebras.
In the light of the above-outlined results we came to the conclusion that it might make sense to search for the generalizations of the zig-zag matrices.One of the results of this search is to be presented in what follows.We will show that and how the numerous user-friendliness features of the zig-zag models can be generalized.
For this purpose 2 let us consider a Hilbert space of finite even dimension 2m, i.e.C 2m .We define a solvable class of non-hermitian model Hamiltonians by with Λ diagonal and N nilpotent of degree 2 .(13) Furthermore, we impose the simplification that N is built entirely from nilpotent 2×2 blocks B such that B 11 = B 21 = B 22 = 0.The matrix N itself need not have to be of an upper or lower block triangular form.Still, any two such matrices multiply to zero.To bring out this 2×2 structure, we label the row and column entries as follows, so that the index i = 1, . . ., m numerates the blocks.With this labelling, the matrix entries are and Abbreviating the matrix entries by λ and n respectively, we may characterize such matrices as H( λ, n).Explicitly, they take the form The odd-dimensional case can be treated in the same fashion via enlarging the matrix H by a zero row and zero column.In this way, it is a special even-dimensional system with λ m− = n im = 0.This class of matrices has interesting properties.Firstly, it is closed under multiplication, since left or right multiplication with a diagonal matrix conserves the structure of N, Even stronger, any pattern of vanishing 2×2 blocks, i.e. some collection of vanishing n ij , is preserved under matrix multiplication.If none of the eigenvalues λ ±i is vanishing, an inverse can be constructed, Generalized zig-zag-matrix Hamiltonians.
Once we move to the applications of the generalized zig-zag-matrices in quantum mechanics, the eigensystem is also found easily.Obviously, the eigenvalues are simply the diagonal entries λ ±i of Λ.Let us arrange the eigenket columns |±i defined (up to normalization) by where the second equation defines an ansatz, which gives Q the same form as H.The choice of the unit matrix as the diagonal part fixes a normalization of the eigenkets.The defining equation 20 for Q is easily solved, Explicitly, the matrix of eigencolumns reads This solution is only viable if λ +i = λ −j whenever n ij is nonzero 3 .In general though, eigenvalue degeneracies are allowed.The special case of n ij = 0 except for j=i and j=i+1 (i.e.only secondary and quaternary sparse upper diagonals) reduces to the zig-zag matrices discussed in [11].They are reproduced by a basis permutation +i ↔ −i, which merely swops the labels in each 2×2 block.The transpose matrix H ⊤ = Λ + N ⊤ is obtained by transposing the 2×2 blocks internally and also blockwise in H, hence N ⊤ −j+i = n ij and others vanish.
with the analog ansatz Q = 1 + N has the solution The quasi-Hermiticity of H defines a physical inner-product metric Θ in our Hilbert space, via In the same manner as above, all of the admissible metrics are given by formula parametrized by 2m nonnegative parameters κ ±i .Collecting these parameters in a diagonal matrix the metric matrix can be written as where N is the off-diagonal part of the solution Q.
As long as N and N⊤ belong to two disjoint classes of nilpotent matrices, this formula in general produces a full 2m×2m matrix.For the class of zig-zag matrices, which are empty beyond the quaternary diagonal, in our basis the metric becomes block tridiagonal, i.e. it is heptadiagonal, but both secondary, tertiary and quaternary diagonals remain sparse.Passing to the 'zig-zag basis' the quaternary diagonals can be removed for the prize of nonsparse secondary ones.