Non-Hermitian Hamiltonians in Quantum Physics pp 235-248 | Cite as

# A Unifying E2-Quasi Exactly Solvable Model

## Abstract

A new non-Hermitian E2-quasi-exactly solvable model is constructed containing two previously known models of this type as limits in one of its three parameters. We identify the optimal finite approximation to the double scaling limit to the complex Mathieu Hamiltonian. A detailed analysis of the vicinity of the exceptional points in the parameter space is provided by discussing the branch cut structures responsible for the chirality when exceptional points are surrounded and the structure of the corresponding energy eigenvalue loops stretching over several Riemann sheets. We compute the Stieltjes measure and momentum functionals for the coefficient functions that are univariate weakly orthogonal polynomials in the energy obeying three-term recurrence relations.

## 1 Introduction

In addition to the interesting mathematical aspect of enlarging the set of \( sl_{2}(\mathbb {C})\) [1, 2] to \(E_{2}\)-quasi-exactly solvable models [3], the latter type also constitutes the natural framework for various physical applications in optics where the formal analogy between the Helmholtz equation and the Schrödinger equation is exploited [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Furthermore, a special case of these systems with a specific representation corresponds to the complex Mathieu equation that finds an interesting application in nonequilibrium statistical mechanics, where it corresponds to the eigenvalue equation for the collision operator in a two-dimensional classical Lorentz gas [14, 15].

Here we are mainly concerned with the extension of quasi-exactly solvable models [3, 16, 17, 18, 19] to non-Hermitian quantum mechanical systems [20, 21, 22, 23] within the above mentioned scheme. So far two different types of \(E_{2}\)-models have been constructed in [3, 24] and the main purpose of this manuscript is to investigate whether it is possible to construct a more general model that unifies the two. We show that this is indeed possible by combining the two models and introducing a new parameter into the system that interpolates between the two. In a similar fashion as the previously constructed models, also this one reduces in the double scaling limit to the complex Mathieu equation. As that equation is not fully explored analytically this limit provides an important option to obtain interesting information about the complex Mathieu system. On the other hand, for some applications it may also be sufficient to study an approximate behaviour for some finite values of the coupling constants. For that purpose we identify the parameter for which the general model is the optimal approximation for the complex Mathieu system.

Our manuscript is organized as follows: In Sect. 2 we introduce the general unifying model involving three parameters. We determine the eigenfunctions by solving the standard three-term recurrence relations for the coefficient functions and determine the energy eigenfunction from the requirement that the three-term recurrence relations reduce to a two-term relation. We devote section three to the study of the exceptional points and their vicinities in the parameter space. The explicit branch cut structure is provided that explains the so-called energy eigenvalue loops. In Sect. 4 we compute the central properties of the weakly orthogonal polynomials entering as coefficient functions in the Ansatz for the eigenfunctions, i.e. their norms, the corresponding Stieltjes measure and the momentum functionals. We state our conclusions in Sect. 5.

## 2 A Unifying E2-Quasi-Exactly Solvable Model

The general notion [1, 2] underlying solvable Hamiltonian systems is that its Hamiltonian operators \(\mathcal {H}\) acting on some graded space \(V_{n}\) as \(\mathcal {H}:V_{n}\mapsto V_{n}\) preserves the flag structure \(V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{n}\subset \cdots \). A distinction is usually made between exactly and quasi-exactly solvable, depending on whether the structure preservation holds for an infinite or a finite flag, respectively. Here we are concerned with the latter. Lie algebraic versions of Hamiltonians in this context are usually taken to be of \(sl_{2}(\mathbb {C})\)-type [1, 2], but as recently proposed [3, 24], they may also be taken to be of a Euclidean Lie algebraic type, thus giving rise to qualitatively new structures.

*u*,

*v*and

*J*that obey the commutation relations

*n*th and \((n-1)\)th order polynomials \(P_{n}(E)\), \(Q_{n}(E)\) in the energies

*E*, respectively. Upon substitution into the Schrödinger equation we obtain the three-term recurrence relations

*N*. For the solutions related to the even fundamental solution in (8) we find

## 3 Exceptional Points and Their Vicinities

*E*as functions of the coupling constants, \(\lambda \) or \(\zeta \) in our case, the corresponding path in the energy plane will inevitably pass through various Riemann sheets due to the branch cut structure. As a consequence one naturally generates eigenvalue loops that stretch over several Riemann sheets. This phenomenon is well studied for a large number of models and we demonstrate here that it also occurs in quasi-exactly solvable models. The basic principle can be demonstrated with the square root singularity occurring in \(E_{2}^{c,\pm }\) with branch cuts from \((-\infty ,-1-1/\zeta )\) and \((1/\zeta -1,\infty )\). The energy loops are generated by computing \(E_{2}^{c,\pm }(\lambda =\tilde{ \lambda }+\rho e^{i\pi \phi },\zeta )\) for some fixed values of \(\zeta \), center \(\tilde{\lambda }\) and the radius \(\rho \) in the \(\lambda \)-plane as functions of \(\phi \) as illustrated in Fig. 1a, b. In panel (a) we simply trace the energy around a point in parameter space that leads to two real eigenvalues. For a small radius ones reaches the starting point by encircling \(\tilde{\lambda }\) just once. However, when the radius is increased one needs to surround \(\tilde{\lambda }\) twice to reach the starting point and when the radius is increased even further one only needs to surround \(\tilde{\lambda }\) once switching, however, between both energy eigenvalues.

Essentially this structure survives when the two eigenvalues merge into an exceptional point. However, since the exceptional point is a branch point we no longer have the option for a closed loop around it produced from only one energy eigenvalue as seen in Fig. 1b.

When more eigenvalues are present the structure will be more intricate. Considering for instance a scenario with four eigenvalues in the form of two complex conjugate eigenvalues and an exceptional point, see Fig. 3a, we need to perform again at least two turns in the \(\lambda \)-plane in order to return to the initial position for the energy loops when surrounding an exceptional point. The two complex conjugate eigenvalues may be enclosed with just one turn, albeit we require again different energy eigenvalues for this. When enlarging the radius the loops will eventually merge as depicted in Fig. 3b for a situation with a degenerate complex eigenvalue and two complex eigenvalues. We observe that for the given values we have to surround the chosen point at least three times to obtain a closed energy loop surrounding the indicated centers.

In the same manner as for the simpler scenario one may understand the nature of these loops from an analysis of the branch cut structure of the energy as seen in Fig. 4. Tracing the indicated radii at \(\rho =4.0\) and \( \rho =8.5\) in Fig. 4 produces the energy loops in Fig. 3 when properly taking care of the analytic continuation at the branch cuts.

As discussed earlier the Hamiltonian \(\mathcal {H}(N,\zeta ,\lambda )\) has the interesting property that in the double scaling limit it reduces to the complex Mathieu equation for which only incomplete information is available, especially concerning the locations of the exceptional points. In comparison with the previously analyzed models \(\mathcal {H}_{E_{2}}^{(1)}\) in [3] and \(\mathcal {H}_{E_{2}}^{(0)}\) in [24] we have now the additional parameter \(\lambda \) at our disposal and we may investigate how the complex Mathieu system is approached. In particular we may address the question of whether there exists a value \(\lambda \) for which this is optimal. Our numerical results are depicted in Fig. 5. We find a similar qualitative behaviour for the other exceptional points, which we do not report here.

Comparing the rate of the approach for different values of \(\lambda \) we conclude that \(\mathcal {H}(N,\zeta ,\lambda =1)\) is the best approximation to the complex Mathieu system for some finite values of *N*.

## 4 Weakly Orthogonal Polynomials

*p*in

*E*as

## 5 Conclusions

Following the principles outlined in [3] we have constructed a new three-parameter quasi-exactly solvable model of \(E_{2}\)-type. One of the parameters can be employed to interpolate between two previously constructed models. With regard to one of the original motivations that triggered the investigation of these models, that is the double scaling limit towards the complex Mathieu equation, we found that for \(\lambda =1\), i.e. \(\mathcal {H} _{E_{2}}^{(1)}\), finite values for *N* best approximate the complex Mathieu system and mimic its qualitative behaviour. We provided a detailed discussion of the determination of the exceptional points and the energy branch cut structure responsible for the intricate energy loop structure stretching over several Riemann sheets. The coefficient functions are shown to possess the standard properties of weakly orthogonal polynomials.

## Notes

### Acknowledgments

I am grateful to Kazuki Kanki for making [15] available to me.

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