A Unifying E2-Quasi Exactly Solvable Model

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)

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.

At present two different types of \(E_{2}\)-quasi-exactly solvable models were identified
$$\begin{aligned} \mathcal {H}_{E_{2}}^{(1)}= & {} J^{2}+\zeta ^{2}(u^{2}-v^{2})^{2}+2i\zeta N(u^{2}-v^{2}),\qquad \qquad \zeta ,N\in \mathbb {R},\quad \end{aligned}$$
(1)
$$\begin{aligned} \mathcal {H}_{E_{2}}^{(0)}= & {} J^{2}+\zeta uvJ+2i\zeta N(u^{2}-v^{2}),\quad \end{aligned}$$
(2)
in [3, 24], respectively. Both Hamiltonians are expressed in terms of the \(E_{2}\)-basis operators u, v and J that obey the commutation relations
$$\begin{aligned} \left[ u,J\right] =iv,\qquad \left[ v,J\right] =-iu,\qquad \left[ u,v\right] =0. \end{aligned}$$
(3)
Except for \(\mathcal {H}_{E_{2}}^{(0)}\) at \(N=1/4\), both Hamiltonians are non-Hermitian, but respect the anti-linear symmetry [25] \(\mathcal {PT} _{3}:\)\(J\rightarrow J\), \(u\rightarrow v\), \(v\rightarrow u\), \(i\rightarrow -i \) as defined in [10]. For the particular representation \( J:=-i\partial _{\theta }\), \(u:=\sin \theta \)\(v:=\cos \theta \) the \(\mathcal { PT}_{3}\)-symmetry is simply \(\mathcal {PT}_{3}:\theta \rightarrow \pi /2-\theta \), \(i\rightarrow -i\), such that the invariant vector spaces over \( \mathbb {R}\) were defined as
$$\begin{aligned} V_{n}^{s}(\phi _{0}): & {} =\text {span}\left\{ \left. \phi _{0}\left[ \sin (2\theta ),i\sin (4\theta ),\ldots ,i^{n+1}\sin (2n\theta )\right] \right| \theta \in \mathbb {R},\mathcal {PT}_{3}(\phi _{0})=\phi _{0}\in L\right\} ,~~~~~~ \end{aligned}$$
(4)
$$\begin{aligned} V_{n}^{c}(\phi _{0}): & {} =\text {span}\left\{ \left. \phi _{0}\left[ 1,i\cos (2\theta ),\ldots ,i^{n}\cos (2n\theta )\right] \right| \theta \in \mathbb {R},\mathcal {PT}_{3}(\phi _{0})=\phi _{0}\in L\right\} . \end{aligned}$$
(5)
In order to construct Hamiltonians that preserve the flag structure one needs to identify the action of the \(E_{2}\)-basis operators and its combinations on these spaces as explained in more detail in [3]. The behaviour found allowed to identify the Hamiltonians \(\mathcal {H} _{E_{2}}^{(1)}\) and \(\mathcal {H}_{E_{2}}^{(0)}\) in (1) and (2) as quasi-exactly solvable. The general structure suggests that there might be a master Hamiltonian that unifies the above Hamiltonians into one preserving the quasi-exact solvability. We demonstrate here that this is possible and study the properties of that model.
Thus we introduce the new Hamiltonian
$$\begin{aligned} \mathcal {H}(N,\zeta ,\lambda )=J^{2}+2(1-\lambda )\zeta uvJ+\lambda \zeta ^{2}(u^{2}-v^{2})^{2}+2i\zeta N(u^{2}-v^{2}),\qquad \lambda ,\zeta ,N\in \mathbb {R}, \end{aligned}$$
(6)
and demonstrate explicitly that it is indeed \(E_{2}\)-quasi-exactly solvable. First we observe that \(\mathcal {H}(N,\zeta ,\lambda )\) interpolates between the two models in (1) and (2) by varying \( \lambda \), since
$$\begin{aligned} \lim _{\lambda \rightarrow 1}\mathcal {H}(N,\zeta ,\lambda )=\mathcal {H} _{E_{2}}^{(1)}\quad \quad \text {and}\quad \quad \lim _{\lambda \rightarrow 0} \mathcal {H}(2N,\zeta /2,\lambda )=\mathcal {H}_{E_{2}}^{(0)}. \end{aligned}$$
(7)
Furthermore, \(\mathcal {H}(N,\zeta ,\lambda )\) reduces to the complex Mathieu Hamiltonian in the double scaling limit \(\lim _{N\rightarrow \infty ,\zeta \rightarrow 0}\mathcal {H}(N,\zeta ,\lambda )=\mathcal {H}_{\text {Mat} }=J^{2}+2ig(u^{2}-v^{2})\) for \(g:=N\zeta <\infty \). We also note that \( \mathcal {H}^{\dagger }(N,\zeta ,\lambda )=\mathcal {H}(1-\lambda -N,\zeta ,\lambda )\), which implies that \(\mathcal {H}(N,\zeta ,\lambda )\) is non-Hermitian unless \(2N=1-\lambda \), with free coupling constant \(\zeta \in \mathbb {R}\).
Given the structure for the vector spaces in (4) and (5) we now make the following Ansätze for the two fundamental solutions of the corresponding Schrödinger equation \(\mathcal {H}_{N}\psi _{N}=E\psi _{N}\)
$$\begin{aligned} \psi _{N}^{c}(\theta )=\phi _{0}\sum _{n=0}^{\infty }i^{n}c_{n}P_{n}(E)\cos (2n\theta ),\quad \text {and} \quad \psi _{N}^{s}(\theta )=\phi _{0}\sum _{n=0}^{\infty }i^{n+1}c_{n}Q_{n}(E)\sin (2n\theta ), \end{aligned}$$
(8)
where the \(\mathcal {PT}_{3}\)-symmetric ground state is taken to be \(\phi _{0}=e^{\frac{i}{2}\zeta \cos (2\theta )}\) and the constant \(c_{n}\) is \( c_{n}=1/\zeta ^{n}(N+\lambda )(1+\lambda )^{n-1}\left[ (1+N+2\lambda )/(1+\lambda )\right] _{n-1}\) with \((a)_{n}:=\Gamma \left( a+n\right) /\Gamma \left( a\right) \) denoting the Pochhammer symbol. The constants are chosen conveniently in order to ensure the simplicity of the to be determined nth 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
$$\begin{aligned} P_{2}= & {} (E-\lambda \zeta ^{2}-4)P_{1}+\mathbf {2}\zeta ^{2}\left[ N-1\right] \left[ N+\lambda \right] P_{0}, \end{aligned}$$
(9)
$$\begin{aligned} P_{n+1}= & {} (E-\lambda \zeta ^{2}-4n^{2})P_{n}+\zeta ^{2}\left[ N+n\lambda +(n-1)\right] \left[ N-(n-1)\lambda -n\right] P_{n-1}, \end{aligned}$$
(10)
$$\begin{aligned} Q_{2}= & {} (E-4-\lambda \zeta ^{2})Q_{1}, \end{aligned}$$
(11)
$$\begin{aligned} Q_{m+1}= & {} (E-\lambda \zeta ^{2}-4m^{2})Q_{m}+\zeta ^{2}\left[ N+m\lambda +(m-1)\right] \left[ N-(m-1)\lambda -m\right] Q_{m-1},~~~~~ \end{aligned}$$
(12)
for \(n=0,2,\ldots \) and for \(m=2,3,4,\ldots \). Note that a more generic Ansatz for the unifying model involving two independent coupling constants \( \mu \), \(\lambda \) in the terms \(\mu \zeta uvJ+\lambda \zeta ^{2}(u^{2}-v^{2})^{2}\) leads to a four term recurrence relation in which the highest term is always proportional to \(\mu +2\lambda -2\). Thus taking this term to zero with the appropriate choice for \(\mu \) reduces this to the desired three term relations that may be solved in complete generality as outlined in [3]. The lowest order polynomials are easily computed in a recursive way. Taking \(P_{0}=1\) we obtain
$$\begin{aligned} P_{1}= & {} E-\lambda \zeta ^{2}, \\ P_{2}= & {} \lambda ^{2}\zeta ^{4}+2\zeta ^{2}\left[ \lambda -\lambda E+N(\lambda +N-1)\right] +(E-4)E, \nonumber \\ P_{3}= & {} -\lambda ^{3}\zeta ^{6}+\lambda \zeta ^{4}\left( \lambda (2\lambda +3E-13)-3N^{2}-3(\lambda -1)N+2\right) +(E-16)(E-4)E \nonumber \\&- \, \zeta ^{2}\left[ 3\lambda E^{2}+E\left( 2\lambda ^{2}-3N^{2}-3\lambda (N+11)+3N+2\right) +32(\lambda +N(\lambda +N-1))\right] , \nonumber \end{aligned}$$
(13)
and likewise with \(Q_{1}=1\) we compute
$$\begin{aligned} Q_{2}= & {} E-4-\lambda \zeta ^{2}, \\ Q_{3}= & {} \lambda ^{2}\zeta ^{4}+\zeta ^{2}\left[ \lambda (15-2\lambda -2E)+N^{2}+(\lambda -1)N-2\right] +(E-16)(E-4), \nonumber \\ Q_{4}= & {} -\lambda ^{3}\zeta ^{6}+\lambda \zeta ^{4}\left[ 8+\lambda (8\lambda +3\text {{ E}}-38)-2N^{2}-2(\lambda -1)N\right] +(\text {{ E}}-36)( \text {{ E}}-16)(\text {{ E}}-4) \nonumber \\&+ \, \zeta ^{2}\left[ -8\left( -12\lambda ^{2}+69\lambda +5\lambda N+5(N-1)N-12\right) \right] \nonumber \\&+ \, \zeta ^{2}\left[ -3\lambda \text {{ E}}^{2}+2\text {{ E}}\left( (47-4\lambda )\lambda +N^{2}+(\lambda -1)N-4\right) \right] . \nonumber \end{aligned}$$
(14)
In both cases we observe the typical feature for quasi-exactly solvable systems that the three term relation can be reset to a two-term relation at a certain level. This is due to the fact that in (10) and (12) the last term vanishes when \(m=n=\hat{n}=-(1+N)/(1+\lambda )\) or \(m=n=\tilde{ n}=(\lambda +N)/(1+\lambda )\). Thus when taking \(N=\tilde{n}+(\tilde{n} -1)\lambda \) we find the typical factorization
$$\begin{aligned} P_{\tilde{n}+\ell }=P_{\tilde{n}}R_{\ell } \qquad \text {and} \qquad Q_{\tilde{n }+\ell }=Q_{\tilde{n}}R_{\ell }. \end{aligned}$$
(15)
The first solutions for the factor \(R_{\ell }\) are easily found from (10) and (12) to
$$\begin{aligned} R_{1}= & {} E-4\tilde{n}^{2}-\lambda \zeta ^{2}, \end{aligned}$$
(16)
$$\begin{aligned} R_{2}= & {} (E-4\tilde{n}^{2}-\lambda \zeta ^{2})(E-4(\tilde{n}+1)^{2}-\lambda \zeta ^{2})-2\tilde{n}(1+\lambda )^{2}\zeta ^{2}. \end{aligned}$$
(17)
Next we compute the energy eigenvalues \(E_{\tilde{n}}\) from the constraints \( P_{\tilde{n}}(E)=0\) and \(Q_{\tilde{n}}(E)=0\) for the lowest values of N. For the solutions related to the even fundamental solution in (8) we find
$$\begin{aligned} N= & {} 1:\qquad \ \ \ \ \ \ \ \ \ E_{1}^{c}=\lambda \zeta ^{2}, \end{aligned}$$
(18)
$$\begin{aligned} N= & {} 2+\lambda :\qquad \ \ E_{2}^{c,\pm }=2+\lambda \zeta ^{2}\pm 2\sqrt{ 1-(1+\lambda )^{2}\zeta ^{2}}, \end{aligned}$$
(19)
$$\begin{aligned} N= & {} 3+2\lambda :\qquad E_{3}^{c,\ell }=\frac{20}{3}+\lambda \zeta ^{2}+ \frac{4\hat{\Omega }}{3}e^{\frac{i\pi \ell }{3}}+\frac{1}{3}\left[ 52-12(1+\lambda )^{2}\zeta ^{2}\right] e^{-\frac{i\pi \ell }{3}}\hat{\Omega } ^{-1},\qquad ~~ \end{aligned}$$
(20)
with \(\hat{\Omega }^{3}:=35+18(\lambda +1)^{2}\zeta ^{2}+\sqrt{\left[ 3(\lambda +1)^{2}\zeta ^{2}-13\right] ^{3}+\left[ 18(\lambda +1)^{2}\zeta ^{2}+35\right] ^{2}}\), \(\ell =0,\pm 2\).
For the solutions related to the odd fundamental solution in (8) we obtain
$$\begin{aligned} N= & {} 2+\lambda :\qquad \ \ \ E_{2}^{s}=4+\lambda \zeta ^{2}, \end{aligned}$$
(21)
$$\begin{aligned} N= & {} 3+2\lambda :\qquad \ E_{3}^{s,\pm }=10+\zeta ^{2}\lambda \pm 2\sqrt{ 9-(\lambda +1)^{2}\zeta ^{2}}, \end{aligned}$$
(22)
$$\begin{aligned} N= & {} 4+3\lambda :\qquad E_{4}^{s,\ell }=\frac{56}{3}+\lambda \zeta ^{2}+ \frac{4\Omega }{3}e^{\frac{i\pi \ell }{3}}+\frac{1}{3}\left[ 196-12(1+\lambda )^{2}\zeta ^{2}\right] e^{-\frac{i\pi \ell }{3}}\Omega ^{-1},\qquad ~~ \end{aligned}$$
(23)
with \(\Omega ^{3}:=143 +18\zeta ^{2}(\lambda +1)^{2}+\sqrt{\left( 3\zeta ^{2}(\lambda +1)^{2}-49\right) ^{3}+\left( 18\zeta ^{2}(\lambda +1)^{2}+143\right) ^{2}}\), \(\ell =0,\pm 2\). Solutions for higher order may of course also be obtained, but are rather lengthy and therefore not reported here.

3 Exceptional Points and Their Vicinities

The special point in parameter space where two real energy eigenvalues viewed as functions of the coupling constants merge and subsequently split into a complex conjugate pair is usually referred to as exceptional point[26, 27, 28, 29]. In our system these points can be computed in an explicit simple and straightforward manner. Using that by definition the discriminant \(\Delta \) equals the product of the squares of the differences of all energy eigenvalues \(E_{i}\) for \(1\le i\le n\), i.e. \( \Delta =\prod \nolimits _{1\le i<j\le n}(E_{i}-E_{j})^{2}\) one obtains the exceptional points from the real zeros of \(\Delta (E)\). For practical purposes one may also exploit the fact [3], that the discriminant equals the determinant of the Sylvester matrix. This viewpoint has the advantage that it does not require the computation of all the eigenvalues and is more efficient when the sole purpose is to find the exceptional points. Thus in our case we have to find the real zeros of the discriminants \(\Delta _{\tilde{n}}^{c}\) and \(\Delta _{\tilde{n}}^{s}\) for the polynomials \( P_{\tilde{n}}(E)\) and \(Q_{\tilde{n}}(E)\), respectively. Extracting overall constant factors \(\kappa \) as \(\Delta =\kappa \tilde{\Delta }\), that do not contribute to the zeros, we obtain for the lowest values of \(\tilde{n}\)
$$\begin{aligned} \tilde{\Delta }_{2}^{c}= & {} \hat{\zeta }^{2}-1, \\ \tilde{\Delta }_{3}^{s}= & {} \hat{\zeta }^{2}-9, \nonumber \\ \tilde{\Delta }_{3}^{c}= & {} \hat{\zeta }^{6}-\hat{\zeta }^{4}+103\hat{\zeta } ^{2}-36, \nonumber \\ \tilde{\Delta }_{4}^{s}= & {} \hat{\zeta }^{6}-37\hat{\zeta }^{4}+991\hat{\zeta } ^{2}-3600, \nonumber \\ \tilde{\Delta }_{4}^{c}= & {} \hat{\zeta }^{12}+2\hat{\zeta }^{10}+385\hat{\zeta } ^{8}-33120\hat{\zeta }^{6}+16128\hat{\zeta }^{4}-732276\hat{\zeta }^{2}+129600, \nonumber \\ \tilde{\Delta }_{5}^{s}= & {} \hat{\zeta }^{12}-94\hat{\zeta }^{10}+7041\hat{\zeta } ^{8}-381600\hat{\zeta }^{6}+6645600\hat{\zeta }^{4}-78318900\hat{\zeta } ^{2}+158760000, \nonumber \end{aligned}$$
(24)
where we abbreviated \(\hat{\zeta }:=\zeta (1+\lambda )\).
There exist many detailed studies about the structures in the coupling constant space in the vicinity of the exceptional points [30, 31, 32, 33, 34]. It is evident that when tracing a complex energy eigenvalue 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.
Fig. 1

Energy eigenvalue loops \(E_{2}^{c,\pm }(\tilde{\lambda }+\rho e^{i\pi \phi },\zeta ) \) around two real eigenvalues panel (a) and around an exceptional point panel (b) as functions of \(\phi \), indicated by the numbers on the loops, for fixed value of \(\zeta =1/2\) at \(\tilde{\lambda }=1/10\) in (a) and \(\tilde{\lambda }=1 \) in (b). The energy eigenvalues for \(\rho =0\) are distinct in panel a as \(E_{2}^{c,-}=0.35\), \(E_{2}^{c,+}=3.70\) and coalesce to an exceptional point in panel b as \(E_{2}^{c,-}=E_{2}^{c,+}=9/4\)

Fig. 2

Energy levels and branch cut structure for \(E_{2}^{c,\pm }\) for fixed \(\zeta =1/2\) as functions of \(\lambda \). The branch cuts extend to the left and right from the exceptional points \(\left( -\infty ,-3\right) \) and \((1,\infty )\)

Fig. 3

Energy eigenvalues \(E_{4}^{c}(\tilde{\lambda }+\rho e^{i\pi \phi },\zeta ) \) as functions of \(\phi \), indicated by the numbers on the loops, for fixed value \(\zeta =1/2\) at \(\tilde{\lambda }=9.5284\) in (a) and \(\tilde{\lambda }=5.2562 +i 9.9526 \) in (b). The energy eigenvalues for \(\rho =0\) in panel a are \(E_{4}^{c,1}=E_{4}^{c,2}=25.6613\), \(E_{4}^{c,3}=(E_{4}^{c,4})^*=7.1029 +i 29.8106\) and \(E_{4}^{c,1}=E_{4}^{c,2}=37.7449-i 8.7611 \), \(E_{4}^{c,3}=9.8103 +i 6.7668\), \(E_{4}^{c,4}=-24.0439 +i 20.7081\) in panel (b). The radii are \(\rho =4.0\) and \(\rho =8.5\) in panels (a) and (b), respectively

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.

This behaviour is easily understood from the structure of the branch cuts as depicted in Fig. 2. Whereas for small radii it is possible to encircle for instance the point \(\tilde{\lambda }=1/10\) without crossing any branch cut, this is not possible when encircling the exceptional point at \( \tilde{\lambda }=1\) where we have to analytically continue from \(E_{2}^{c,-}\) to \(E_{2}^{c,+}\) when crossing a cut. This structure is the same for intermediate radii. For large radii we cross the first cut already at a half circle turn, such that one returns back to the original value already after one complete turn.
Fig. 4

Energy levels and branch cut structure for \(E_{2}^{c,1,2,3,4}\) for fixed \(\zeta =1/2\) as functions of \(\lambda \)

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.

If one is exclusively interested in the computation of the exceptional point it is most efficient to carry out the double scaling limit already for the three-term relation (10) and (12) as explained in [3, 24].
Fig. 5

Double scaling limit of \(\lim _{N\rightarrow \infty ,\zeta \rightarrow 0}\mathcal {H}(N,\zeta ,\lambda )=\mathcal {H}_{\text {Mat}}\) to the smallest exceptional point at \(\zeta _M = 1.46877\) with \(\Delta (n)= \zeta _0 N(n) -\zeta _M\), \(N(n)=(n+1)+n \lambda \) for \(n = 1,2,3, \ldots \)

4 Weakly Orthogonal Polynomials

It is well known from Favard’s theorem [35, 36] that polynomials \(\Phi _{n}(E)\) constructed from three-term relations in the way mentioned above possess a norm \(N_{n}^{\Phi }\)
$$\begin{aligned} \mathcal {L}(\Phi _{n}\Phi _{m})=N_{n}^{\Phi }\delta _{nm}. \end{aligned}$$
(25)
defined by the action of a linear functional \(\mathcal {L}\) acting on arbitrary polynomials p in E as
$$\begin{aligned} \mathcal {L}(p)=\int \limits _{-\infty }^{\infty }p(E)\omega (E)dE,\qquad \mathcal {L}(1)=1. \end{aligned}$$
(26)
This norm may be computed in two alternative ways. The simplest way is to multiply the three-term relation by \(\Phi _{n-1}\) and act subsequently on the resulting equation with \(\mathcal {L}\). Using the property \(N_{n}^{\Phi }= \mathcal {L}(\Phi _{n}^{2})=\mathcal {L}(E\Phi _{n-1}\Phi _{n})\) together with (25) then simply yields \(N_{n}^{\Phi }=\)\(\prod \nolimits _{k=1}^{n}b_{k}\), where the \(b_{k}\) are the negative coefficients in front of \(\Phi _{n-1}\). Whereas the first method simply assumes that the functional exist the second method goes further and actually provides an explicit expressions for the measure. As argued in [37] the concrete formulae for \(\omega (E)\) may be computed from
$$\begin{aligned} \omega (E)=\sum \limits _{k=1}^{\ell }\omega _{k}\delta (E-E_{k}), \end{aligned}$$
(27)
where the energies \(E_{k}\) are the \(\ell \) roots of the polynomial \(\Phi (E)\) . The \(\ell \) constants \(\omega _{k}\) can be determined by the \(\ell \) equations
$$\begin{aligned} \sum \limits _{k=1}^{\ell }\omega _{k}\Phi _{n}(E_{k})=\delta _{n0}\text {,} \qquad \text {for }n\in \mathbb {N}_{0}. \end{aligned}$$
(28)
In our case the integer \(\ell \) are determined from \(N=\ell +(\ell -1)\lambda \) and \(N=(\ell +1)+\ell \lambda \) for the \(P_{\ell }(E)\) and \( Q_{\ell +1}(E)\), respectively.
Using the first method we obtain
$$\begin{aligned} N_{n}^{P}= & {} 2\zeta ^{2n}(1+\lambda )^{2n}\left( \frac{1-N}{1+\lambda } \right) _{n}\left( \frac{\lambda +N}{1+\lambda }\right) _{n},~~\quad n=1,2,3,... \end{aligned}$$
(29)
$$\begin{aligned} N_{n}^{Q}= & {} \frac{1}{2(N+\lambda )(1-N)}N_{n}^{P},\qquad \qquad ~\quad \ \ \ \ \ \ \ \ \ \ \ \ \ n=2,3,4,... \end{aligned}$$
(30)
with \(N_{0}^{P}=N_{1}^{Q}=1\). Due to the non-Hermitian nature of the Hamiltonian this norm is in general not positive definite. For instance for \( N=4+3\lambda \) we have
$$\begin{aligned} N_{0}^{P}=1,~~~N_{1}^{P}=-24\zeta ^{2}(1+\lambda )^{2},~~~N_{2}^{P}=240\zeta ^{4}(1+\lambda )^{4},~~~N_{3}^{P}=-1440\zeta ^{6}(1+\lambda )^{6}. \end{aligned}$$
(31)
The exception is the class of models where the Hamiltonian becomes Hermitian, i.e. when \(\lambda =1-2N\) holds. For this value of \(\lambda \) the expressions in (29) and (30) become positive definite
$$\begin{aligned} N_{n}^{P}=2^{1+2n}\zeta ^{2n}(N-1)^{2n}\left( \frac{1}{2}\right) _{n}^{2}=2\zeta ^{2}(N-1)^{2}N_{n}^{Q}. \end{aligned}$$
(32)
Let us now consider the second method and compute explicitly the measure for a few examples. For \(N=2+\lambda \) and \(N=3+2\lambda \) we solve (28) for the even and odd solutions, respectively, to
$$\begin{aligned} \omega _{\pm }^{c}=\frac{1}{2}\pm \frac{1}{2\sqrt{1-(1+\lambda )^{2}\zeta ^{2}}},\quad \text {and} \quad \omega _{\pm }^{s}=\frac{1}{2}\pm \frac{3}{2 \sqrt{9-(1+\lambda )^{2}\zeta ^{2}}}. \end{aligned}$$
(33)
Computing now (25) with (26) agrees with (29) and (30)
$$\begin{aligned} N_{0}^{P}= & {} \mathcal {L}(P_{0}^{2})=\omega _{+}^{c}+\omega _{-}^{c}=1 \end{aligned}$$
(34)
$$\begin{aligned} N_{1}^{P}= & {} \mathcal {L}(P_{1}^{2})=\omega _{+}^{c}\left( E_{2}^{c,+}-\lambda \zeta ^{2}\right) ^{2}+\omega _{-}^{c}\left( E_{2}^{c,-}-\lambda \zeta ^{2}\right) ^{2}=-4\hat{\zeta }^{2}, \end{aligned}$$
(35)
$$\begin{aligned} N_{2}^{Q}= & {} \mathcal {L}(Q_{2}^{2})=\omega _{+}^{s}\left( E_{3}^{s,+}-4-\lambda \zeta ^{2}\right) ^{2}+\omega _{-}^{s}\left( E_{3}^{s,-}-4-\lambda \zeta ^{2}\right) ^{2}=-4\hat{\zeta }^{2}.~~~ \end{aligned}$$
(36)
Similarly we compute for \(N=3+2\lambda \)
$$\begin{aligned} \omega _{1}^{c}= & {} \frac{1}{3}-\frac{\left( 260-60\hat{\zeta }^{2}\right) \Omega +\left( 3\hat{\zeta }^{2}+4\right) \Omega ^{2}+20\Omega ^{3}}{12\left[ \left( 13-3\hat{\zeta }^{2}\right) ^{2}+\left( 13-3\hat{\zeta }^{2}\right) \Omega ^{2}+\Omega ^{4}\right] },\qquad \omega _{2}^{c}=\chi _{-2},\qquad \omega _{3}^{c}=\chi _{2}, \\ \chi _{\ell }= & {} \frac{1}{3}+\frac{\left( 3\hat{\zeta }^{2}-20\Omega +4\right) \left( 1+2e^{\frac{i\pi \ell }{3}}\right) }{36(3\hat{\zeta } ^{2}+\Omega ^{2}-13)}+\frac{4+3\hat{\zeta }^{2}-20e^{\frac{i\pi \ell }{3} }\Omega }{12\left( 1+2e^{\frac{i\pi \ell }{3}}\right) \left( 3\hat{\zeta } ^{2}-13\right) +\left( 1-e^{\frac{i\pi \ell }{3}}\right) \Omega ^{2}} \nonumber \end{aligned}$$
(37)
and confirm that
$$\begin{aligned} N_{0}^{P}= & {} \mathcal {L}(P_{0}^{2})=\omega _{1}^{c}+\omega _{2}^{c}+\omega _{3}^{c}=1, \\ N_{1}^{P}= & {} \mathcal {L}(P_{1}^{2})=\omega _{1}^{c}P_{1}^{2}(E_{3}^{c,0})+\omega _{2}^{c}P_{1}^{2}(E_{3}^{c,-2})+\omega _{3}^{c}P_{1}^{2}(E_{3}^{c,2})=-12\hat{\zeta }^{2}, \nonumber \\ N_{2}^{P}= & {} \mathcal {L}(P_{2}^{2})=\omega _{1}^{c}P_{2}^{2}(E_{3}^{c,0})+\omega _{2}^{c}P_{2}^{2}(E_{3}^{c,-2})+\omega _{3}^{c}P_{2}^{2}(E_{3}^{c,2})=48\hat{\zeta }^{4} \nonumber \\ \mathcal {L}(P_{1}P_{2})= & {} \omega _{1}^{c}P_{1}(E_{3}^{c,0})P_{2}(E_{3}^{c,0})+\omega _{2}^{c}P_{1}(E_{3}^{c,-2})P_{2}(E_{3}^{c,-2})+\omega _{3}^{c}P_{1}(E_{3}^{c,2})P_{2}(E_{3}^{c,2})=0. \nonumber \end{aligned}$$
(38)
Note that the last relation in (38) does not follow from the first method.
As the final quantity we also compute the moment functionals defined in [35, 36] as
$$\begin{aligned} \mu _{n}:=\mathcal {L}(E^{n})=\sum \limits _{k=1}^{\ell }\omega _{k}E_{k}^{n}=\sum \limits _{k=0}^{n-1}\nu _{k}^{(n)}\mu _{k}, \end{aligned}$$
(39)
Once again also these quantities can be obtained in two alternative ways, that is either from the computation of the integrals or directly from the original polynomials \(P_{n}\) and \(Q_{n}\) without the knowledge of the constants \(\omega _{k}\). In the last equation the coefficients \(\nu _{k}^{(n)}\) are defined through the expansion \(P_{n}(E)=2^{n-1}E^{n}-\sum \nolimits _{k=0}^{n-1}\nu _{k}^{(n)}E^{k}\) and \(Q_{n}(E)=2^{n-1}E^{n-1}- \sum \nolimits _{k=0}^{n-2}\nu _{k}^{(n)}E^{k}\) for our even and odd solutions, respectively. For the even solutions with \(N=2+\lambda \) we obtain
$$\begin{aligned} \mu _{0}^{P}= & {} 1, \end{aligned}$$
(40)
$$\begin{aligned} \mu _{1}^{P}= & {} \lambda \zeta ^{2}, \end{aligned}$$
(41)
$$\begin{aligned} \mu _{2}^{P}= & {} \lambda ^{2}\zeta ^{4}-4\hat{\zeta }^{2}, \end{aligned}$$
(42)
$$\begin{aligned} \mu _{3}^{P}= & {} \lambda ^{3}\zeta ^{6}-12\lambda \zeta ^{2}\hat{\zeta }^{2}-16 \hat{\zeta }^{2}, \end{aligned}$$
(43)
$$\begin{aligned} \mu _{4}^{P}= & {} \lambda ^{4}\zeta ^{8}-24\lambda ^{2}\zeta ^{4}\hat{\zeta } ^{2}+16\left( \zeta ^{2}-1\right) ^{2}\zeta ^{4}-64\hat{\zeta }^{2}, \end{aligned}$$
(44)
and similarly for the odd solutions with \(N=3+2\lambda \) we compute for instance
$$\begin{aligned} \mu _{0}^{Q}= & {} 1, \end{aligned}$$
(45)
$$\begin{aligned} \mu _{1}^{Q}= & {} 4+\lambda \zeta ^{2}, \end{aligned}$$
(46)
$$\begin{aligned} \mu _{2}^{Q}= & {} 16-4\hat{\zeta }^{2}+\lambda ^{2}\zeta ^{4}, \end{aligned}$$
(47)
$$\begin{aligned} \mu _{3}^{Q}= & {} \lambda ^{3}\zeta ^{6}-12\left( \lambda ^{3}+\lambda ^{2}+\lambda \right) \zeta ^{4}-48(2\lambda ^{2}+3\lambda +2)\zeta ^{2}+64. \end{aligned}$$
(48)
Thus \(\mathcal {H}(N,\zeta ,\lambda )\) possesses indeed all the standard features of a quasi-exactly solvable model of \(E_{2}\)-type.

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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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsCity University LondonLondonUK

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