1 Introduction

Quantum systems described by non-Hermitian interactions have been recently discovered in studies of coupled optical resonators [1]. Here we investigate a new quantization procedure for a previously studied two dimensional system of harmonic oscillators coupled by a parity-time (𝓟𝓣) symmetric non-Hermitian perturbation potential [2] parametrized by the strength parameter g. The Schrödinger equation for this system reduces to:

$$ -\frac{\partial^{2}\psi}{\partial x^{2}} - \frac{\partial^{2}\psi}{\partial y^{2}} + \left( x^{2} + y^{2} + igx^{2} y \right) \psi = 2 E \psi . $$
(1)

Parity-time invariant potentials, for which V ∗(x, y) = −V(x, y), can display an unbroken symmetry for a certain range of its parameters, when several low-lying energy levels (or even all) are real eigenvalues, or a broken 𝓟𝓣 symmetry regime with complex eigenvalues. The partial parity symmetry of (1) V(−x, y) = V(x, y) defines two classes of solutions: x-odd and x-even wave functions.

Direct numerical diagonalization of this system has been utilized to provide results for small values of the strength parameter g ≲ 0.4 [3]. An alternative algebraic method exploiting the power moment structure of this system was previously reported, referred to as the Orthogonal Polynomial Projection Quantization (OPPQ) method [4]. Excellent results were obtained consistent with the intricate analysis of the earlier studies.

The OPPQ analysis is intrinsically global since it depends on the power moments. It consists of weighted polynomial expansions,

$$\begin{array}{@{}rcl@{}} {\Psi}(x,y) = \sum\limits_{n_{1},n_{2}} {\Omega}_{n_{1},n_{2}} P_{n_{1},n_{2}}(x,y) R(x,y), \end{array} $$
(2)

for different types of Freudian weights, R(x, y) = e x p(−|x|q1 − |y|q2) > 0, etc., and their orthogonal polynomials: \(\langle P_{m_{1},m_{2}}|R|P_{n_{1},n_{2}}\rangle = \delta _{m_{1},n_{1}}\delta _{m_{2},n_{2}}\). The projection coefficients are generated algebraically, from the power moments of Ψ, which in turn satisfy a linear recursion relation involving the energy as a parameter (i.e. the moment equation). Pure OPPQ quantization is achieved by approximating the condition \(\lim _{n_{1},n_{2} \rightarrow \infty } {\Omega }_{n_{1},n_{2}} = 0\).

It is known, at least in one dimension, that for appropriate weights, these expansions are point-wise convergent [5]. In one space dimension, it is possible to modify the traditional OPPQ analysis into one that couples the power moments to the local structure (at the origin) of the wavefunction. This is referred to as a global-local analysis. The ability of this approach to reproduce the energies obtained through a pure OPPQ analysis then suggests the adequacy of the weighted polynomial expansion in recovering the local structure of the wavefunction.

In this work we report results that show how one combines local scale (Taylor expansion) information that forms the basis of Hill determinant method, which is not always reliable, with a global representation like OPPQ, to obtain in a robust way convergent eigenstates, accurate at both energy and wave function level. Therefore, the advantage of the local-global approach is two folded: firstly, it ensure the correct asymptotic behavior of the wave function, both at the origin and at infinity, leading to increase accuracy for the determination of wave functions, and secondly, it improves the convergence rate when compared with the original OPPQ solutions. Both of these advantages are demonstrated in great detail in this paper [6].

2 Results

The target local structure is that embodied in the traditional Hill determinant representation for the weight R(x, y), amply reviewed in reference [7]. For a two dimensional system, the Hill determinant method assumes a power series expansion

$$ \psi(x,y) = \sum\limits_{j,k \in \mathcal{A}_{C}} C_{jk}\ x^{j} y^{k} R(x,y) $$
(3)

for a convenient choice of the reference function R(x, y), which is traditionally chosen to be the known asymptotic form of the wave function far from the origin. For this investigation we chose R(x, y) = exp(−(x 2 + y 2)/2). Coupling this expansion with Schrödinger (1) results in a recursion for the power series coefficients

$$ -(j+1)(j+2)C_{j+2,k} - (k+1)(k+2) C_{j,k+2} + (2j + 2k + 2 - 2 E)C_{j,k} + i g C_{j-2,k-1} = 0 , $$
(4)

which allows calculation of all C coefficients from two sets of independent ones (which initiate the recursive structure), for example C 0,k and C 1,k , for a range of k = 0, 1, … , P. With this choice, all coefficients \(k,j \in \mathcal {A}_{C}\) for which k + 2⌊j/2⌋ ≤ P can be calculated, where the floor function ⌊x⌋ represents the largest integer not greater then x.

The Hill determinant quantization method, which is also algebraic in structure, involves terminating the power series (3) at a some finite level, imposed by requiring that C j k = 0 for appropriate combinations of indices j and k. This results in a finite (truncated) set of linear constraints for the impacted independent C’s whose determinant becomes a polynomial equation in energy E . The roots of this determinant can generate solutions convergent (in the truncation order) to the eigenfunctions of (1).

Despite the elegance and early successes for certain one-dimensional problems, the original Hill determinant method is limited, and even fails to provide correct results [8], mainly because the expansion (3) does not distinguished among physical and un-physical solutions (non-square integrable), leading to the promotion of the dominant solution by the recursion (4), which in some cases is not desirable [9].

A more robust alternative to the Hill determinant method is obtained by employing the power moments of the wave function instead of its Taylor’s coefficients C. The advantage of this approach is that the power moments defined for the present 2D system as

$$\mu_{p,q} = {\int}_{\!\!\!\!-\infty}^{\infty}{\int}_{\!\!\!\!-\infty}^{\infty} \psi(x,y) x^{p} y^{q} \; dx dy $$

exist only for square integrable wave functions, and therefore the non-physical solutions are automatically eliminated. By multiplying (1) with x p y q and integrating over the whole x − y plane one obtains a moment equation

$$ -p(p-1) \mu_{p-2,q} - q(q-1) \mu_{p,q-2} + \mu_{p+2, q} + \mu_{p,q+2} + i g\mu_{p+2,q+1} - 2 E \mu_{p,q} = 0 $$
(5)

that can be used to recursively calculate higher order moments from a set of initial, or missing, moments. For example, if the set of moments μ p, 0 and μ p, 1, with p = 0,1, … , N is chosen then all moments μ p q with p + 2q ≤ N are calculated as a function of the missing 2(N + 1) moments and energy E. This information alone is not sufficient to determine the eigenvalues of (1).

For exactly and quasi-exactly solvable one dimensional systems the consistency of the algebraic structure of the moment equations is enough to provide the eigenvalues [10].

In those cases where the desired solution is known to be nonnegative (i.e. the bosonic ground state, and other states depending on the wavefunction representation chosen), then the power moments must satisfy additional constraints consistent with the underlying nonnegative character of the solution. This usually takes on the form of the Hankel-Hadamard determinantal positivity inequalities from the well known moment problem in mathematics. This type of analysis involves convex optimization and its first applications define the first introduction of semidefinite programming type analysis to quantum operators, resulting in converging lower and upper bounds to the physical energies [11]. Its first important application was to the singular perturbation strong coupling problem associated with the quadratic Zeeman effect for superstrong magnetic fields [12, 13]. These works implemented a linear programming analysis of the underlying semidefinite programming problem. The next important application was its use on complex potentials, accurately predicting the onset of PT-symmetry breaking for the V(x) = i x 3 + i a x potential [14].

Instead of using positivity requirements, a different set of conditions is obtained by representing the wave function as

$$ \psi(x,y) = \sum\limits_{p,q \in \mathcal{A}_{\mu}} {\Omega}_{pq} P_{p}(x) P_{q}(y) R(x,y) $$
(6)

where summation goes over all elements in the set \(\mathcal {A}_{\mu }\) of moments that can be calculated from the initial set of 2(N + 1) missing moments, R(x, y) is the reference function, exp(−(x 2 + y 2)/2) in the present case, and P p (x) are a set of polynomials orthogonal with respect to the chosen reference function, such that

$${\int}_{\!\!\!\!-\infty}^{\infty} P_{i}(x) P_{j}(x) R(x)\; dx = \mathcal{N}_{i} \delta_{ij} . $$

Here R(x) = exp(−x 2/2) and \(\mathcal {N}\) represents the normalization factor. Up to a multiplicative factor, polynomials \(P_{k}(x) \sim H_{k}(x/\sqrt {2})\) relate to classical Hermite orthogonal polynomials H k . The OPPQ condition requires that Ω p q = 0 for a set of indices p and q. One generates a truncated set of linear relations in the independent, or missing, moments. The resulting determinantal roots in the energy variable are generated, which rapidly converge to the physical values, as the truncation order is increased. An added advantage is that unlike the Hill determinant approach which tends to become unstable when the reference function, or weight, is chosen to mimic the asymptotic form of the physical solutions, the OPPQ approach has no such instabilities. The better the reference function mimics the true asymptotic behavior, the faster the convergence of the energy approximants. This principle was demonstrated with good results for a two-dimensional dipole problem [15] in addition to the problem at hand, as given in [4].

We note that if the polynomials are given by \(P_{s}(x) = {\sum }_{j} \pi _{s,j} x^{j}\) then the Ω p q coefficients result from

$$\mathcal{N}_{p}\mathcal{N}_{q} {\Omega}_{pq} = {\int}_{\!\!\!\!-\infty}^{\infty} {\int}_{\!\!\!\!-\infty}^{\infty} {\Psi}(x,y) P_{p}(x) P_{q}(y)\; dx dy = \sum\limits_{j} \sum\limits_{k} \pi_{p,j} \pi_{q,k} \mu_{p,q} $$

without explicit knowledge of the wave function Ψ(x, y). Their calculation requires only the knowledge of power moments μ p q , which are related linearly, via recursions in (5), to the smaller set of 2(N + 1) missing moments. Therefore, by requiring that Ω p q = 0 for a set of 2(N + 1) indices given by p + 2q = N and p + 2q = N − 1, with p = 0, 1, … N, one obtains a homogenous system of linear equations with the determinant given as a polynomial in energy. The roots of this determinant are converging approximations to the eigenvalues of (1). Because the set of power moments represents global properties of the wave function, OPPQ and any other method that relies on the use of moments, inherits certain global properties.

As noted at the outset, we wish to test the efficacy by which the OPPQ representation might recover the local structure of the wave function at the origin. Our approach is to combine the global features of the moment representation analysis previously given, with the local approximation character of the representation in (6). Instead of imposing OPPQ conditions, we report the preliminary results obtained by combining the local representation (3) with the global representation (6), which requires the identity

$$ \sum\limits_{p,q \in \mathcal{A}_{\mu}} {\Omega}_{pq} P_{p}(x) P_{q}(y) = \sum\limits_{j,k \in \mathcal{A}_{C}} C_{jk} x^{j} y^{k} $$
(7)

In turn, this results in power-by-power equalities

$$ \sum\limits_{p,q \in \mathcal{A}_{C}} {\Omega}_{p,q}\pi_{p,j}\pi_{q,k} = C_{jk} . $$
(8)

By using Taylor’s coefficient recursion (4), and the recursion for moments (5), Eq. (8) become a homogeneous linear system of 2P + 2N + 4 equations that has non-trivial solutions only when its energy dependent determinant is zero. In situations when the number of equations in (8) does not match the number of unknowns, we complemented the system (8) with OPPQ type conditions Ω p q = 0 from the set of indices p + 2q = N and p + 2q = N − 1 until the match is satisfied.

Since the OPPQ representation is expected to be pointwise convergent, and not uniformly convergent, our strategy requires that a fixed (but arbitrary) number of Cs be chosen (dependent on a subset of independent C-variables), with an increasing set of independent moment variables. We constrain the independent C’s and missing moments through (8) as well as additional constraints from the original OPPQ conditions, in order to close the system and generate a finite dimensional determinant.

Figure 1 shows the results obtained for the first 10 eigenvalues with parameters N = 31 (for the moments) and P = 4 (for the C’s), as a function of strength parameter g. Energies with the same symmetry (x-even or x-odd) can evolve to display merging at critical g values, when they become complex conjugate of each other, or avoided crossing. Energy from different symmetries have real crossings. These results are consistent with those reported in [4] upon rescaling the g-parameter. For this calculation, given the agreement between the eigenvalues, we are confident that the OPPQ representation can recover the local structure of the wave function, provided sufficient numbers of moment variables are used.

Fig. 1
figure 1

Real part of the lowest energy levels for the coupled 𝓟𝓣 invariant harmonic oscillator system as a function of the strength parameter g. Solid lines represent states of even x-parity, while dashed lines are states odd under x-parity. In the g → 0 limit the system decouples and energy levels are degenerate. 𝓟𝓣-phase transitions are evident at critical values of g when the 𝓟𝓣 symmetry breaks and pairs of eigenvalues merge and become complex conjugate

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