Symmetry, Integrability and Geometry: Methods and Applications Two-Dimensional Supersymmetric Quantum Mechanics: Two Fixed Centers of Force ⋆

The problem of building supersymmetry in the quantum mechanics of two Coulombian centers of force is analyzed. It is shown that there are essentially two ways of proceeding. The spectral problems of the SUSY (scalar) Hamiltonians are quite similar and become tantamount to solving entangled families of Razavy and Whittaker-Hill equations in the first approach. When the two centers have the same strength, the Whittaker-Hill equations reduce to Mathieu equations. In the second approach, the spectral problems are much more difficult to solve but one can still find the zero-energy ground states.


Introduction
Supersymmetry is a bold idea which arose in the seventies in string and field theory. It was immediately realized that mechanisms of spontaneous supersymmetry breaking should be investigated searching for explanations of the apparent lack of supersymmetry in nature. In a series of papers [1,2,3] Witten proposed to analyze this phenomenon in the simplest possible setting: supersymmetric quantum mechanics. A new area of research in quantum mechanics was born with far-reaching consequences both in mathematics and physics.
Of course, there were antecedents in ordinary quantum mechanics (nihil novum sub sole), and indeed even before. The track can be followed back to some work by Clifford on the Laplacian operator, see [4], quoted in [5]. Recast in modern SUSY language, the Clifford supercharge is: where i, j, k are the imaginary unit quaternions and are the components of the gradient modified by the components of the electromagnetic vector potential. The SUSY Hamiltonian is: the Laplacian plus Pauli terms. Needless to say, an identical construction relates the Dirac operator in electromagnetic and/or gravitational fields backgrounds with the Klein-Gordon operator. The factorization method of identifying the spectra of Schrodinger operators, see [6] for a review, is another antecedent of supersymmetric quantum mechanics that can also be traced back to the 19th century through the Darboux theorem. In its modern version, supersymmetric quantum mechanics prompted the study of many one-dimensional systems from a physical point of view. A good deal of this work can be found in [7,8,9,10]. A frequent starting point in this framework is the following problem: given a non-SUSY one-dimensional quantum Hamiltonian, is it possible to build a supersymmetric extension? The answer to this question is positive when one finds a solution to the Riccati equation that identifies the -a priori unknown -"superpotential" W (x) from the -given -potential V (x) . Several examples of this strategy have been worked out in [11]. The formalism of physical supersymmetric systems with more than one degree of freedom was first developed by Andrianov, Ioffe and coworkers in a series of papers [12,13], published in the eighties. The same authors, almost simultaneously, considered higher than one-dimensional SUSY quantum mechanics from the point of view of the factorization of N -dimensional quantum systems [14,15]. Factorability, even though essential in N -dimensional SUSY quantum mechanics, is not so effective as compared with the one-dimensional situation. Some degree of separability is also necessary to achieve analytical results. For this reason we started a program of research in the two-dimensional supersymmetric classical mechanics of Liouville systems [16]; i.e., those separable in elliptic, polar, parabolic, or Cartesian coordinates, see the papers [17] and [18]. We followed this path in the quantum domain for Type I Liouville models in [19], whereas Ioffe et al. also studied the interplay between supersymmetry and integrability in quantum and classical settings in other type of models in [20,21]. In these papers, a new structure was introduced: second-order supercharges provided intertwined scalar Hamiltonians even in the two-dimensional case, see [23] for a review. This higher-order SUSY algebra allows for new forms of non-conventional separability in two dimensions. There are two possibilities: (1) a similarity transformation performs separation of variables in the supercharges and some eigenfunctions (partial solvability) can be found, see [24,25]. (2) One of the two intertwined Hamiltonian allows for exact separability: the spectrum of the other is consequently known [26,27].
The second difficulty with the jump in dimensions is the identification of the superpotential. Instead of the Riccati equation one must solve the PDE: In our case, we look for solutions of this PDE when V ( x) is the potential energy of the two Coulombian centers. We do not know how to solve it in general, but two different strategies should help us. First, following the work in [38] and [39] on the supersymmetric Coulomb problem, we shall choose the superpotential as the solution of the Poisson equation: The superpotential will be the solution of another Riccati-like PDE where a classical piece must be added to the potential of the two centers. Second, as in [41,34] the selection of superpotential requires the solution of the Hamilton-Jacobi equation: Again the superpotential must solve a third Riccati-like PDE, where now a quantum piece must be added to the potential of the two centers. The organization of the paper is as follows: We start by briefly recalling the non-SUSY classical, Newtonian [40], and quantal, Coulombian [42,33], two-center problem. We shall constrain the particle to move in one plane containing the two centers. The third coordinate is cyclic and it would be easy to extend our results to three dimensions. In Section 2 the formalism of two-dimensional SUSY quantum mechanics is developed, and the superpotential of the first Type is identified for two Coulombian centers. Bosonic zero-energy ground states are also found. Sections 3 and 4 are devoted to formulating the SUSY system in elliptic coordinates where the problem is separable in order to find fermionic zero-energy ground states. It is also shown that the spectral problem is tantamount to families of two ODE's of Razavy [43], and Whittaker-Hill type [44], see also [28]. Since these systems are quasi-exactly solvable, several eigenvalues are found following the work in [32]. In Section 5 two centers of the same strength are studied and some eigenfunctions are also found. Section 6 is fully devoted to the analysis of the Manton-Heumann approach applied to the two Coulombian centers. Finally, a summary is offered in Section 7.

The classical problem of two Newtonian/Coulombian centers
d d x 1 x 2 r 2 r 1 Figure 1. Location of the two centers and distances to the particle from the centers.
The classical action for a system of a light particle moving in a plane around two heavy bodies which are sources of static Newtonian/Coulombian forces is: The centers are located at the points (x 1 = −d, x 2 = 0), (x 1 = d, x 2 = 0), their strengths are α 1 = α ≥ α 2 = δα > 0, δ ∈ (0, 1], and are the distances from the particle to the centers. In the following formulas we show the dimensions of the coupling constants and parameters and define non-dimensional variables: In the rest of the paper we shall use non-dimensional variables. From the non-dimensional actioñ the linear momenta and Hamiltonian are defined: This system is completely integrable because there exists a "second invariant" in involution with the Hamiltonian:

The quantum problem of two Coulombian centers of force
If √ mαd is of the order of the Planck constant , the system is of quantum nature. Canonical quantization in terms of the non-dimensional¯ constant, converts the dynamical variables into operators. The quantum Hamiltonian,Ĥ = α dĤ , and the quantum symmetry operator,Î 2 = (mdα)Î 2 , are mutually commuting operators: 2 Two-dimensional N = 2 SUSY quantum mechanics We now describe how to build a non-specific system in two-dimensional N = 2 SUSY quantum mechanics. Besides commuting -non-commuting -operators there are anti-commuting -nonanti-commuting operators to be referred respectively as "bosonic" and "fermionic" by analogy with QFT. The Fermi operators are represented on Euclidean spinors in R 4 by the Hermitian 4 × 4 gamma matrices: The building blocks of the SUSY system are the two (N = 2) quantum Hermitian supercharges: It is convenient to define the non-Hermitian superchargesQ ± =Q 1 ± iQ 2 , because their anti-commutator determines the HamiltonianĤ S of the supersymmetric system: The explicit form of the quantum SUSY Hamiltonian is enlightened by the "Fermi" number It has a block diagonal structure acting on the sub-spaces of the Hilbert space of Fermi numbers 0, 1, and 2. In the sub-spaces of Fermi numbers evenĤ S acts as ordinary differential Schrödinger operators. The scalar Hamiltonians are: In the sub-space of Fermi number 1, however,Ĥ S is a matrix of differential operators, the 2 × 2 matrix Hamiltonian: This is exactly the structure unveiled in [12,13]. All the interactions expressed inĤ S come from the as yet unspecified function W (x 1 , x 2 ), which is thus called the superpotential.

The superpotential I for the two-center problem
To build a supersymmetric system containing the interactions due to two Coulombian centers of force, we must start by identifying the superpotential. One possible choice, inspired by [38], is having the two-center potential energy inĥ (0) in the term proportional to¯ . We must therefore solve the Poisson equation to find the superpotential I: Note that the anticommutator between the supercharges induces a¯ factor in front of the Laplacian. This fact, in turns, forces the singularity of the superpotential (henceforth, also of the potential) at the classical limit¯ = 0. The same singularity arises in the bound state spectra of atoms, e.g., in the energy levels of the hydrogen atom. The potential energies in the scalar sectors are accordingly: We stress that the superpotential I is a solution of the Riccati-like PDE's: I and the scalar Hamiltonians for two SUSY Coulombian centers read:

Bosonic zero modes I
The bosonic zero modeŝ if normalizable, are the bosonic ground states of the system: The norm of the true bosonic ground state Ψ (0) 0 is finite and given in terms of Bessel functions: In Fig. 3 plots of the zero energy bosonic ground state probability density of finding the particle in some area of the plane are shown for several values of¯ . The drawings reveal the physical meaning of the¯ = 0 singularity: for¯ = 0.2 we see the particle probability density peaked around the center on the right with a very small probability. Exactly at¯ = 0, e − 4r 1 2 is finite only for r 1 = 0 and zero otherwise, giving probability of finding the particle exactly in the center. e − 4δr 2 2 , however, is zero ∀ r 2 meaning that the probability of this state is zero at the classical limit; in classical mechanics there are no isolated discrete states.

Two-dimensional N = 2 SUSY quantum mechanics in elliptic coordinates
The search for more eigenfunctions of the SUSY Hamiltonian requires the use of the separability in elliptic coordinates of the problem at hand. This, in turn, needs the translation of our twodimensional N = 2 SUSY system to elliptic coordinates. A general reference (in Russian) where SUSY quantum mechanics is formulated in curvilinear coordinates is [36], see also [37] to find a more geometric version of SUSY QM on Riemannian manifolds. The change from Cartesian to elliptic coordinates, induces a map from the plane to the infinite elliptic strip: . This map also induces a non-Euclidean (but flat) metric in E 2 : with Christoffel symbols: , Using the zweig-bein chosen in this form, we now define "elliptic" spinors, "elliptic" Fermi operators, and "elliptic" supercharges: To obtain the supercharges in Cartesian coordinates from the supercharges in elliptic coordinates, besides expressing u and v in terms of x 1 and x 2 , one needs to act by conjugation with the idempotent, Hermitian matrix: is separable: with separation constant κ. The general solution of equation (2) depends on two integration constants (besides an unimportant additive constant): We find thus a three-parametric family of supersymmetric models for which the potentials in the scalar sectors are: We shall restrict ourselves in the sequel (as before) to the simplest choice κ = C 1 = C 2 = 0 such that we shall work with the "elliptic" superpotential Î because this election is significative and contains enough complexity. Considering families of superpotentials related to the same physical system in our 2D framework differs from a similar analysis on the 1D SUSY oscillator, see e.g. [29], in two aspects: (a) Because we solve the Poisson equation, not the Riccati equation, the family of superpotentials induces different families of potentials in bothĥ (0) andĥ (2) . (b)ĥ (0) andĥ (2) are not iso-spectral because are not directly intertwined. The spectrum ofĥ (1) is the union of the spectra ofĥ (0) andĥ (2) .

Fermionic zero modes I
The fermionic zero modeŝ are fermionic ground states if normalizable. Because the norm is: it is finite if either A 1 = 0 or A 2 = 0. With our choice of sign in F (u) the fermionic ground state is: and the norm is also given in terms of Bessel functions: It is also possible to give the fermionic ground state in R 2 using the S matrix:

The bosonic spectral problem I
The spectral problem for the scalar Hamiltonians is also separable in elliptic coordinates. Plugging in the separation ansatẑ in the above spectral equation we find: where I is the eigenvalue of the symmetry operatorÎ = −{ĥ ( (0) (2) ) +Î Research on the solution of these two ODE's by power series expansions will be published elsewhere. Here, we shall describe how another change of variables transmutes the first ODE into Razavy equation [43,28]: with parameters: Simili modo, another change of variables leads from the second ODE to the Whittaker-Hill or Razavy trigonometric [44,28], equation with parameters: Therefore, the spectral problem in the scalar sectors is tantamount to the solving of two entangled sets -one per each pair (E,I) -of Razavy and Whittaker-Hill equations. If M ± = n ± 1 + 1, n ± 1 ∈ N + , the Razavy equation is QES; i.e., there are known n + 1 finite eigenfunctions with an eigenvalue, see [32]: This means that for those values of E ± one expects bound eigenstates of the SUSY Hamiltonian, although the v-dependence cannot be identified. If N ± = n ± 2 + 1, n ± 2 ∈ N + , there exist finite eigenfunctions of the Whittaker-Hill equation, with eigenvalues: Again, one expects these values of E ± to be eigenvalues of the SUSY Hamiltonian, although their eigenfunctions are expected to be non-normalizable (except the n + 2 = 0 case) because the u-dependent part of the eigenfunction is non finite and u is a non-compact variable. In any case, n + 1 = n + 2 = 0 gives the bosonic zero mode.

The fermionic spectrum I
The eigenfunctions of the matrix Hamiltonian, h except the fermionic ground states, follows easily from the SUSY algebra. The fermionic eigenfunctions in elliptic coordinates,

Two centers of the same strength
If the two centers have the same strength, δ = 1, the spectral problem in the scalar sectors becomes tantamount to the two ODE's: Identical changes of variables as those performed in the 0 ≤ δ ≤ 1 cases now lead to the Razavy and Mathieu equations [35], with parameters: We now select the three lowest energy levels from finite solutions of the Razavy equation: The corresponding eigenfunctions of the Razavy Hamiltonians for n ± 1 = 0, 1, 2 are: The energy degeneracy is labeled by the eigenvalues of the symmetry operator . . , n + 1, In this rotationally non-invariant system the symmetry operatorÎ replaces the orbital angular momentum in providing a basis of common eigenfunctions withH in each degenerate in energy sub-space of the Hilbert space in such a way that a quantum number reminiscent of the orbital angular momentum arises. Next we next consider even/odd in v solutions of the Mathieu equations ξ nm ±even (v) = c 1 2 C a nm ± , q n , arccos(v) + C a nm ± , q n , arccos(−v) The reason is that the invariance of the problem under r 1 ↔ r 2 ≡ v ↔ −v -the exchange between the two centers -forces even or odd eigenfunctions in v. The parameters of the Mathieu equations determined by the spectral problem are: To fit in with the parameters of the Razavy Hamiltonians set by n ± 1 = 0, 1, 2 we must choose: Therefore, is a set of bound states of non-zero energy of the scalar Hamiltonian of two SUSY Coulombian centers of the same strength. The paired fermionic eigenstates are obtained through the action of the appropriate supercharge. Since every non-zero-energy state come in bosonic-fermionic pairs, the criterion for spontaneous supersymmetry breaking is the existence of a Fermi-Bose pair ground state of positive energy connected one with each other by one of the supercharges.

Bosonic and fermionic ground states for two centers of the same strength
For comparison, we also plot the bosonic and fermionic ground states for several values of¯ .   6 The superpotential II for the two-center problem There is another possibility to find the potential energy of two Coulombian centers in a 2D N = 2 SUSY Hamiltonian. The superpotential must satisfy the Hamilton-Jacobi equation rather than the Poisson equation (1): Note that in this case the two centers must be repulsive to guarantee a real W . This point of view, which follows the path shown in [41] and [34] for the Coulomb problem, amounts to the quantization of a classical supersymmetric system, only semi-positive definite for repulsive potentials. Again using elliptic coordinates, the Hamilton-Jacobi equation separates by plugging in (3) the ansatz: The quadratures show that the separation constant κ is constrained in order to find real F a (u) and G b (v): 2(1 − δ) ≤ κ ≤ 2(1 + δ). Note that there are two different possibilities: a = b and a = b.
A different global sign only exchanges the (0) with the (2) and the (1)1 with the (1)2 sectors. The superpotential II is thus given in terms of incomplete and complete elliptic integrals of the first and second type, see [30,31]: The scalar Hamiltonians, both in elliptic and Cartesian coordinates, read:

Type IIa and Type IIb two-center SUSY quantum mechanics
We now present the graphics of the scalar potential for a = b, a system that we shall call Type IIa N = 2 SUSY two Coulombian centers. By the same token, the system arising from the superpotential I will be called Type I N = 2 SUSY two Coulombian centers. If a = b, we shall call the system Type IIb N = 2 SUSY two Coulombian centers. The scalar potential is drawn in the graphics below for several values of¯ .
The differences between the Type IIa and Type IIb potentials, increasing with¯ , are shown in the next figure.
Below we shall discuss the coincidences and differences of the three distinct points of view.

The spectral problem
Both for the Type IIa and Type IIb systems the spectral problem in the scalar sectors is separable in elliptic coordinates: The separated ODE's are, however, much more difficult (non-linear) than in the Type I system and there is no hope of finding explicit eigenvalues and eigenfunctions.

Bosonic ground states
We shall therefore concentrate on searching for the ground states. First, the bosonic zero modes: The separation ansatz makes these equations equivalent to: The solutions, i.e. the bosonic zero modes, are: It is not possible to calculate analytically the norm in these cases, but we offer a numerical integration of the F = 0 bosonic ground states: in the next figures. We observe that there is one normalizable bosonic ground state of zero energy of Type IIa and one of Type IIb once the value of a is set to be one. Remarkably, the Type IIb zero mode disappears (the norm becomes infinity) at the classical limit.

Fermionic ground states
Second, the fermionic ground states of zero energy: Unlike the bosonic case where the logic is or instead of and, note that both equations must be satisfied by the fermionic zero modes. The separation ansatz makes these equations tantamount to The fermionic zero modes have the form of the linear combination:

Because the norm is
only A 1 = 0 or A 2 = 0 are normalizable, depending on the choice of a and b. Again, these integrals cannot be computed analytically but the outcome of numerical calculations is shown in the next figures for several values of¯ .
In this case we see that the fermionic zero mode of Type IIa does not have a classical limit whereas the Type IIb fermionic ground state behaves smoothly near¯ = 0.

Summary
In this paper we have built and studied two types of supersymmetric quantum mechanical systems starting from two Coulombian centers of force. Our theoretical analysis could be of interest in molecular physics seeking supersymmetric spectra closely related to the spectra of homonuclear or heteronuclear diatomic molecular ions, e.g., the hydrogen molecular ion or the same system with a proton replaced by a deuteron. In H + 2 for instance, the first case, the value of the non-dimensional quantization parameter is checked to be¯ = 0.7: = 1.05 · 10 −34 kg · m 2 · s −1 , √ mdα = 1.493 · 10 −34 kg · m 2 · s −1 , using the international system of units (SI). The first type is defined from a superpotential that solves the Poisson equation with the potential of the two centers as the source. In this case, the spectral problem is shown to be equivalent to entangled families of Razavy and Whittaker-Hill equations. Using the property of the quasi-exact solvability of these systems, many eigenvalues of the SUSY system corresponding to bound states have been identified when the strengths of the two centers are different. If the strengths are equal things become easier and some bound states are also found. In summary, for our simplest choice of Type I superpotential the main features of the spectrum are the following: • There is an infinite set of discrete energy eigenvalues in the F = 0 Bose sub-space of the Hilbert space: The ionization energy is: E ∞ = 2 mα 2 2 (1 + δ) 2 , the threshold of the continuous spectrum. • There is a sub-space of dimension n + 1 of degenerate eigenfunctions with energy E n . The eigenvalues of the symmetry operatorÎ I nm = 2 4 (λ nm − (n + 1) 2 , m = 1, 2, . . . , n + 1, where λ nm are the n + 1 roots of the polynomial that solves the Razavy equation, label a basis of eigefunctions in each energy sub-space.
• In the case δ = 1 the eigenfunctions can be explicitly found. Moreover, the system enjoys a discrete symmetry under center exchange: v ↔ −v (r 1 ↔ r 2 ). The energy eigenfunctions come in even/odd pairs of functions of v with respect to this reflection. This fact suggests that the purely bosonic two fixed centers Hamiltonian enjoys a hidden supersymmetry (if the constant C is greater than 8 mα 2 2 ) of the kind recently unveiled in [45] for classy one-dimensional systems. This hidden supersymmetry is spontaneously broken because the even/odd ground states have positive energy: E ± = C − 8 mα 2 2 > 0. • There are eigenfunctions in the F = 1 Fermi sub-space for the same eigenvalues E n , n > 0.
Analytically, the Fermi eigenfunctions are obtained from the Bose eigenfunctions through the action ofQ + .
The second type starts from superpotentials solving the Hamilton-Jacobi equation. There are two non-equivalent sign combinations giving two sub-classes. Both Type IIa and Type IIb superpotentials are defined in terms of incomplete and complete elliptic integrals of the first and second kind. The superpotential in this approach is no more than the Hamilton's characteristic function for zero energy and flipped potential. The separability of the HJ equation in elliptic coordinates means that we can find a "complete" solution of this equation. The spectral problem is, however, hopeless for this Type.
All the zero-energy ground states, bosonic and fermionic, Type I and Type IIa/IIb, different strengths and equal strengths, have been obtained. The cross section (x 2 = 0) of the probability density of some of the ground states are shown, for the sake of comparison, in these final Tables. It is remarkable that, despite of being analytically very different, Type I and Type II zero modes show similar patterns.