New Generalization of Supersymmetric Quantum Mechanics to Arbitrary Dimensionality or Number of Distinguishable Particles

We present here a new approach to generalize supersymmetric quantum mechanics to treat multiparticle and multidimensional systems. We do this by introducing a vector superpotential in an orthogonal hyperspace. In the case of N distinguishable particles in three dimensions this results in a vector superpotential with 3N orthogonal components. The original scalar Schrödinger operator can be factored using a 3N-component gradient operator and introducing vector “charge” operators: Q⃗ 1 and Q⃗ 1 †. Using these operators, we can write the original (scalar) Hamiltonian as H 1 = Q⃗ 1 †·Q⃗ 1 + E 0 (1), where E 0 (1) is the ground-state energy. The second sector Hamiltonian is a tensor given by H⃡ 2 = Q⃗ 1 Q⃗ 1 † + E 0 (1) and is isospectral with H 1. The vector ground state of sector 2, ψ⃗0 (2), can be used with the charge operator Q⃗ 1 † to obtain the excited-state wave function of the first sector. In addition, we show that H⃡ 2 can also be factored in terms of a sector 2 vector superpotential with components W 2j = −(∂ ln ψ0j (2))/∂x j . Here ψ0j (2) is the jth component of ψ⃗0 (2). Then one obtains charge operators Q⃗ 2 and Q⃗ 2 † so that the second sector Hamiltonian can be written as H⃡ 2 = Q⃗ 2 † Q⃗ 2 + E 0 (2). This allows us to define a third sector Hamiltonian which is a scalar, H 3 = Q⃗ 2·Q⃗ 2 † + E 0 (2). This prescription continues with the sector Hamiltonians alternating between scalar and tensor forms, both of which can be treated by the variational method to obtain approximate solutions to both scalar and tensor sectors. We demonstrate the approach with examples of a pair of separable 1D harmonic oscillators and the example of a nonseparable 2D anharmonic oscillator (or equivalently a pair of coupled 1D oscillators). We consider both degenerate and nondegenerate cases. We also present a generalization to arbitrary curvilinear coordinate systems in the Appendix.