[Paper Review] Factorization Method in Curvilinear Coordinates and Pairing of Levels for Matrix Potentials

A. A. Andrianov, M. V. Ioffe+1 more|arXiv (Cornell University)|Jan 4, 2011

Quantum Mechanics and Non-Hermitian Physics5 references3 citations

TL;DR

This paper extends the multidimensional factorization method to arbitrary curvilinear coordinates, enabling the construction of matrix potentials with hidden symmetries that induce double or equidistant energy level degeneracy. By applying the method in polar and spherical coordinates, it derives new classes of matrix potentials—particularly in 2D and 3D—where spectra exhibit fourfold degeneracy due to both supersymmetric equivalence and additional hidden symmetry, confirmed via Darboux transformations and explicit solutions to the Laplace equation for the superpotential.

ABSTRACT

Multidimensional factorization method is formulated in arbitrary curvilinear coordinates. Particular cases of polar and spherical coordinates are considered and matrix potentials with separating variables are constructed. A new class of matrix potentials is obtained which reveals a double degeneracy or equidistant splitting of energy levels (hidden symmetry).

Motivation & Objective

• To generalize the multidimensional factorization method to arbitrary curvilinear coordinates, particularly polar and spherical systems, to enable the construction of matrix potentials with separable variables.
• To identify new classes of matrix potentials that exhibit double degeneracy or equidistant level splitting, indicating hidden symmetry beyond standard supersymmetry.
• To demonstrate that such degeneracies arise from intertwining relations and solutions to the Laplace equation for the superpotential χ in curvilinear geometries.
• To establish a framework for generating Hamiltonians with fourfold degeneracy by combining supersymmetric equivalence with additional spectral degeneracy from constant or zero potential shifts.

Proposed method

• Formulates the factorization method in curvilinear coordinates using the Laplace-Beltrami operator expressed via the metric tensor $ g_{ik} $ and covariant derivatives $ \nabla_i $.
• Defines the intertwining operators $ Q_l^\pm $ in terms of the superpotential $ \chi $, with $ Q_l^\pm = \frac{1}{\sqrt{2}} ( \mp \nabla_l + \partial_l \chi ) $, ensuring factorization of the scalar Hamiltonian $ H^{(0)} $.
• Constructs matrix Hamiltonians $ H^{(1)}_{ik} $ via Darboux transformations using operators $ P_l^\pm = \sqrt{g} \epsilon_{lk} Q^{k\mp} $, preserving spectral interrelations.
• Solves the equation $ \triangle \chi = \varepsilon $ (constant) or $ \triangle \chi = 0 $ in 2D and 3D to generate potentials with equidistant or degenerate spectra.
• Uses spherical harmonics and Fourier-like expansions in angular variables to construct explicit forms of matrix potentials in 3D, particularly for $ \chi $ with radial and angular dependence.
• Derives explicit expressions for $ V^{(1)}_{11} $, $ V^{(1)}_{22} $, and off-diagonal terms in 2D, showing dependence on $ \alpha_m $, $ \delta_m $, and $ \rho $.

Experimental results

Research questions

• RQ1 How can the multidimensional factorization method be generalized to arbitrary curvilinear coordinates, particularly polar and spherical systems?
• RQ2 What class of matrix potentials can be constructed in curvilinear coordinates that allow separation of variables and exhibit hidden symmetry?
• RQ3 Under what conditions does the factorization method lead to additional degeneracy beyond standard supersymmetry, such as fourfold level degeneracy?
• RQ4 How do constant or zero potential shifts ($ \delta V = \varepsilon $ or $ \delta V = 0 $) affect the spectral structure of matrix Hamiltonians?
• RQ5 What role do solutions to $ \triangle \chi = \varepsilon $ or $ \triangle \chi = 0 $ play in generating equidistant or doubly degenerate energy levels?

Key findings

• In two dimensions, matrix potentials with double degeneracy are constructed via $ \chi $ satisfying $ \triangle \chi = \varepsilon $, leading to $ E_n^{(1)} = E_n^{(0)} $ and $ E_n^{(2)} = E_n^{(0)} + \varepsilon $, resulting in equidistant splitting.
• The matrix potential $ V^{(1)}_{ik} $ in 2D takes the form $ \begin{pmatrix} V^{(1)}_{11} & \sum m(m-1)\alpha_m \rho^{m-3} \cos(m\varphi + \delta_m) \\ \sum m(m-1)\alpha_m \rho^{m-1} \cos(m\varphi + \delta_m) & V^{(1)}_{22} \end{pmatrix} $, with $ V^{(1)}_{11} $ and $ V^{(1)}_{22} $ differing by a sign in the angular-dependent term.
• For $ \delta V = \varepsilon = \text{const} $, the resulting matrix Hamiltonian exhibits equidistant level splitting: $ E_{2n-1}^{(1)} = E_n^{(0)} $, $ E_{2n}^{(1)} = E_n^{(0)} + \varepsilon $, confirming a new class of potentials with constant-level spacing.
• In three dimensions, solutions to $ \triangle \chi = 0 $ using spherical harmonics $ Y_j(\theta,\varphi) $ generate matrix potentials with partial fourfold degeneracy, where $ H^{(1)} $ and $ H^{(2)} $ share degenerate levels due to hidden symmetry.
• The joint Hamiltonian $ \widehat{H} = \text{diag}(H^{(0)}, H^{(1)}, H^{(2)}) $ exhibits fourfold degeneracy for a subset of energy levels when $ \delta V = 0 $, due to combined supersymmetric and hidden symmetry contributions.
• Explicit forms of $ V^{(0)} $, $ V^{(1)}_{11} $, and $ V^{(1)}_{22} $ are derived for 2D, including harmonic and anharmonic potentials such as $ V^{(0)} = \alpha \rho^2 + \beta \rho^4 + 4\alpha\beta \rho^2 \cos(\varphi + \delta) $, with bound states guaranteed by asymptotic behavior.