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q-Deformed Sturm-Liouville and SUq(1, 1)

Shawn Eastmond

Published May 4, 2026• 0 citations
Publisher

Abstract

We present a mathematical analysis of the one-dimensional infinitely deep square quantum well (1D IDSQW) within the paradigm of q-deformed quantum mechanics. The central technical contribution is a realization of the naive application of Sturm-Liouville theory (S-L theory): we prove that the eigenvalue parameter of the q-hypergeometric differential equation is a formal spectral parameter that does not directly correspond to physical en- ergies. Physical eigenvalues are extracted only after imposing q-Dirichlet boundary conditions on the Jackson lattice, which forces a truncation of the q-hypergeometric series, leading to q-quantized energy levels Eq,n = π2􏰀2 [n]2. We establish that the spectrum-generating algebra is the non- 2mL2 q compact quantum group SUq(1,1), whose positive discrete series D+(κ) furnishes the complete set of bound states. The theory is illustrated with precise numerical calculations of q-deformed quantum numbers, energy spec- tra, and wavefunctions, demonstrating the breakdown of the classical Sturm- Liouville eigenvalue-readoff technique.

DOI
10.5281/zenodo.20015647

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We present a mathematical analysis of the one-dimensional infinitely deep square quantum well (1D IDSQW) within the paradigm of q-deformed quantum mechanics. The central technical contribution is a realization of the naive application of Sturm-Liouville theory (S-L theory): we prove that the eigenvalue parameter of the q-hypergeometric differential equation is a formal spectral parameter that does not directly correspond to physical en- ergies. Physical eigenvalues are extracted only after imposing q-Dirichlet boundary conditions on the Jackson lattice, which forces a truncation of the q-hypergeometric series, leading to q-quantized energy levels Eq,n = π2􏰀2 [n]2. We establish that the spectrum-generating algebra is the non- 2mL2 q compact quantum group SUq(1,1), whose positive discrete series D+(κ) furnishes the complete set of bound states. The theory is illustrated with precise numerical calculations of q-deformed quantum numbers, energy spec- tra, and wavefunctions, demonstrating the breakdown of the classical Sturm- Liouville eigenvalue-readoff technique.
q-deformed harmonic oscillator is established for any arbitrary complex q-deformation parameter (with real and imaginary q as its special cases) and the formalism so developed is used to construct the quantum group SUq(2). The eigenvalues of the Casimir of SUq(2) are now real, even for complex q-deformation (q=ea+ib). This happens for (a,b)≪1. Thus, the use of real part of energies in the earlier works of Gupta and collaborators7–14 is mathematically rigorous, since only small values of (a, b) are involved in all the nuclear and molecular physics problems studied7–14,18–20 so far.
The Heisenberg algebra is first deformed with the set of parameters q,l,λ{q, l, λ}q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p,q){\cal R}(p,q)R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual qqq- and (p,q)(p,q)(p,q)-differentiation and integration. A Hopf algebra structure compatible with the R(p,q){\cal R}(p,q)R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called R(p,q){\cal R}(p,q)R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous R(p,q){\cal R}(p,q)R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.
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We found hermitian realizations of the position vector r⃗\vec{r}r, angular momentum Λ⃗\vecΛΛ and linear momentum p⃗\vec{p}p​ behaving like vectors with respect to the SUq(2)SU_q(2)SUq​(2) algebra, generated by L0L_0L0​ and L±L_\pmL±​. They are used to write the qqq deformed Schrödinger equation, whose solution for Coulomb and oscillator potential are briefly discussed.