Orthogonal Polynomial Applications in Quantum Mechanics

Orthogonal polynomials have long played a central role in quantum mechanics by furnishing exact eigenfunctions for a wide class of solvable models. Classical families such as Hermite, Laguerre and Jacobi polynomials appear in the harmonic oscillator, radial potentials and angular momentum problems, respectively. Beyond these canonical cases, the discovery of exceptional and multi-indexed orthogonal polynomials has enabled the construction of rationally extended and quasi-exactly solvable potentials, preserving shape invariance while introducing novel spectral features. In parallel, discrete quantum mechanical systems with pure imaginary shifts employ families such as the continuous Hahn and Meixner–Pollaczek polynomials, leading to complete orthogonal bases in weighted Hilbert spaces. These advances have deepened our understanding of ladder operators, polynomial Heisenberg algebras and superintegrable systems, and have broad implications for wavefunction engineering, spectral design and the study of many-body interactions.

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Orthogonal Polynomial Applications in Quantum Mechanics publication trend

The graph below shows the total number of articles in orthogonal polynomial applications in quantum mechanics across all publications each year (not limited to Nature Index journals).

Technical terms

Orthogonal polynomial: A sequence of polynomials orthogonal under a specific weight function over a given domain, often serving as eigenfunctions in quantum models.

Exceptional orthogonal polynomial: A generalisation of classical orthogonal polynomials featuring gaps in degree sequence, arising from rational extensions and new solvable potentials.

Shape-invariant potential: A potential whose parameters transform in a way that preserves its form under supersymmetric partner operations, enabling algebraic solution of energy spectra.

Multi-indexed polynomial: An orthogonal polynomial family labelled by multiple integer indices, allowing the construction of extended spectral models with richer algebraic structure.

References

1. Infinitely many shape invariant potentials and new orthogonal polynomials. Physics Letters B (2009).
2. Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials. Physics Letters B (2011).
3. Exactly solvable discrete quantum mechanical systems and multi-indexed orthogonal polynomials of the continuous Hahn and Meixner–Pollaczek types. Progress of Theoretical and Experimental Physics (2019).
4. New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics. Progress of Theoretical and Experimental Physics (2017).
5. New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies. Journal of Physics Conference Series (2015).
6. Ladder operators for solvable potentials connected with exceptional orthogonal polynomials. Journal of Physics Conference Series (2015).
7. Rational extension of many particle systems. Acta Polytechnica (2022).