Propagators of isochronous an-harmonic oscillators and Mehler formula for the x-Hermite polynomials

2015/02/06 by Andrey M. Pupasov-Maksimov, Pupasov-Maksimov, Andrey M.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1502.01778

arxiv created 2015/02/06 · openalex publication_date 2015/02/06 · arxiv updated 2015/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that fundamental solutions K^σ(x,y;t)=⟨ x|e^{-i Hσt}|y⟩ of the non-stationary Schrödinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator H^σ=H_{\rm osc}+ΔV^σ are expressed in terms of elementary functions only. An algorithm to calculate explicitly K_σ for an arbitrary σ∈ ℕ is given, compact expressions for K^\1,2\ and K^\2,3\ are presented. A generalization of the Mehler's formula to the case of exceptional Hermite polynomials is given.

Citations

• Space-Time Approach to Non-Relativistic Quantum Mechanics

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