Inverted Rosen–Morse Potential

• The inverted Rosen–Morse potential is a hyperbolic quantum model that reverses classic well configurations into barriers, enabling detailed scattering and resonance analyses.
• It employs exact solution techniques such as the NU method, SUSYQM, and exact WKB, offering insights into bound states and quasi-stationary resonances.
• Applications span quantum mechanics, molecular physics, and non-Hermitian systems, highlighting its relevance in studying tunneling, decay processes, and optical solitons.

The inverted Rosen–Morse potential constitutes a non-polynomial, hyperbolic model relevant in quantum mechanics, quantum field theory, molecular physics, and non-Hermitian systems. It arises by reversing the sign or specific parameters in the classical Rosen–Morse potential, transforming a conventional well into a barrier or asymmetric scattering configuration. Recent advances facilitate exact analytic solutions, resonance calculations, and symmetry analysis for both relativistic and non-relativistic regimes.

1. Definition, Forms, and Basic Structure

The canonical Rosen–Morse potential is given (in one-dimensional form) by

$$V(x) = -V_1\,\mathrm{sech}^2(a x) + V_2\,\tanh(a x),$$

where $V_1$, $V_2$ are depth/asymmetry coefficients and $a$ sets the range. The "inverted" Rosen–Morse potential is defined by reversal of the sign of either $V_1$ or $V_2$, or more generally via analytic continuation such as $a \to i a$. For instance:

• $$V_\text{inv}(x) = +V_1\,\mathrm{sech}^2(a x) - V_2\,\tanh(a x)$$,
• or $$V_\text{inv}(x) = -V_3\,\sec^2(a x) - V_4\,i\,\tan(a x)$$ (analytic continuation in $a$).

The inverted variant often produces barrier-type behaviors, non-standard scattering properties, and may shift the bound state structure or generate resonances associated with quasi-stationary states (Morikawa et al., 24 Mar 2025).

2. Spectral Theory and Resonances

For the inverted Rosen–Morse potential, exact solutions are tractable due to hypergeometric and special function techniques. The Schrödinger equation or Dirac equation with $V_\text{inv}$ can be mapped to differential equations whose solutions are given in terms of Jacobi, Legendre, or Romanovski polynomials (with complex parameters), or hypergeometric functions. For resonance analysis, the boundary conditions are set to outgoing waves (Siegert), and complex energy eigenvalues emerge: $$E^\textrm{R}_n = \frac{\hbar^2\beta^2}{8m} \left[ \sqrt{\frac{8mU_0}{\beta^2\hbar^2} - 1} - i(2n+1) \right]^2,$$ with $U_0$ the barrier height parameter and $\beta$ the range (Morikawa et al., 24 Mar 2025).

The poles of the S-matrix for the Rosen–Morse II potential reflect bound, redundant, and anti-bound states, which extend to the inverted case via sign changes and parameter analytic continuation (Millán et al., 2023).

3. Solution Methods: NU Method, SUSYQM, Exact WKB

Nikiforov–Uvarov (NU) Method

The inverted Rosen–Morse potential fits within the generalized inverted hyperbolic family: $V(r) = -a V_0\,\coth(a r) - c V_2\,\cosech^2(a r),$ with $E_{n\ell}$ and $R_{n\ell}(r)$ solvable in closed form via NU reduction to hypergeometric-type equations (Ikot et al., 2011), even in nonzero angular momentum sectors.

Supersymmetric Quantum Mechanics (SUSYQM)

Shape invariance permits exact algebraic solution for both bound and rationally extended spectra. Rational extensions preserve or modify the number of bound states, and the transformation of sign structures (e.g., $V_2\rightarrow -V_2$) inverts the well to a barrier without loss of functional solvability (Oyewumi et al., 2010, Quesne, 2012). Enlarged shape invariance involving multi-parameter shifts applies to extended cases (Quesne, 2012).

Exact WKB Analysis

In the non-perturbative regime, barrier resonances are precisely characterized by the exact WKB method. Quantization conditions derive from monodromies along Stokes cycles in the complex plane: $$A = \exp\left\{\oint S_{\mathrm{odd}}(x,\hbar)\,dx\right\},$$ Resonance energies correspond to $A=1$ or $A=-1$ in leading order, and the resonance widths follow from the imaginary part of $E$. There is explicit consistency between the exact analytic solutions and WKB predictions (Morikawa et al., 24 Mar 2025).

4. PT-Symmetry, Non-Hermiticity, and Spectral Properties

The inverted Rosen–Morse potential is central to the analysis of non-Hermitian quantum systems, where PT symmetry ensures a real spectrum under certain parameterizations despite non-Hermiticity (Falaye et al., 2011). Analytic continuation ($a\to ia$) and complex coefficients ($V_4$ imaginary, for example) allow construction of potentials that exhibit real spectra and stable states, or complex spectra in regimes of broken PT symmetry.

For "asymmetric Rosen–Morse" or modified forms in dissipative systems (such as optical solitons described by complex Ginzburg–Landau equations), inversion manifests in reversed gain/loss profiles and energy flow characteristics essential for nonlinear optics (Hari et al., 2019).

5. Applications: Resonances, Scattering, Quantum Wells, and Soliton Physics

Barrier Resonance

The inverted Rosen–Morse barrier produces quasi-bound (resonant) states with finite lifetimes, relevant for modeling decay processes, molecular dissociation, and quantum tunneling phenomena (Morikawa et al., 24 Mar 2025).

Scattering Theory

The spectrum of transmission and reflection coefficients can be computed via generalized Legendre functions. Explicit formulas: $$R = \frac{\sin^2(\pi\alpha) + \sinh^2 \left(\frac{\pi}{2}[k - \sqrt{k^2 - 4\beta}] \right)}{\sin^2(\pi\alpha) + \sinh^2 \left(\frac{\pi}{2}[k + \sqrt{k^2 - 4\beta}] \right)}, \quad T = \frac{\sinh(\pi k) \sinh(\pi \sqrt{k^2 - 4\beta})}{\sin^2(\pi\alpha) + \sinh^2 \left(\frac{\pi}{2}[k + \sqrt{k^2 - 4\beta}] \right)},$$ provide full classical scattering solutions (Freitas, 2023).

Quantum Wells and Position-Dependent Mass

Via point canonical transformations, the inverted Rosen–Morse potential enables construction of semi-infinite quantum wells with position-dependent mass backgrounds (mass diverges at a boundary), leading to non-rectangular profiles and zero modes absent in certain gravitating soliton models (Quesne, 2022, Wang et al., 23 Sep 2024).

Graphene and Dirac Systems

In gapped graphene, the Rosen–Morse (and inverted variants) applied to the Dirac equation gives rise to energy band gaps, modulated by magnetic field and potential asymmetry (Kalani et al., 30 Aug 2024).

Field Theory, Kinks, Gravity, and Shape Modes

In models of gravitating kinks with noncanonical scalar fields, the inverted Rosen–Morse form arises in stability analysis. The spectrum exhibits absence of zero modes, reflective vs. reflectionless behavior, and resonance phenomena (Wang et al., 23 Sep 2024, Gordillo-Núñez et al., 2023).

6. Mathematical Tools: Special Functions, Ladder Operators, Recurrence Relations

The solution space of the inverted Rosen–Morse potential heavily leverages fractional calculus for ladder operator construction (Weyl fractional integrals), orthogonal polynomial extensions (Romanovski, Jacobi, generalized Legendre), and recurrence relations. These techniques enable efficient computation of polynomial coefficients, facilitate analytic continuation in parameter space, and provide a unified approach for resonances, bound-state spectral engineering, and normalization of states (Freitas, 2018, Quesne, 2013).

7. Interrelations, Generalizations, and Open Directions

The inverted Rosen–Morse potential is embedded within a broader family of hyperbolic-function potentials (e.g., Manning–Rosen, Eckart, Scarf II), with exact solutions derived via the NU method or SUSYQM. Rational and multi-parameter extensions demonstrate enlarged shape invariance, broadening the set of exactly solvable models (Quesne, 2012). Coupled systems—such as Morse–Rosen–Morse models on curved manifolds—exhibit superintegrability with explicit polynomial integrals of motion (Acosta et al., 2020). Scenarios involving inversion hint at new physics in resonance theory, quantum decay, and non-Hermitian symmetry breaking.

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This synthesis provides an authoritative view of the inverted Rosen–Morse potential, its analytic solutions, and its role in resonance, scattering, solvable models, and advanced quantum theory as extracted from the cited research literature [(Oyewumi et al., 2010, Ikot et al., 2011, Falaye et al., 2011, Quesne, 2012, Morikawa et al., 24 Mar 2025), and others].