Pöschl-Teller Potential in Quantum Systems

• Pöschl-Teller Potential is a family of exactly and quasi-exactly solvable quantum potentials defined in trigonometric and hyperbolic forms for modeling molecular vibrations and quantum spectra.
• Its analytic solvability enables closed-form energy eigenvalues and wavefunctions, serving as a benchmark in spectral theory, supersymmetry, and quantum information.
• The potential offers versatile applications from molecular spectroscopy to quantum field theory, with extensions into relativistic and approximate numerical models.

The Pöschl-Teller potential encompasses a class of exactly and quasi-exactly solvable potentials in quantum mechanics, distinguished by their trigonometric and hyperbolic forms. Originally introduced to model vibrational spectra of diatomic molecules, these potentials now play a central role across quantum spectral theory, supersymmetric quantum mechanics, test cases for approximate methods, and quantum field theory. Their analytic solvability, rich algebraic structure, and spectral flexibility make them benchmark systems for theoretical and mathematical physics.

1. Formulations of the Pöschl-Teller Potential

The archetypal Pöschl-Teller potentials appear in both trigonometric and hyperbolic variants, and in several parameterizations.

Trigonometric Pöschl-Teller (PT) Potential

Defined for $0<r<\pi/(2a)$: $$V(r) = \frac{V_1}{\sin^2(a r)} + \frac{V_2}{\cos^2(a r)},$$ where $V_1, V_2 > 0$ set the well-depths and $a > 0$ sets the spatial range. The potential diverges at the boundaries, producing confinement in a finite interval (Hamzavi et al., 2012). A closely related form is

$$V(x) = \frac{p^2 - \tfrac14}{\sin^2 x} + \frac{v^2 - \tfrac14}{\cos^2 x}, \quad x \in (0, \tfrac{\pi}{2}),$$

which admits singularities at each boundary and supports a variety of self-adjoint extensions (Fucci et al., 26 Nov 2024).

Hyperbolic (Modified) Pöschl-Teller Potential

On the real line: $V(x) = -\frac{\lambda(\lambda-1)}{\cosh^2{x}}$ or more generally,

$$V(x) = \frac{A}{\sinh^2{(a x)}} - \frac{B}{\cosh^2{(a x)}}$$

with $A, B, a > 0$ (Zlatev, 2013, Cevik et al., 2016, Ikhdair et al., 2013). The hyperbolic PT family can also be parameterized in terms of dimensionless coupling constants $g_s$, $g_c$ to incorporate inverse-square singularities and generalized wells or barriers (Silva et al., 2023).

2. Exact Spectral Solutions and Algebraic Structure

The Schrödinger equation with the PT potential is exactly solvable for a wide range of scenarios.

Trigonometric PT: S-Wave Exact Solutions via the Nikiforov–Uvarov Method

For the $\ell=0$ (s-wave) case, the radial Schrödinger equation is transformed using the substitution $s = \sin^2(a r)$ to an equation of hypergeometric type. The NU method gives energy eigenvalues (Hamzavi et al., 2012): $$E_n = \frac{1}{2m} \Bigg\{ a^2(n+2)^2 + (2n+1)\Big[ \sqrt{a^4 + 8ma^2 V_1} + \sqrt{a^4 + 8ma^2 V_2} \Big] + V_1 + V_2 \Bigg\}.$$ Wavefunctions are expressed as

$$R_n(r) = N_n [\sin(a r)]^\alpha [\cos(a r)]^\beta P_n^{(\alpha - 1/2,\, \beta - 1/2)} (\cos 2a r),$$

where $\alpha, \beta$ are functions of $V_1,V_2,a,m$, and $P_n^{(p,q)}$ are Jacobi polynomials (normalization is in terms of Gamma functions) (Hamzavi et al., 2012). For $a \to 0$, the entire discrete spectrum collapses to $V_1 + V_2 + 2\sqrt{V_1 V_2}$, reproducing Kratzer-like degeneracy.

Hyperbolic PT: Bound, Antibound, and Resonance Spectra

For the hyperbolic case $V(x) = -\lambda(\lambda-1)/\cosh^2{x}$, three spectral regimes emerge (Cevik et al., 2016):

• Potential-well: $\lambda>1$—finite number of bound states, infinite ladder of antibound (virtual) poles.
• Low-barrier: $1/2 \leq \lambda < 1$—two ladders of antibound poles, no bound or resonance states.
• High-barrier: $\lambda=1/2+i\ell$, $\ell>0$—infinite resonance pole ladder in the lower-half $k$-plane (complex energies), corresponding to Gamow (quasi-normal) states.

The associated wavefunctions involve hypergeometric functions, with explicit transfer matrix (S-matrix) analysis yielding the poles and residues for all spectral categories (Cevik et al., 2016). Ladder operators acting on these states close a $\mathfrak{su}(1,1)$ algebra.

Approximate and Generalized Solutions

For non-s-wave ($\ell>0$) or four-parameter forms, exact reduction is typically precluded by the centrifugal barrier. Two main strategies are employed:

• Approximate reduction: Approximations such as $1/r^2 \approx a^2 [d_0 + 1/\sin^2{(a r)}]$, $d_0 = 1/12$, retain solvability with closed-form results for the energy spectrum and eigenfunctions in terms of Jacobi polynomials (Hamzavi et al., 2012, Zlatev, 2013).
• Asymptotic and numerical methods: The uniform asymptotic approximation (UAA) provides high-accuracy analytic solutions across parameter regimes, with relative errors as low as $0.15\%$ in first order and systematic improvement at higher order (Pan et al., 2023).

Relativistic Extensions

The PT potential supports closed-form solutions for relativistic equations such as the Dirac and Klein-Gordon equations, including the PT-symmetric complexifications and q-deformations (Oyewumi et al., 2010, Ikhdair et al., 2012).

3. Quantum-Information, Coherent States, and Algebraic Methods

The PT potential is a testbed for quantum-information measures and coherent-state construction.

• Position-momentum and Fisher-information inequalities: The $\ell$-state hyperbolic PT potential exhibits exact analytic expressions for $\langle r^2 \rangle$, $\langle p^2 \rangle$, Fisher information, always satisfies the Heisenberg, Fisher-product, and Cramér–Rao inequalities for all $\ell,n$ (Yahya et al., 2014).
• Coherent states: Coherent states minimising specialized uncertainty relations can be constructed algebraically, with explicit resolution of the identity and covariant quantization of classical observables (Bergeron et al., 2010).
• su(1,1) representations and coherent states: In the Feinberg-Horodecki equation with the trigonometric PT potential, Barut–Girardello coherent states associated with $su(1,1)$ ladder operators admit closed analytic construction, eigenstate decomposition, and resolution of the identity (Arda et al., 2017).

4. Spectral and Analytic Invariants: Zeta Functions and Boundary Sensitivity

The spectral zeta function $\zeta(s)$ for Schrödinger operators with PT potentials has been thoroughly analyzed (Fucci et al., 26 Nov 2024). For PT potentials with appropriate self-adjoint extensions, $\zeta(s)$ displays a singular analytic landscape:

• Universal structure: Simple pole at $s=1/2$, logarithmic branch points at all nonpositive integers.
• Parameter-sensitive poles and branch points: Rational values of the boundary coupling parameters ($v$ rational) induce infinitely many additional poles and branch points at nonclassical locations.
• Sensitivity to smooth perturbations: Even compactly supported, smooth perturbations can profoundly alter the meromorphic structure of $\zeta(s)$, shifting or eliminating poles and generating new branch points, as detailed via explicit comparison to the Bessel potential.

These features strongly affect derived invariants, such as the determinant, heat-kernel coefficients, and trace formulas.

5. Supersymmetry, Self-Adjoint Extensions, and Renormalization

The PT potential is a cornerstone of exactly solvable models in supersymmetric quantum mechanics:

• Partner Hamiltonians: For suitable parameter choices, the PT potential admits factorization into partner pairs $Q^\dagger Q$ and $Q Q^\dagger$, with superpotential $W(x)$ determined algebraically (Cevik et al., 2016, Filippo et al., 2016).
• Spontaneous SUSY breaking: SUSY is spontaneously broken when the ground-state wavefunction becomes non-normalizable due to the singularity structure at $x=0$. The emergent anomalous scale $L$ (from self-adjoint extension/renormalization) controls both SUSY and conformal symmetry breaking, resulting in nontrivial RG flows, appearance of “walking” phases, and modification of the spectrum (Silva et al., 2023).
• S-matrix structure: All bound, anti-bound, and resonance states are classified via the analytic structure of the S-matrix, with explicit dependence on the renormalization/extension parameters.

6. Applications and Physical Context

• Molecular spectroscopy: The trigonometric PT potential provides analytic and highly accurate vibrational-rotational spectra for a wide class of diatomic molecules. In the long-range (small $a$) limit, it reduces to the Kratzer potential (Hamzavi et al., 2012, Hamzavi et al., 2012, Ikhdair et al., 2013).
• Electronic and wavepropagation band structure: Periodic arrays of PT wells result in solvable models for band-structure analysis, with Bloch wavefunctions constructed using supersymmetric techniques. The band gap and spectrum are controlled by the depth and spacing of PT units (Filippo et al., 2016).
• Cosmological and black hole perturbations: The PT potential underlies analytic approximations for mode equations in quantum cosmology and black hole quasi-normal mode analysis, via both exact and uniform asymptotic approximations (Pan et al., 2023).
• Quantum finance: The hyperbolic PT potential is isospectral to financial Hamiltonians with quadratic normal volatility, connecting the volatility smile to quantum spectral properties, spectral curvature, and integrable quantum systems (Saucedo, 16 Jul 2025).

---

References:

• (Hamzavi et al., 2012): Exact S-Wave Solution of the Trigonometric Poschl-Teller Potential
• (Cevik et al., 2016): Resonances and antibound states of Pöschl-Teller potential: Ladder operators and SUSY partners
• (Arda et al., 2017): Feinberg-Horodecki Equation with Pöschl-Teller Potential: Space-like Coherent States
• (Filippo et al., 2016): Exact electronic bands for a periodic Pöschl-Teller potential
• (Silva et al., 2023): Renormalization group and spectra of the generalized Pöschl-Teller potential
• (Bergeron et al., 2010): Semi-classical behavior of Pöschl-Teller coherent states
• (Fucci et al., 26 Nov 2024): The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential
• (Yahya et al., 2014): Quantum Information Entropies for the $\ell$-state Pöschl-Teller-type potential
• (Oyewumi et al., 2010): Relativistic Treatment of the Spin-Zero Particles Subject to the q-Deformed Hyperbolic Modified Pöschl-Teller Potential
• (Pan et al., 2023): Uniform Asymptotic Approximation Method with Pöschl-Teller Potential
• (Ikhdair et al., 2012): Approximate Dirac solutions of complex -symmetric Pöschl-Teller potential in view of spin and pseudospin symmetries
• (Saucedo, 16 Jul 2025): Quadratic Volatility from the Pöschl-Teller Potential and Hyperbolic Geometry
• (Ikhdair et al., 2013): Approximate Analytical Solutions to Relativistic and Nonrelativistic Pöschl-Teller Potential with its Thermodynamic Properties
• (Hamzavi et al., 2012): Approximate L-State Solution of the Rotating Trigonometric Pöschl-Teller Potential
• (Zlatev, 2013): Pekeris-type approximation for the $l$-wave in a Pöschl-Teller potential