Spectral and thermodynamic properties of supersymmetric quantum systems with self-adjoint deformed momentum

Technical Summary: Spectral and Thermodynamic Properties of Supersymmetric Quantum Systems with Self-Adjoint Deformed Momentum

Problem Statement
Standard quantum mechanics relies on the canonical commutation relation $[x, p] = i\hbar$, which assumes a flat phase space. However, theories of quantum gravity (e.g., string theory, loop quantum gravity) and non-commutative geometries suggest modifications to this relation, often introducing a position-dependent deformation function $\mu(x)$. While such "extended" momentum operators have been studied, previous approaches (specifically the Position-Dependent Mass or PDM framework by Bagchi et al.) often rely on boundary conditions to ensure hermiticity rather than intrinsic self-adjointness. Furthermore, existing literature has treated linear and quadratic deformations separately and has largely neglected the thermodynamic consequences of these geometric deformations. This work addresses the need for a rigorous, unified framework that ensures the strict self-adjointness of the deformed momentum operator and provides a complete thermodynamic characterization of such systems.

Methodology
The authors construct a rigorous framework using the Generalized Extended Momentum Operator (GEMO), defined as:
$$p = -i\hbar \left(1 + \mu(x)\right) \frac{d}{dx} - i\hbar \frac{1}{2} \frac{d\mu(x)}{dx}$$
This operator ensures intrinsic self-adjointness, guaranteeing real spectra and probability conservation without relying solely on boundary conditions. The system under study is a one-dimensional quantum oscillator subjected to a mixed potential comprising a harmonic term, a linear electric field, and an inversely quadratic interaction, evolving under the deformation function $\mu(x)$.

The spectral analysis employs a non-Hermitian supersymmetric quantum mechanics (SUSYQM) factorization scheme. Unlike standard SUSYQM, the ladder operators ($A$ and $B$) are not mutually adjoint. However, the resulting Hamiltonians remain Hermitian. This approach allows for the exact analytical solution of the Schrödinger equation via shape invariance. The study systematically analyzes two specific deformation cases:

1. Linear deformation: $\mu(x) = \alpha x$
2. Quadratic deformation: $\mu(x) = \alpha x^2$

For thermodynamic analysis, the authors derive the partition function $Z(\beta)$ from the exact energy spectra. Due to the complexity of the discrete sums, they utilize the Euler–Maclaurin approximation to convert the sum into an integral corrected by boundary terms. This semi-analytic approach yields expressions for the partition function involving error functions (erf and erfi), from which internal energy, free energy, entropy, and heat capacity are derived.

Key Contributions and Results

• Unified Self-Adjoint Formalism: The paper establishes a unified non-Hermitian SUSY factorization scheme that ensures intrinsic self-adjointness for both linear and quadratic deformations. This contrasts with previous works that treated these cases separately or relied on boundary-condition hermiticity.
• Exact Spectral Solutions:
  • For linear deformation ($\mu(x) = \alpha x$), the energy spectrum is found to be non-equidistant, containing a quadratic term in the quantum number $n$. The spectrum exhibits a maximum energy $E_{max}$, leading to a concave parabolic structure.
  • For quadratic deformation ($\mu(x) = \alpha x^2$), the spectrum grows quadratically with $n$ ($E_n \propto n^2$). The analysis reveals hidden $\mathfrak{su}(1,1)$ symmetry structures that are inaccessible to perturbative methods.
• Thermodynamic Characterization:
  • Linear Deformation: The density of states $\rho(E)$ diverges as $E$ approaches the maximum energy ($E_{max}$), following $\rho(E) \propto 1/\sqrt{E_{max} - E}$. This accumulation of states near the energy ceiling results in a divergent peak in heat capacity ($C$) at specific temperatures, a signature of the upper-bounded spectrum.
  • Quadratic Deformation: The density of states decays algebraically as $\rho(E) \propto 1/\sqrt{E}$. This limits the number of thermally accessible states at high temperatures. Consequently, the heat capacity saturates at high temperatures, approaching a value of $C/k_B \approx 0.6$. This is a distinct deviation from the standard Dulong–Petit limit ($C/k_B \to 1$) observed in standard harmonic oscillators.
• Numerical Validation: Numerical simulations confirm that while linear deformation allows for a smoother transition between states, quadratic deformation induces a "stiffening" of the energy spectrum, leading to more pronounced and less gradual thermodynamic effects.

Significance and Claims
The authors claim this work provides the first complete thermodynamic characterization of quantum systems based on the GEMO formalism. By moving beyond spectral analysis to include thermodynamic quantities, the paper demonstrates that geometric deformation acts as a tunable parameter for engineering quantum thermodynamic responses.

The study highlights that the modified density of states is the microscopic origin of anomalous thermal behaviors. Specifically, the saturation of heat capacity in quadratic deformations and the divergent peaks in linear deformations offer potential experimental fingerprints for geometric quantum effects. The authors suggest these results are relevant for curved nanostructures, such as carbon nanotubes and spherical quantum dots, where effective geometry influences thermal transport.

The paper concludes that the synergy between strict self-adjointness, exact spectral solutions via non-Hermitian SUSY, and analytical thermodynamics provides a robust foundation for future explorations into driven-dissipative nanostructures and non-equilibrium quantum thermal machines, though the current study is limited to static fields to preserve exact solvability.