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

We establish a rigorous framework for quantum systems with geometric deformations by constructing a strictly self-adjoint deformed momentum operator through the generalized extended momentum operator (GEMO) formalism. Unlike previous approaches relying on boundary-condition hermiticity, our method ensures intrinsic self-adjointness for both linear ($\mu (x)=\alpha x$) and quadratic ($\mu (x)=\alpha x^2$) deformations within a unified non-Hermitian supersymmetric factorization scheme. This yields exact analytical spectra while revealing hidden $\mathfrak{s}\mathfrak{u}(1,1)$ symmetry structures. Crucially, we provide the first complete thermodynamic characterization of such systems by analytically evaluating the partition function via the Euler–Maclaurin approximation. Geometric deformation fundamentally reshapes the density of states $\rho (E)$, producing distinct thermal signatures: a divergent heat capacity peak for linear deformation due to state accumulation near a maximal energy, and a saturation $C\mathrm{/}{k}_{\mathrm{B}}\to 0.6$ (below the Dulong–Petit limit) for quadratic deformation. These results establish geometric deformation as a tunable parameter for engineering quantum thermodynamic responses in curved nanostructures.