Analysis of Quantum-Deformed Fractional Schrödinger Equations Based on Generalized Hermite Polynomials

We investigate the fractional Schrödinger equation driven by the generalized (q,τ)-Caputo nabla derivative and an energy operator diagonalized by (q,τ)-Hermite polynomials. A modal decomposition approach is adopted, allowing the solution to be expanded in terms of an orthogonal polynomial basis with deformed weights and spectral parameters. We derive a Lyapunov-type energy functional and prove a decay estimate under small initial energy using a generalized (q,τ)-Grönwall inequality. A complete nonlinear analysis is presented, including bounds on quartic interaction terms, global existence of solutions in both L² and Sobolev spaces, and a corollary on uniqueness and modal regularity. Furthermore, we formulate a fractional energy balance and provide numerical illustrations demonstrating long-time decay in truncated-mode expansions. These results highlight the stability and regularity of quantum states under (q,τ)-deformed fractional dynamics and show that (q,τ)-Hermite polynomials serve as a powerful analytical tool in modeling memory-dependent quantum systems.