Abstract
<jats:p>We introduce the Fractional Memory Helmholtz Operator (FMHO), a generalized spectral operator constructed from Romero’s AFA operator for the description of wave phenomena with temporal memory. The proposed formulation extends the classical Helmholtz framework by embedding fractional temporal dynamics directly into the spectral operator rather than modifying the governing equations. A generalized Helmholtz eigenvalue problem is formulated and shown to admit, under separable solutions, a decomposition of its spectrum into a classical Helmholtz contribution and a memory-induced fractional correction. This result provides a natural interpretation of temporal memory as an intrinsic spectral property of the operator. As a direct consequence, a generalized Schr¨odinger–Helmholtz equation is obtained and applied to the one-dimensional infinite quantum well. The analysis demonstrates that the spatial eigenfunctions coincide with the classical solutions, while the temporal evolution is governed by the Mittag–Leffler function, producing a shift of the energy spectrum associated with temporal memory. The proposed framework establishes a unified operator-based formulation for generalized Helmholtz problems and opens new perspectives for the mathematical modeling of non-Markovian phenomena in quantum mechanics, wave propagation, optics, acoustics, and other complex systems.</jats:p>