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Spectral theory of deformed differential operators

Establish the spectral theory for deformed differential operators induced by conformable derivatives, including characterization of spectra, eigenfunctions, and resolvent properties, to support rigorous analysis of conformable Ginzburg–Landau-type equations and related models.

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Background

The conformable framework replaces classical derivatives with temperature-weighted operators to capture critical scaling, enabling closed-form modeling of observables. However, a comprehensive spectral theory for these deformed operators is absent.

A rigorous spectral theory would enable stability analysis, solution decomposition, and operator-theoretic methods for conformable dynamics, paralleling tools available for classical and fractional operators.

References

Beyond phenomenological applications, several open questions remain in the rigorous mathematical formulation of the conformable framework. These include the development of a well-defined operator semigroup structure, the spectral theory of deformed differential operators, and the formulation of variational principles compatible with conformable dynamics Godinho et al. [19].

— Conformable Scaling and Critical Dynamics: A Unified Framework for Phase Transitions (2507.11782 - Weberszpil, 15 Jul 2025) in Section IX (Conclusions and Outlook)