Spectral Projection, Perturbative Response, and Spectral Action for Quantum Hamiltonians

We present a unified operator-theoretic formulation of spectral projection, perturbative response, and spectral action for self-adjoint quantum Hamiltonians. The framework is based on Riesz projections associated with isolated ground states and introduces a trace-class interaction functional comparing spectral projectors. The construction is related to standard perturbation theory, resolvent identities, and determinant formulations, providing a common trace-based description of spectral response. In the presence of Dirac-type operators coupled to gauge connections, the spectral action expansion recovers curvature-dependent contributions through heat kernel asymptotics, including terms consistent with Yang–Mills functionals. The results are structural and organize established tools in spectral theory and mathematical physics, clarifying their role in quantum Hamiltonian systems and spectral action formulations.