Microlocal Energy Dissipation in Semiclassical Pseudodifferential Equations: From Wavefront Sets to Hamiltonian Flows

This paper develops a rigorous semiclassical and microlocal framework to analyze energy dissipation and singularity propagation in systems governed by dissipative pseudodifferential operators. We consider semiclassical operators \( P_h \) with symbols \( a(x, ξ) ∈ S^m(R^n) \), where \( m < 0 \), and study families of solutions \( u_h \) to \( P_h u_h = 0 \). By constructing microlocal partitions of unity, we derive asymptotic formulas for the local and total dissipation rate \( ε_h \), proving its convergence to an explicit phase-space integral as \( h → 0^+ \). Our analysis reveals that the semiclassical wavefront set \( WF_h(u_h) \) propagates along the generalized bicharacteristics of the real part of the principal symbol \( p_0(x, ξ) \), while the imaginary part \( Im\, p_0(x, ξ) ≤ 0 \) governs the amplitude decay along these flows. This establishes a precise, quantitative relationship between geometric propagation and physical dissipation, unifying the description of singularity transport and energy loss in the semiclassical regime. The framework bridges deterministic and statistical perspectives, offering a robust tool for investigating multiscale energy dynamics in turbulent systems and beyond.