Operator Calculus and Spectral Theory

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This work explores various advanced topics in mathematical physics and operator theory, including the heat equation asymptotics of a generalized Ahlfors Laplacian on manifolds with boundaries and the contrasting behaviors of recurrent versus diffusive quantum systems under time-dependent Hamiltonians. It delves into spectral measures for Feller operators and provides a global perspective on locating quantum resonances. The text presents estimates for eigenvalues in elliptic problems and discusses quantum scattering in the presence of long-range magnetic fields. It addresses spectral invariance and submultiplicativity in Fréchet algebras, with applications to pseudo-differential operators and *-quantization. Key results include exponential decay of eigenfunctions for Kac's operator in higher dimensions and second-order perturbations of divergence-type operators exhibiting a spectral gap. The analysis extends to Weyl quantized relativistic Hamiltonians and spectral asymptotics for commuting operator families. Additionally, it examines pseudo-differential operators with negative definite symbols, one-dimensional Schrödinger operators with high barriers, and boundary value problems in angular domains. Insights into non-linear equations in cylindrical domains, stable asymptotics for Dirichlet problems, and Maslov operator calculus are also presented, alongside discussions on trace formulas and functional calculus for boundary value