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International audienceThe structure of the lower part (i.e. $\varepsilon $-away below the two-boson threshold) spectrum of Fröhlich's polaron Hamiltonian in the weak coupling regime is obtained in spatial dimension $d\geq 3$. It contains a single polaron branch defined for total momentum $p\in G^{\left( 0\right) } $, where $G^{\left( 0\right) }\subset {\mathbb R}^d$ is a bounded domain, and, for any $p\in {\mathbb R}^d$, a manifold of polaron + one-boson states with boson momentum $q$ in a bounded domain depending on $p$. The polaron becomes unstable and dissolves into the one boson manifold at the boundary of $G^{\left( 0\right) }$. The dispersion laws and generalized eigenfunctions are calculated

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The trace-norm convergence of the Hille-Phillips perturbation series is proved for the whole perturbation class of the generator of a Gibbs semigroup.

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The problem of the independence of the thermodynamic limit on the boundary conditions is considered in the framework of functional integration. For every domain and every boundary condition in a sufficiently large class a functional measure is constructed and the Feynman-Kac-like formula for the statistical operator written down. Making use of some volume-independent estimates for the Green function of the heat equation, the thermodynamic limit along convex domains for general boundary conditions is proved to exist and to be equal to that for Dirichlet conditions.

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