The authors study accurate semiclassical estimates of the resolvent of a class of pseudodifferential operators that include the Kramers-Fokker-Planck operator \[ P=v\cdot h\partial_x-V'(x)\cdot h\partial_v+{\gamma\over{2}}(-(h\partial_v)^2+v^2-hn), \] in \({\mathbb R}^{2n},\) where \(V\) is a smooth potential, \(x,v\in{\mathbb R}^n\) and \(h>0\) is essentially the temperature (so that in this case the semiclassical limit is given by a low-temperature limit). The class they consider is made of pseudodifferential operators that are neither elliptic nor self-adjoint, but that satisfy certain subelliptic conditions: if \(p=p_1+ip_2\) is the symbol of \(p^w(x,D)\) (semiclassical Weyl-Hörmander quantization), it is assumed that \(p_1\geq 0\) and that:
\(\bullet\) \(p\in S(\lambda^2,g_0),\) \(\partial p\in S(\lambda,g_0),\) \(\partial^2p_1\in S(1,g_0)\) and \(\partial H_{p_2}p_1\in S(\lambda,g_0)\), where \(1\leq\lambda(x,\xi)=\lambda\in C^\infty\) is a \(g_0\)-admissible weight such that \(\lambda\in S(\lambda,g_0)\) and \(\partial\lambda\in S(1,g_0)\), where \(g_0=| dx| ^2+\lambda^{-2}| d\xi| ^2\) is an admissible Hörmander’s metric on \({\mathbb R}^{2n}\), and where \(H_{p_2}\) is the Hamilton vector-field associated with \(p_2\);
\(\bullet\) \(p\) has finitely many critical points \(C=\{\rho_1,\ldots,\rho_N\}\) (where, for simplicity only, \(p(\rho_j)=0\)) such that, with \(c_0>0\) sufficiently small, in a fixed ball \(B\) containing \(C\) one has \[ p_1+c_0H_{p_2}^2p_1\approx\text{ dist}_C^2; \]
\(\bullet\) outside \(B\) one has \[ p_1+c_0H_{p_2}^2p_1\approx\lambda^2. \] Under these assumptions, the authors obtain precise resolvent estimates inside the pseudo-spectrum and, when \(p^w+Ch^2\) is \(m\)-accretive, a precise description of the spectrum in a ball centered at the origin and radius \(Ch.\) They are then able to apply the resolvent estimates and the description of the eigenspaces to the large time behavior of the semigroup \(e^{-tp^w/h}\), \(t\geq 0,\) associated with the Cauchy problem \[ (h\partial_t+p^w(x,D))u=0,\,\,\,\, u\bigl| _{t=0}=u_0. \] Near the critical points of \(p\), the main technical tool is the use of the FBI-transform and of suitable weighted \(L^2\)-spaces of holomorphic functions; whereas away from the critical points, it is the use of the semiclassical Weyl-Hörmander calculus.