Circular symmetry and the trace formula

A Semi-classical Trace Formula for Schrödinger Operators in the Case of a Critical Energy Level

“…For more generic Hamiltonians, the semiclassical trace formula gives us one of the most important tools in this direction. There has been many mathematical publications, leading to a rigorous version of this formula [19,14,4,18,21,20,7,8], many years after Gutzwiller has introduced it [15]. Let us now present a detailed description of this result.…”

“…The contributions coming from the periods of the linearized flow appears as a regularization of the Duistermaat Guillemin density which has been defined in [11,14,3] and is studied in Appendix A. This regularization appears explicitly in (8).…”

Semiclassical spectral estimates for Toeplitz operators

“…More precisely we will study the trace of the Fourier coefficients of the operator (p(T -E{-i9e)), where E is a constant and ^ a smooth function with compact ly supported Fourier transform. (This regime corresponds to the semi-classical analogue of the procedure of reduction of the classical Hamiltonian system (Z, <7r) with respect to the 5' 1 action, [18].) We now discuss how this setting and problem arise in semi-classical analysis.…”

“…5, we now have only to calculate the result of the Lagrangian symbol map A. But now we can proceed as in the trace formula calculation in the presence of circular symmetry in [18]. Namely, we move the calculation down to X by considering symplectic reduction under the nd the measures dp,t are those appearing in Theorem 5.6 of [18].…”

“…But now we can proceed as in the trace formula calculation in the presence of circular symmetry in [18]. Namely, we move the calculation down to X by considering symplectic reduction under the nd the measures dp,t are those appearing in Theorem 5.6 of [18]. As the construction of the d^t was described in detail in this paper, we will not repeat these details here.…”