A semi-classical trace formula for Schrödinger operators in the case of a critical energy level
The author considers a semiclassical Schrödinger operator with smooth potential \(V\) and discrete spectrum, and studies the smoothed spectral density: \[ \gamma(E,h)= \sum_{E-\lambda\leq E_k\leq E+\lambda} \varphi\Biggl({E_k- E\over h}\Biggr), \] where \(\lambda>0\) is fixed, \(h\) is the semiclassical parameter, the \(E_k\)s are the eigenvalues of the operator, \(\varphi\) is a Schwartz function with compactly supported Fourier transform, and \(E\) is a critical energy level, that is the energy surface associated to \(E\) is assumed to contain the critical set \(\theta= \{(x,0); \nabla V(x)=0\}\). Under additional geometrical assumptions, the author shows the existence of a full asymptotic expansion of \(\gamma(E,h)\) as \(h\) goes to \(0\), in powers of \(\sqrt h\) and \(\ln h\). This asymptotics includes the contributions of the periodic orbits of the linearized flow on \(\theta\).
scientific article; zbMATH DE number 1077101
scientific article