On the quantum expected values of integrable metric forms
Let \((M^n,g)\) be a compact Riemannian manifold and \(p(x,\xi)= (g^{ij}(x) \xi_i\xi_j)^{1/2}\) be the Hamiltonian function for the geodesic flow on \(T^*M\). Suppose there exist \(C^\infty\) functions \(p_0,p_1,\dots, p_{n-1}\) on \(T^*M\) with the property that \(dp_0,dp_1, \dots,dp_{n-1}\) are linearly independent almost everywhere and that \(\{p_k,p_\ell\} =0\) for \(k,\ell= 0,1, \dots, n-1\), the Poisson product \(\{ , \}\) being taken with respect to the canonical symplectic form on \(T^*M\). Suppose that the level set \(\Sigma_E= \{z \in T^*M;\;p_0(z)-E_0= \cdots= p_{n-1}(z)-E_{n-1} =0\}\) contains a finite number of non-degenerate unstable periodic geodesics \(\gamma_1,\dots, \gamma_k\) and that \(\Sigma_E\) is smooth outside a union of tubular neighborhoods of the \(\gamma_j\)'s. The author first proves the existence of a convergent joint quantum Birkhoff normal form for the quantum integrals \(p_0,p_1, \dots,p_{n-1}\) under a certain joint non-resonance condition \((H_1)\) and the forward limit sets condition \((H_2)\) for \(\gamma_1,\dots, \gamma_k\). This is done by establishing a convergent classical Birkhoff normal form near each of the \(\gamma_j\) using a result of \textit{H. Ito} [Comment. Math. Helv. 64, 412-461 (1989; Zbl 0686.58021)]. After examining several examples and studying the time asymptotics of the classical geodesic flow, the author carries out the microlocalization near the \(\gamma_j\)'s to use the quantum Birkhoff normal form construction above. Finally the author proves the main theorem: Under the condition \((H_1)\), the bicharacteristics \(\gamma_j\)'s always support eigenfunction mass in the semiclassical limit (the precise description is given, in the introduction, as Theorem 1). The author asserts that it follows, from the main theorem, that in many integrable examples including Euler tops and geodesic flow on quadrics, one can find instable periodic bicharacteristics that support eigenfunction mass.
nondegenerate unstable periodic geodesics
scientific article; zbMATH DE number 1547119