Semi-classical counting function with application to quantum current (Q2774114)

Let \(A(h)=a_0+a_1h+a_2h^2+\ldots\) be an \(h\)-admissible pseudodifferential operator (\(h\psi\)do). Under suitable assumptions on \(a_0,\) the author studies the counting function NEWLINE\[NEWLINEN^{P,Q}(\lambda,h):=\sum_{\lambda_j(h)\leq\lambda}\Bigl(P(h)\phi_j(h),Q(h)\phi_j(h) \Bigr)_{L^2({\mathbb R}^n)},NEWLINE\]NEWLINE as \(h\to 0,\) where \(\lambda<\lambda_0\) (\(a_0^{-1}((-\infty,\lambda_0])\) non-empty and compact) denotes a fixed regular value of \(a_0,\) \(P(h),\) \(Q(h)\) denote two \(h\psi\)dos with compactly supported symbols (i.e. \(P(h)=p_0+p_1h+\ldots,\) where \(p_j\in C_0^\infty({\mathbb R}^{2n})\) for all \(j\in{\mathbb Z}_+;\) likewise for \(Q(h)\)), and \(\lambda_j(h)\), \(\phi_j(h)\) denote the eigenvalues and corresponding orthonormalized eigenfunctions of \(A(h)\). The sum is counted according to the multiplicity of each eigenvalue. The main result is a new two-term asymptotic formula for \(N^{P,Q}(\lambda,h),\) as \(h\to 0,\) that allows the author to reobtain, as a corollary, recent results concerning the clustering of eigenvalues [\textit{V. Petkov} and \textit{G. Popov}, Ann. Inst. Henri Poincaré, Phys. Théor. 68, No. 1, 17-83 (1998; Zbl 0919.35095)] and, as an application, to study the semiclassical behavior of the quantum current [\textit{S. Fournais}, Commun. Partial Differ. Equ. 23, No.3-4, 601--628 (1998; Zbl 1054.81501)].