Equidistribution of phase shifts in semiclassical potential scattering. (English) Zbl 1323.35121

Let \(H=h^2\Delta+V-E\) be a Hamiltonian (Schrödinger operator) at energy \(E>0\), where \(\Delta\) is the positive Laplacian, \(V\in C_c^\infty(\mathbb R^d)\) a potential, and \(h>0\) the semiclassical parameter (Planck’s constant). In this review, we assume \(E=1\) for simplicity. The scattering operator \(S_h\) is unitary on \(L^2(\mathbb S^{d-1})\) and differs from the identity by a trace class operator. Therefore, the spectrum of \(S_h\) is contained in the unit circle \(\mathbb S^1\) and discrete except at 1. It is shown that, in the semiclassical limit at \(\mathbb S^1\setminus\{1\}\), eigenvalues equidistribute: If \(f(z)=(z-1)g(z)\), \(g:\mathbb S^1\to\mathbb C\) continuous, then (Theorem 5.1) \[ \lim_{h\downarrow 0} (2\pi h)^{d-1} \sum_{e^{i\beta}\in\text{spec}(S_h)} f(e^{i\beta}) =\mathrm{vol}(\mathcal{I}) (2\pi)^{-1}\int_0^{2\pi} f(e^{i\phi})\,d\phi. \] It is assumed that the classical dynamics is non-trapping and that the interacting fixed point set of the reduced scattering map \(\mathcal{S}:T^\ast\mathbb S^{d-1}\to T^\ast\mathbb S^{d-1}\) is negligible with respect to the Liouville measure. The interaction region \(\mathcal{I}\subset T^\ast \mathbb S^{d-1}\) consists of those \((\omega,\eta)\) such that each trajectory which satisfies \(x(t)=2t\omega+\eta\) for \(-t\gg 1\) intersects the support of \(V\). The interacting fixed point set is the fixed point set of \(\mathcal{S}\) intersected with \(\mathcal{I}\). Also, the interacting fixed point sets of all integer powers of \(\mathcal{S}\) are assumed to be of zero Liouville measure.
The trace of \(S_h-\text{Id}\) is computed using a trace formula, proved in an appendix, for semiclassical FIOs associated with Legendre submanifolds. This gives the equidistribution formula for \(f(z)=z-1\). The proof of the general result proceeds to the general case via polynomials \(f(z)\), \(f(1)=0\), and by a density argument.


35P25 Scattering theory for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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