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Approximation and equidistribution of phase shifts: spherical symmetry. (English) Zbl 1293.35320

The authors of this very interesting paper consider a dynamical system with a Hamiltonian \[ H_{V,h}\equiv h^2\Delta +V-E, \] where \(h>0\) is a semiclassical parameter, \(\Delta =-\sum\limits_{i=1}^{d}\partial_i^2\) is the positive Laplacian on \(\mathbb{R}^d\) (\(d\geq 2\)), \(V(x)\) is a smooth, compactly supported central potential function, and \(E>0\) is an energy level. The scattering matrix for a semiclassical potential scattering problem with spherical symmetry on \(\mathbb{R}^d\) (\(d\geq 2\)) is investigated. We define the scattering matrix \(S_h(E)\) for the above stated Hamiltonian in terms of the asymptotics of generalized eigenfunctions of the operator \(H_{V,h}\). Thus, for each \textit\mathrm{in}-function \(q_{\mathrm{in}}\in C^{\infty }(\mathbb{S}^{d-1})\), there is a unique solution to the equation \(H_{V,h}u = 0\), \[ u = r^{-(d-1)/2}\biggl(e^{-i\sqrt{E}r/h}q_{\mathrm{in}}(\omega ) + e^{+i\sqrt{E}r/h}q_{\mathrm{out}}(-\omega )\biggr) + O(r^{-(d+1)/2}), \] as \(r\to\infty \); \(q_{\mathrm{out}}\in C^{\infty }(\mathbb{S}^{d-1})\). Actually, the scattering matrix \(S_{h}(E)\) is represented by the map \(q_{\mathrm{in}}\to e^{+i\pi (d-1)/2} q_{\mathrm{out}}\) according to the definition. It seems that the scattering matrix in the case of central potentials would be of great interest. Then the eigenfunctions of the scattering matrix are spherical harmonics and the generalized eigenfunctions take the form \(u=r^{-(d-2)/2}f(r)Y_{l}^{m}\), where \(f(r)\) is defined by Hankel functions. Introducing the number \(R\) such that \(B(0,R)\) is the smallest ball containing the support of \(V\), \(V\) is nontrapping at the energy \(E\) in the interaction region. Here a condition concerning the scattering angle is imposed. Two important results are shown by the authors.
The first statement is that if a function \(G(\alpha )\) of a real argument \(\alpha \) by \(dG/d\alpha =\Sigma (\alpha )\), \(G(\alpha )=0\) for \(\alpha \geq R\) (\(\Sigma \) is the scattering angle function) is defined, and if the dimension \(d\) is even, then there exists a \(C=C(d)\) such that \(\forall l\in\mathbb{N}\) satisfying \(l h \leq R\) there follows the estimate \[ |e^{i\beta_{l,h}}-\exp\{ih^{-1}G\bigl((l+0.5(d-2))h\bigr)\}|\leq Ch . \] If \(d>2\) is odd, then \(\forall\epsilon >0\) there exists a \(C=C(\epsilon ,d)\) such that the same estimate holds whenever \(\alpha =lh\geq \epsilon \) is at distance at least \(\epsilon \) from the set \(\{\alpha : \Sigma (\alpha )\in \{\pi k\}_{k\in\mathbb{Z}}\}\).
The second statement concerns the eigenvalues \(e^{i\beta_{l,h}}\) for which \(l\leq R\sqrt{E}/h\) as \(h\to 0\), counted with certain multiplicity.

MSC:

35Q70 PDEs in connection with mechanics of particles and systems of particles
35P25 Scattering theory for PDEs
81U20 \(S\)-matrix theory, etc. in quantum theory
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