## 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.

### MSC:

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

### Keywords:

potential scattering; phase shift; semiclassical FIO
Full Text:

### References:

 [1] Abramowitz, Handbook of mathematical functions with formulas, graphs, and mathematical tables (1964) · Zbl 0171.38503 [2] Alexandrova, Structure of the semi-classical amplitude for general scattering relations, Comm. Partial Differential Equations 30 pp 1505– (2005) · Zbl 1088.81091 [3] Alexandrova, Resolvent and scattering matrix at the maximum of the potential, Serdica Math. J. 34 pp 267– (2008) · Zbl 1199.35269 [4] Birman, Asymptotics of the spectrum of the s-matrix in potential scattering, Soviet Phys. Dokl. 25 pp 989– (1980) [5] Birman, Asymptotic behavior of limit phases for scattering by potentials without spherical symmetry, Theoret. Math. Phys. 51 pp 344– (1982) [6] Bulger, The spectral density of the scattering matrix for high energies, Comm. Math. Phys. 316 pp 693– (2012) · Zbl 1264.47015 [7] Bulger, The spectral density of the scattering matrix of the magnetic Schrödinger operator for high energies, J. Spectr. Theory pp 517– (2013) · Zbl 1295.81128 [8] Buslaev, The scattering matrix and associated formulas in Hamiltonian mechanics, Comm. Math. Phys. 293 pp 563– (2010) · Zbl 1218.37072 [9] Chazarain, Sur le comportement semi-classique de l’amplitude de diffusion d’un hamiltonien quantique (French) [On the semiclassical behavior of the scattering amplitude of a quantum Hamiltonian], Goulaouic-Meyer-Schwartz Seminar, 1980-1981 (1981) [10] Datchev, Approximation and equidistribution of phase shifts: spherical symmetry, Comm. Math. Phys. 326 pp 209– (2014) · Zbl 1293.35320 [11] Dimassi, Spectral asymptotics in the semi-classical limit (1999) · Zbl 0926.35002 [12] Doron, Semiclassical quantization of chaotic billiards: a scattering theory approach, Nonlinearity 5 pp 1055– (1992) · Zbl 0770.58043 [13] Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 pp 207– (1974) · Zbl 0285.35010 [14] Gérard, Breit-Wigner formulas for the scattering phase and the total scattering cross-section in the semi-classical limit, Comm. Math. Phys. 121 pp 323– (1989) · Zbl 0704.35114 [15] Grigis, An introduction, Microlocal analysis for differential operators (1994) · Zbl 0804.35001 [16] Guillemin, Sojourn times and asymptotic properties of the scattering matrix, Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976) 12 pp 69– (1976/77 supplement) · Zbl 0381.35064 [17] Guillemin, Semi-classical analysis (2010) [18] Hassell, The spectral projections and the resolvent for scattering metrics, J. Anal. Math. 79 pp 241– (1999) · Zbl 0981.58025 [19] Hassell, The semiclassical resolvent and the propagator for non-trapping scattering metrics, Adv. Math. 217 pp 586– (2008) · Zbl 1131.58018 [20] Hunziker, The S-matrix in classical mechanics, Comm. Math. Phys. 8 pp 282– (1968) · Zbl 0159.55001 [21] Keller, Determination of the potential from scattering data, Phys. Rev. 102 ((2)) pp 557– (1956) · Zbl 0070.21905 [22] Klingenberg, Generic properties of geodesic flows, Math. Ann. 197 pp 323– (1972) · Zbl 0225.58006 [23] Majda, High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering, Comm. Pure Appl. Math. 29 pp 261– (1976) · Zbl 0463.35048 [24] Melrose, Geometric scattering theory (1995) · Zbl 0849.58071 [25] Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) pp 85– (1994) [26] Mijatović, Classical inverse problem for finite scattering region, Phys. Rev. A 38 ((3)) pp 5038– (1988) [27] Nakamura, Spectral shift function for trapping energies in the semiclassical limit, Comm. Math. Phys. 208 pp 173– (1999) · Zbl 0945.47053 [28] Nakamura, The spectrum of the scattering matrix near resonant energies in the semiclassical limit, Trans. Amer. Math. Soc. 366 pp 1725– (2014) · Zbl 1283.81115 [29] Narnhofer, Canonical scattering transformation in classical mechanics, Phys. Rev. A 23 ((3)) pp 1688– (1981) [30] Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat. 37 pp 141– (1999) · Zbl 1088.70009 [31] Petkov, Spectrum of the Poincaré map for periodic reflecting rays in generic domains, Math. Z. 194 pp 505– (1987) · Zbl 0673.58035 [32] Petkov, Geometry of reflecting rays and inverse spectral problems (1992) · Zbl 0761.35077 [33] Reed, Methods of modern mathematical physics. III (1979) · Zbl 0405.47007 [34] Robert, Semiclassical bounds for resolvents of Schrödinger operators and asymptotics for scattering phases, Comm. Partial Differential Equations 9 pp 1017– (1984) · Zbl 0561.35021 [35] Robert, Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits, Ann. Inst. Fourier (Grenoble) 39 pp 155– (1989) · Zbl 0659.35026 [36] Simon, Wave operators for classical particle scattering, Comm. Math. Phys. 23 pp 37– (1971) · Zbl 0238.70012 [37] Sobolev, Phase analysis in the problem of scattering by a radial potential, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 147 pp 155– (1985) · Zbl 0588.47011 [38] Taylor, Partial differential equations II. Qualitative studies of linear equations (2011) · Zbl 1206.35003 [39] Vaınberg, Quasiclassical approximation in stationary scattering problems, Funktsional. Anal. i Priložhen. 11 pp 6– (1977) [40] Vasy, Semiclassical estimates in asymptotically Euclidean scattering, Comm. Math. Phys. 212 pp 205– (2000) · Zbl 0955.58023 [41] Wang, Time-delay operators in semiclassical limit. II. Short-range potentials, Trans. Amer. Math. Soc. 322 pp 395– (1990) · Zbl 0714.35064 [42] Yafaev, Mathematical scattering theory: analytic theory (2010) · Zbl 1197.35006 [43] Zelditch, Kuznecov sum formulae and Szego limit formulae on manifolds, Comm. Partial Differential Equations 17 pp 221– (1992) · Zbl 0749.58062 [44] Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 pp 305– (1997) · Zbl 0865.47018 [45] Zworski, Semiclassical analysis (2012) · Zbl 1252.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.