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Semiclassical spectral asymptotics. (English) Zbl 0839.35092
Robert, D. (ed.), Méthodes semi-classiques. Vol. 1. École d’été (Nantes, juin 1991). Paris: Société Mathématique de France, Astérisque. 207, 7-33 (1992).
Summary: These lectures are devoted to semiclassical spectral asymptotics with accurate remainder estimates and their applications to spectral asymptotics of other types.
In the introduction a brief description of the hyperbolic operator method is given. In part 1 we show how starting from rather classical theorems concerning LSSA (local semiclassical spectral asymptotics) one can weaken their conditions. In 2 we show how LSSA yield asymptotics of eigenvalues tending to \(+\infty\) for operators on compact manifolds and for operators on \(\mathbb{R}^d\) with potentials increasing at infinity and asymptotics of eigenvalues tending to \(-0\) for operators in \(\mathbb{R}^d\) with potentials decreasing at infinity. In 3 we present basic ideas permitting us to use the hyperbolic operator method for general matrix operators and for operators on manifolds with boundary. In 4 we apply the short-time propagation of singularities in order to justify the previous section construction; then in 5 we derive LSSA.
We treat long-time propagation of singularities in order to improve the remainder estimate in LSSA in 6 and in 7. In 8 we split LSSA and Lieb-Cwickel-Rozenbljum eigenvalue estimate and derive estimates above and below for the number of the eigenvalues for the Schrödinger operator. In 9 a more advanced development of the theory is presented.
For the entire collection see [Zbl 0773.00029].

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35J10 Schrödinger operator, Schrödinger equation
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