Abstract
We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum b0 on a closed curve. We derive a classical and quantum normal form which gives uniform eigenvalue asymptotics in a window (−∞,b0+ϵ) for ϵ>0 independent on the semiclassical parameter. These asymptotics are obtained in two complementary settings: either an approximate invariance of the system under translation along the curve, which produces oscillating eigenvalues, or a Morse hypothesis reminiscent of Helffer–Sjöstrand’s “miniwell” situation.
Highlights
Let (M, ω) be a symplectic surface without boundary
As we will see below, this is the universal form of a non-degenerate well. This normal form is not sufficient to describe the semiclassical quantization of our setting, because assumption (1) is not stable under perturbation
We are reduced to Proposition 2.7: because of the volume considerations, one can extend the symplectic normal form given by Proposition 2.1 to a hamiltonian change of variables, equal to identity outside of {(x, ξ) ∈ R2, R ≤ x2 + ξ2 ≤ 9R}
Summary
Let (M, ω) be a symplectic surface without boundary. When introducing quantization, we will assume for simplicity that M = T ∗R or M = T ∗S1. By the Morse-Bott lemma, there exists a neighborhood Ω ⊂ Ω of γ, and coordinates (x, t) : Ω → γ ×[−δ, δ] such that γ = {t = 0} and p = b0 +t2q(x), for some smooth, non-vanishing function q on γ An example of such a p can be obtained in the following way. Under a non-degeneracy assumption on the mini-wells, one has a complete expansion, as well as sharp decay estimates, for the lowest energy eigenfunction of P This result generalizes to any Morse-Bott principal symbol which reaches its minimum on a compact isotropic submanifold, see [4] for a treatment in the Berezin-Toeplitz setting. This paper is organized as follows: Section 2 contains a classical normal form for functions admitting a non-degenerate well on a closed loop and a reminder on the invariant I0.
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