Abstract
In the 1980s, Helffer and Sjöstrand examined in a series of articles the concentration of the ground state of a Schrödinger operator in the semiclassical limit. In a similar spirit, and using the asymptotics for the Szegő kernel, we show a theorem about the localization properties of the ground state of a Toeplitz operator, when the minimal set of the symbol is a finite set of non-degenerate critical points. Under the same condition on the symbol, for any integer $K$ we describe the first $K$ eigenvalues of the operator.
Highlights
In classical mechanics, the minimum of the energy, when it exists, is a critical value, and any point in phase space achieving this minimum corresponds to a stationary trajectory
This is an instance of what is called quantum selection: not all points in phase space where the classical energy is minimal are equivalent in quantum mechanics
We propose to study the Kähler quantization, which associates to a symbol on a phase space a Toeplitz operator
Summary
The minimum of the energy, when it exists, is a critical value, and any point in phase space achieving this minimum corresponds to a stationary trajectory. It is easy to prove that the ground state concentrates on this submanifold From this fact, a formal calculus leads to the study of a Schrödinger operator, on the submanifold, with an effective potential that depends on the 2-jet behaviour of V near the submanifold. On the contrary, when the minimal submanifold corresponds to a symmetry of V , the ground state is spread out on the submanifold This is an instance of what is called quantum selection: not all points in phase space where the classical energy is minimal are equivalent in quantum mechanics. The arguments used by Helffer and Sjöstrand depend strongly on the fact that they deal with Schrödinger operators, when the phase space is T ∗Rn. it is a priori not clear to which extent the quantum selection can be generalised to a quantization of compact phase spaces. Exponential estimates will be the object of a separate investigation
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