587 research outputs found
Quantum selection for spin systems
We report mathematical results on the process by which quantum order by
disorder takes place for spin systems. The selection rules follow the influence
of several competing contributions. Moreover there is no link between quantum
selection and thermal selection. We present work in the general setting as well
as toy models and examples for which quantum selection has an interesting
behaviour
Low-energy spectrum of Toeplitz operators: the case of wells
In the 1980s, Helffer and Sj\"ostrand examined in a series of articles the
concentration of the ground state of a Schr\"odinger operator in the
semiclassical limit. In a similar spirit, and using the asymptotics for the
Szeg\"o 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
WKB eigenmode construction for analytic Toeplitz operators
We provide almost eigenfunctions for Toeplitz operators with real-analytic
symbols, at the bottom of non-degenerate wells. These almost eigenfunctions
follow the WKB ansatz; the error is O(exp(--cN)), where c > 0 and N
+ is the inverse semiclassical parameter
The Szeg\H{o} kernel in analytic regularity and analytic Fourier Integral Operators
We build a general theory of microlocal (homogeneous) Fourier Integral
Operators in real-analytic regularity, following the general construction in
the smooth case by H\"ormander and Duistermaat. In particular, we prove that
the Boutet-Sj\"ostrand parametrix for the Szeg\H{o} projector at the boundary
of a strongly pseudo-convex real-analytic domain can be realised by an analytic
Fourier Integral Operator. We then study some applications, such as FBI-type
transforms on compact, real-analytic Riemannian manifolds and propagators of
one-homogeneous (pseudo)differential operators
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