Related papers: Spectral Geometry and the Semiclassical Approach
In the past decade there has been a flurry of activity at the intersection of spectral theory and symplectic geometry. In this paper we review recent results on semiclassical spectral theory for commuting Berezin-Toeplitz and…
We address the question as to why, in the semiclassical limit, classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of Random Matrix Theory. To do so, we use a semiclassical resummation…
Perturbation theory, the quasiclassical approximation and the quantum surface of section method are combined for the first time. This gives a new solution of the the long standing problem of quantizing the resonances generically appearing…
In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral…
In this paper we propose a general framework to study the quantum geometry of $\sigma$-models when they are effectively localized to small quantum fluctuations around constant maps. Such effective theories have surprising exact descriptions…
We study the evolution of the non-equilibrium quantum fields from a highly excited initial state in two approaches: the standard Keldysh-Schwinger diagram technique and the semiclassical expansion. We demonstrate explicitly that these two…
We present new theoretical results on the spectrum of the quantum field theory of the Double Sine Gordon model. This non-integrable model displays different varieties of kink excitations and bound states thereof. Their mass can be obtained…
A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace…
We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a…
The spectral geometry of negatively curved manifolds has received more attention than its positive curvature counterpart. In this paper we will survey a variety of spectral geometry results that are known to hold in the context of…
We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting…
We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential…
In this short paper we discuss the precise relationship between the semiclassical and standard pseudodifferential algebras and explore implications such as for large spectral parameter elliptic estimates, even in the case of…
A quantitative test for the validity of the semi-classical approximation in gravity is given. The criterion proposed is that solutions to the semi-classical Einstein equations should be stable to linearized perturbations, in the sense that…
Perturbation theory, the quasiclassical approximation and the quantum surface of section method are combined for the first time. This solves the long standing problem of quantizing the resonances and chaotic regions generically appearing in…
The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis…
The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have…
The classical sine-Gordon model is a two-dimensional integrable field theory, with particle like solutions the so-called solitons. Using its integrability one can define its quantum version without the process of canonical quantization.…
Bifurcations of classical orbits introduce divergences into semiclassical spectra which have to be smoothed with the help of uniform approximations. We develop a technique to extract individual energy levels from semiclassical spectra…
This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an…