A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes in flat domains while preserving distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the LapalceBeltrami operator. The Laplace-B...

In the past few years,much attention has been devoted to the behavior of eigenfunctions of classically chaotic quantum systems. One aspect of this topic concerns their value distribution and specifically their extreme values (see [Be], [AS], [S1], [IS], [HeR], [ABST]). Our aim is to explore this topic for one of the best-studied models in quantum chaotic dynamics—the quantized cat map (see [HB]...

A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes into flat domains, while preserving the distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the Laplace–Beltrami operator. The L...

Quantum chaos concerns eigenfunctions of the Laplace operator in a domain where a billiard ball would bounce chaotically. Such chaotic eigenfunctions have been conjectured to share statistical properties of their nodal domains with a simple percolation model, from which many interesting quantities can be computed analytically. We numerically test conjectures on the number and size of nodal doma...

Affine Fourier transform (AFT) also called as the canonical. transform. It generalizes the fractional Fourier transform (FRFT), Fresnel transform, scaling operation, etc., and is a very useful tool for signal processing. In this paper, we will derive the eigenfunctions of AFT. The eigenfunctions seems hard to be derived, but since AFT can be represented by the time-frequency matrix (TF matrix),...

The method of Λ-operators developed by S. Derkachov, G. Korchemsky, A. Ma-nashov is applied to a derivation of eigenfunctions for the open Toda chain. The Sklyanin measure is reproduced using diagram technique developed for these Λ-operators. The properties of the Λ-operators are studied. This approach to the open Toda chain eigenfunctions reproduces Gauss-Givental representation for these eige...

We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...

ior of small cuspidal eigenpairs of ΔS. In Theorem 1.7, we describe limiting behavior of these eigenpairs on surfaces Sm ∈Mg,n when (Sm) converges to a point in M̄g,n. Then, we consider the ith cuspidal eigenvalue, λc i (S), of S∈Mg,n. Since noncuspidal eigenfunctions (residual eigenfunctions or generalized eigenfunctions) may converge to cuspidal eigenfunctions, it is not known if λc i (S) is a...

This article is devoted to the analysis of eigenfunctions (modes) and approximate eigenfunctions (quasi-modes) of the Laplacian on a compact manifold (M, g) with completely integrable geodesic flow. We give a new proof of the main result of [TZ] that (M, g) with integrable Laplacians and with uniformly bounded eigenfunctions must be flat. The proof is based on the use of Birkhoff normal forms a...