1 Introduction The inverse spectral problem on a Riemannian manifold (M, g), possibly with boundary, is to determine as much as possible of the geometry of (M, g) from the spectrum of its Laplacian ∆ g (with some given boundary conditions). The special inverse problem of Kac is to determine a Euclidean domain Ω ⊂ R n up to isometry from the spectrum Spec B (Ω) of its Laplacian ∆ B with Dirichlet, Neumann or more general boundary conditions B. The physical motivation is to identify physical objects from the light or sound they emit, which may be all that is observable of remote objects such as stars or atoms. The inverse spectral problem is just one among many kinds of inverse problems whose goal is to determine a metric, domain or scatterer from physically relevant invariants. A comparison with other inverse problems shows just how small a set of invariants the spectrum is. For instance, the boundary inverse problem asks to determine the metric g on a fixed bounded domain Ω ⊂ M of a Riemannian manifold (M, g) from the spectrum of the Dirichlet Laplacian on L 2 (Ω), and from the Cauchy data ∂φ λ ∂ν | ∂Ω of its eigenfunctions, or equivalently from its Dirichlet-to-Neumann operator [BK, KKL, LU]. The inverse scattering problem seeks to determine an obstacle from its scattering amplitude [Ma, Ma2], or from its scattering length spectrum [St, St2], a set of lengths parametrized by S n−1 × S n−1. By comparison, the inverse spectral problem has to make do with just a discrete 'unformatted' set of eigenvalues (or resonance poles in the open case). 2 It is probably a consequence of this relative poverty of invariants that most of the results in inverse spectral theory over the last two decades (especially since the appearance of Sunada's aricle [Su]) are 'negative results', showing that one cannot determine metrics or domains by their spectra. The collection of non-isometric isospectral pairs of Riemannian manifolds would require a lengthy survey of its own and for that we refer to the lectures of C. Gordon (cf. [Gor, Gor2]). By comparison, the number of 'positive results' showing that one can indeed recover a domain or metric is rather small. This survey is devoted to the positive results. The emphasis is on relations between the spectrum of the Laplacian and the dynamics of the geodesic flow G t : …

https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2004/0009/0001/SDG-2004-0009-0001-a012.pdf

The inverse spectral problem

Many differential equations arising in engineering and the physical sciences, particularly in the study of vibratory or oscillatory phenomena, have the form 



(1)

in which αn, αn-1,…, αl, αo and f are given functions of x. Such an equation is known as a linear ordinary differential equation of order n. In this chapter it will sometimes be convenient to write a linear equation in the abbreviated form 



(2)

where L denotes a linear differential operator: 



The linearity property of L is expressed by the easily confirmed result 



where c1 and c2 are any constants and y and z any functions of x possessing derivatives up to the nth order.

Ordinary Differential Equations of Order > 1

This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete inverse spectral problems.

https://www.intlpress.com/site/pub/files/_fulltext/journals/iccm/2014/0002/0002/ICCM-2014-0002-0002-a001.pdf

Survey on the Inverse Spectral Problem

Trace formulae provide one of the most elegant descriptions of the classical-quantum correspondence. One side of a formula is given by a trace of a quantum object, typically derived from a quantum Hamiltonian, and the other side is described in terms of closed orbits of the corresponding classical Hamiltonian. In algebraic situations, such as the original Selberg trace formula, the identities are exact, while in general they hold only in semi-classical or high-energy limits. We refer to a recent survey \cite{Ur} for an introduction and references. 
In this paper we present an intermediate trace formula in which the original trace is expressed in terms of traces of quantum monodromy operators directly related to the classical dynamics. The usual trace formulae follow and in addition this approach allows handling effective Hamiltonians.

Quantum monodromy and semi-classical trace formulae

In this work we construct logarithms and Birkhoff normal forms for elliptic Fourier integral operators in the semi-classical limit under more general assumptions than in aprevious work by the first author. The methods are similar but slightly different.

Birkhoff normal forms for Fourier integral operators II

We prove that a Hamiltonian p ∈ C1(TRn) is locally integrable near a non-degenerate critical point �0 of the energy, provided that the fundamental matrix at �0 has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in the C 1 sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that when p is holomorphic near �0 ∈ TC n , then Re p becomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge, i.e., p may not be integrable. These normal forms also hold in the semi-classical frame.

Michel Rouleux

Papers

Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form