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This text deals with inverse spectral theory in a semiclassical setting. Given a quantum system, the haunting question is “What interesting quantities can be discovered on the spectrum that can help to characterize the system ?” The general framework will be semiclassical analysis, and the issue is to recover the classical dynamics from the quantum spectrum. The coupling of a spin and an oscillator is a fundamental example in physics where some nontrivial explicit calculations can be done.

Spectrum and generation of solutions of the Toda lattice.

Rolanía, D.Barrios, Márquez, J.R.Gascón (2009)

Discrete Dynamics in Nature and Society

Structure theory for second order 2D superintegrable systems with 1-parameter potentials.

Kalnins, Ernest G., Kress, Jonathan M., Jr., Willard Miller, Post, Sarah (2009)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Supersymmetric representations and integrable fermionic extensions of the Burgers and Boussinesq equations.

Kiselev, Arthemy V., Wolf, Thomas (2006)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Supersymmetry of affine Toda models as fermionic symmetry flows of the extended mkdv hierarchy.

Schmidtt, David M. (2010)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Symmetries of the nonlinear Schrödinger equation

Benoît Grébert, Thomas Kappeler (2002)

Bulletin de la Société Mathématique de France

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $\dots < λ_{k}^{-} \leq λ_{k}^{+} < λ_{k + 1}^{-} \leq \dots$ of a Zakharov-Shabat operator is symmetric,i.e. $λ_{k}^{\pm} = - λ_{- k}^{\mp}$ for all $k$ , if and only if the sequence ${(γ_{k})}_{k \in ℤ}$ of gap lengths, $γ_{k} : = λ_{k}^{+} - λ_{k}^{-}$ , is symmetric with respect to $k = 0$ .

Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body

Zapiski naucnych seminarov POMI

Systèmes hamiltoniens complètement intégrables et déformations d'algèbres de Lie.

Faouzi Ammar (1994)

Publicacions Matemàtiques

Some of the completely integrable Hamiltonian systems obtained through Adler-Kostant-Symes theorem rely on two distinct Lie algebra structures on the same underlying vector space. We study here the cases when two structures are linked together by deformations.

Systèmes intégrables à $m$ -corps sur la droite

J.-P. Françoise (1991)

Mémoires de la Société Mathématique de France

Systems of Toda Type, Inverse Spectral Problems, and Representation Theory.

W.W. Symes (1980)

Inventiones mathematicae

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