The semiclassical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence in 2-D for families of Hamiltonians $(H_\lambda(x,\xi), {\cal H}_\lambda(x,\xi))$, when ${\cal H}_\lambda(x,\xi)$ is the perturbation of completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming the Weyl $h$-PDO $H^w_\lambda$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}={\cal E}\}$ is separated by some pairwise disjoint lagrangian tori, we show that most of eigenvalues for $\hat H_\lambda$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere

Fourier integrals and a new representation of Maslov's canonical operator near caustics

We suggest a new representation of Maslov's canonical operator in a neighborhood of the caustics using a special class of coordinate systems (\eikonal coordinates") on Lagrangian manifolds

Semi-classical Green functions

Let $H(x,p)\sim H_0(x,p)+hH_1(x,p)+\cdots$ be a semi-classical Hamiltonian on $T^*{\bf R}^n$, and $\Sigma_E=\{H_0(x,p)=E\}$ a non critical energy surface. Consider $f_h$ a semi-classical distribution (the "source") microlocalized on a Lagrangian manifold $\Lambda$ which intersects cleanly the flow-out $\Lambda_+$ of the Hamilton vector field $X_{H_0}$ in $\Sigma_E$. Using Maslov canonical operator, we look for a semi-classical distribution $u_h$ satisfying the limiting absorption principle and $H^w(x,hD_x)u_h=f_h$ (semi-classical Green kernel). In this report, we elaborate (still at an early stage) on some results announced in [Doklady Akad. Nauk, Vol. 76, No1, p.1-5, 2017] and provide some examples, in particular from the theory of wave beams.Comment: Conference Days of Diffraction 2018, St Petersbur

The semi-classical Maupertuis-Jacobi correspondence: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians $(H_\lambda(x,\xi), {\cal H}_\lambda(x,\xi))$, when ${\cal H}_\lambda(x,\xi)$ is the perturbation of a completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming $\hat H_\lambda$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}_0={\cal E}\}$ is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for $\hat H_\lambda$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.Comment: Conference Days of Diffraction 2012, St Petersbur

The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory

~40 pagesInternational audienceWe extend to the semi-classical setting the Maupertuis-Jacobi correspondance for a pair of hamiltonians $(H(x,hD_x), {\cal H}(x,hD_x)$. If ${\cal H}(p,x)$ is completely integrable, or has merely has invariant diohantine torus $\Lambda$ in energy surface ${\cal E}$, then we can construct a family of quasi-modes for $H(x,hD_x)$ at the corresponding energy $E$. This applies in particular to the theory of water-waves in shallow water, and determines trapped modes by an island, from the knowledge of Liouville metrics

Tunneling, librations and normal forms in quantum double well with magnetic field

Non UBCUnreviewedAuthor affiliation: A.Ishlinski Institute for Problem in Mechanics of RASFacult