Reduced Gutzwiller formula with symmetry: case of a Lie group

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a Lie group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_{\chi}$ to the symmetry subspace $L^2\_{\chi}(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of $\hat{H}\_{\chi}$ in an interval of $\mathbb{R}$, and interpret it geometrically in terms of dynamics in the reduced space $\mathbb{R}^{2d}/G$. Besides, oscillations of the spectral density of $\hat{H}\_{\chi}$ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of $\mathbb{R}^{2d}$.Comment: 23 page

Semi-classical trace formula, isochronous case. Application to conservative systems

Under conditions of clean flow we compute the leading term in the STF when the set of periods of the energy surface is discrete. Comparing to the case of non-degenerate periodic orbits, we obtain a supplementary term which is given in terms of the linearized flow. As particular cases, we give a STF for quadratic Hamiltonians and we obtain the Berry-Tabor formula for integrable systems. For conservative systems (i.e. systems with several first integrals), we give practical conditions to get a clean flow and interpret the leading term of the STF for a compact symmetry. We give several examples to illustrate our computation.Comment: 24 page

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

AbstractLet G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator res○A0○ext:Cc∞(X)→L2(X) is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L2(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem

Reduced Gutzwiller formula with symmetry: case of a finite group

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_\chi$ to the symmetry subspace $L^2\_\chi(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming from the decomposition of Peter-Weyl. We give reduced semi-classical asymptotics of a regularised spectral density describing the spectrum of $\hat{H}\_\chi$ near a non critical energy $E\in\mathbb{R}$. If $\Sigma\_E:=\{H=E \}$ is compact, assuming that periodic orbits are non-degenerate in $\Sigma\_E/G$, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space $\Sigma\_E/G$ appear. The method is based upon the use of coherent states, whose propagation was given in the work of M. Combescure and D. Robert.Comment: 20 page

The Ground State Energy of Heavy Atoms According to Brown and Ravenhall: Absence of Relativistic Effects in Leading Order

It is shown that the ground state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas-Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order $\alpha^2$Ry

Hamiltoniens quantiques et symétries

Rapporteurs : Thierry Paul, San Vu Ngoc. Jury : Gilles Carron, Monique Combescure, Bernard Helffer (président), François Laudenbach, Thierry Paul, Georgi Popov, Didier Robert, San Vu NgocWe study the semi-classical behavior of a quantum Hamiltonian whose Weyl symbol has some symmetries coming from a compact group G. The quantum reduction is done by restricting the operator to subspaces of L^2(R^n) called symmetry subspaces, coming from the Peter-Weyl decomposition. The restrictions are called the reduced quantum Hamiltonians. For a finite group, we give a Gutzwiller formula for the reduced Hamiltonian, involving the symmetry of periodic orbits of the energy shell. We interpret this formula in the classical reduced space when G acts freely. For a compact Lie group, we give a Weyl asymptotic formula of the eigenvalue counting function of the reduced Hamiltonian, for which we calculate the first term. Oscillations of the spectral density are also described by a Gutzwiller formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of the euclidian space.On étudie le comportement semi-classique d'hamiltoniens quantiques dont le symbole de Weyl est invariant par un groupe de symétries. La réduction quantique consiste à restreindre le hamiltonien aux sous-espaces de symétrie de L^2(R^n) donnés par la décomposition de Peter-Weyl. Les opérateurs restreints sont appelés hamiltoniens quantiques réduits. Pour un groupe fini, on donne une formule de Gutzwiller pour le hamiltonien réduit qui fait intervenir la symétrie d'orbites périodiques classiques du niveau d'énergie étudié. On l'interprète dans l'espace de phase réduit lorsque le groupe agit librement. Pour un groupe de Lie compact, on donne une asymptotique de Weyl de la fonction de comptage des valeurs propres du hamiltonien réduit. On interprète géométriquement le premier terme. On obtient ici aussi une formule de type Gutzwiller impliquant des orbites périodiques de l'espace de phase réduit qui correspondent à des orbites quasi-périodiques de l'espace euclidien

Hamiltoniens quantiques et symétries

On étudie le comportement semi-classique d'hamiltoniens quantiques dont le symbole de Weyl est invariant par un groupe de symétries. La réduction quantique consiste à restreindre le hamiltonien aux sous-espaces de symétrie de L 2 (R n ) donnés par la décomposition de Peter-Weyl. Les opérateurs restreints sont appelés hamiltoniens quantiques réduits. Pour un groupe fini, on donne une formule de Gutzwiller pour le hamiltonien réduit qui fait intervenir la symétrie d'orbites périodiques classiques du niveau d'énergie étudié. On l'interprète dans l'espace de phase réduit lorsque le groupe agit librement. Pour un groupe de Lie compact, on donne une asymptotique de Weyl de la fonction de comptage des valeurs propres du hamiltonien réduit. On interprète géométriquement le premier terme. On obtient ici aussi une formule de type Gutzwiller impliquant des orbites périodiques de l'espace de phase réduit qui correspondent à des orbites quasi-périodiques de R 2n .We study the semi-classical behavior of a quantum Hamiltonian whose Weyl symbol has some symmetries coming from a compact group G. The quantum reduction is done by restricting the operator to subspaces of L 2 (R n ) The quantum reduction is done by restricting the operator to subspaces of L 2 (R n ) called symmetry subspaces, coming from the Peter Weyl decomposition. The restrictions are called the reduced quantum Hamiltonians. For a finite group, we give a Gutzwiller formula for the reduced Hamiltonian, involving the symmetry of periodic orbits of the energy shell. We interpret this formula in the classical reduced space when G acts freely. For a compact Lie group, we give a Weyl asymptotic formula of the eigenvalue counting function of the reduced Hamiltonian, for which we calculate the first term. Oscillations of the spectral density are also described by a Gutzwiller formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of R 2n .NANTES-BU Sciences (441092104) / SudocSudocFranceF