arXiv subjects

Roch Cassanas

Publications and source records attributed to Roch Cassanas.

5 recordsLinked to original sources

arXiv2007

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

Let $G\subset Ø(n)$ be a compact group of isometries acting on $n$-dimensional Euclidean space $\R^n$, and ${\bf{X}}$ a bounded domain in $\R^n$ which is transformed into itself under the action of $G$. Consider a symmetric, classical pseudodifferential operator $A_0$ in $Ł^2(\R^n)$ that commutes with the regular representation of $G$, and assume that it is elliptic on $\bf{X}$. We show that the spectrum of the Friedrichs extension $A$ of the operator $\mathrm{res} \circ A_0 \circ \mathrm{ext}: \CT({\bf{X}}) \to Ł^2({\bf{X}})$ is discrete, and using the method of the stationary phase, we derive asymptotics for the number $N_χ(λ)$ of eigenvalues of $A$ equal or less than $λ$ and with eigenfunctions in the $χ$-isotypic component of $Ł^2({\bf{X}})$ as $λ\to \infty$, 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.

math.AP↗

arXiv2007

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.

math-ph↗

arXiv2006

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 $α^2$Ry.

math-ph↗

arXiv2005

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 $χ$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_χ$ to the symmetry subspace $L^2\_χ(\mathbb{R}^d)$ 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 $\hat{H}\_χ$ 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 $\hat{H}\_χ$ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of $\mathbb{R}^{2d}$.

math-ph↗

arXiv2005

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 $χ$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_χ$ to the symmetry subspace $L^2\_χ(\mathbb{R}^d)$ 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 $\hat{H}\_χ$ near a non critical energy $E\in\mathbb{R}$. If $Σ\_E:=\{H=E \}$ is compact, assuming that periodic orbits are non-degenerate in $Σ\_E/G$, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space $Σ\_E/G$ appear. The method is based upon the use of coherent states, whose propagation was given in the work of M. Combescure and D. Robert.

math-ph↗