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Brice Camus

Publications and source records attributed to Brice Camus.

14 recordsLinked to original sources
arXiv2015

Higher dimensional non standard eigenvalue asymptotics

In this article we extend B. Simon's construction and results for leading order eigenvalue asymptotics to $n$-dimensional Schrödinger operators with non-confining potentials given by: $H^α_n=-Δ+\prod\limits_{i=1}^n |x_i|^{α_i}$ on $\mathbb{R}^n$ ($n>2$), $α:=(α_1,\cdots,α_n)\in (\mathbb{R}_{+}^*)^n$. We apply the results to also derive the leading order spectral asymptotics in the case of the Dirchlet Laplacian $-Δ^D$ on domains $Ω^α_n=\{x\in\mathbb{R}^n: \prod\limits_{j=1}^n |x_j|^{\frac{α_j}{α_n}}<1 \}$. keywords : Trace formulae; Schrödinger operators; Singular asymptotics.

math.SP↗
arXiv2013

Determination of the Genus of Surfaces from the Spectrum of Schrödinger Operators attached to height functions. (An inverse spectral problem for surfaces)

Using results on inverse spectral problems, in particular the so-called new wave invariants attached to a classical equilibrium, we show that it is possible to determine the Morse index of height functions. For compact Riemannian surfaces $M\subset \mathbb{R}^3$ this imply that we can retrieve the topology (via the genus). Our results are independent from the choice of a metric on $M$ and can be obtained from the choice of a 'generic' height-function. For surfaces of genus zero, diffeomorphic to a 2-sphere, the method allows to detect the convexity, or the local convexity of the surface. Keywords : Micro-local analysis; Schrödinger operators; Inverse spectral problems.

math.AP↗
arXiv2013

Inverse spectral problems for Schrödinger and pseudo-differential operators

Starting from the semi-classical spectrum of Schrödinger operators $-h^2Δ+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express spectral statistics in terms of different invariants: \begin{itemize} \item Geometry of energy surfaces (heat invariant like). \item Classical orbits (wave invariants). \item But also classical equilibria (new wave invariants). \end{itemize} Any critical point of $V$ with zero momentum is an equilibrium of the flow and generates many singularities in the semi-classical distribution of eigenvalues. Via sharp spectral estimates, this phenomena indicates the presence of a critical energy level and the information contained in this singularity allows to reconstruct partially the local shape of $V$. Several generalizations of this approach are also proposed. Keywords : Spectral analysis, P.D.E., Micro-local analysis; Schrödinger operators; Inverse spectral problems.

math.AP↗
arXiv2006

A semi-classical trace formula at a totally degenerate critical level

We study the semi-classical trace formula at a critical energy level for an $h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has a totally degenerate critical point for that energy. This problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point.

math.AP↗
arXiv2006

Equilibrium and eigenfunctions estimates in the semi-classical regime

We establish eigenfunctions estimates, in the semi-classical regime, for critical energy levels associated to an isolated singularity. For Schrödinger operators, the asymptotic repartition of eigenvectors is the same as in the regular case, excepted in dimension 1 where a concentration at the critical point occurs. This principle extends to pseudo-differential operators and the limit measure is the Liouville measure as long as the singularity remains integrable.

math.AP↗
arXiv2006

Spectral fluctuations of Schrödinger operators generated by critical points of the potential

Starting from the spectrum of Schrödinger operators on $\mathbb{R}^n$, we propose a method to detect critical points of the potential. We argue semi-classically on the basis of a mathematically rigorous version of Gutzwiller's trace formula which expresses spectral statistics in term of classical orbits. A critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum. Via sharp spectral estimates, this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential. Some generalizations of this approach are also proposed.\medskip keywords : Semi-classical analysis; Schrödinger operators; Equilibriums in classical mechanics.

math-ph↗
arXiv2006

A semi-classical trace formula at a non-degenerate critical level

We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.

math.AP↗
arXiv2005

Fundamental solutions of homogeneous elliptic differential operators

We compute fundamental solutions of homogeneous elliptic differential operators, with constant coefficients, on $\mathbb{R}^n$ by mean of analytic continuation of distributions. The result obtained is valid in any dimension, for any degree and can be extended to pseudodifferential operators of the same type.

math.AP↗
arXiv2005

Semiclassical spectral estimates for Schrödinger operators at a critical energy level. Case of a degenerate potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local minimum. The main result, which computes the contribution of this equilibrium, is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point.

math-ph↗
arXiv2005

Spectral estimates for degenerate critical levels

We establish spectral estimates at a critical energy level for $h$-pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in term of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. $h$ of the spectral distributions are bigger than in the regular setting.

math.AP↗
arXiv2005

Asymptotic approximation of degenerate fiber integrals

We study asymptotics of fiber integrals depending on a large parameter. When the critical fiber is singular, full-asymptotic expansions are established in two different cases : local extremum and isolated real principal type singularities. The main coefficients are computed and invariantly expressed. In the most singular cases it is shown that the leading term of the expansion is related to invariant measures on the spherical blow-up of the singularity. The results can be applied to certain degenerate oscillatory integrals which occur in spectral analysis and quantum mechanics.

math.GM↗
arXiv2005

Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of $\mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.

math.SP↗
arXiv2004

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of "real principal type". The new contribution to the trace formula is valid for all time in a compact subset of $\mathbb{R}$ but the result is modest since we have restrictions on the dimension.

math.FA↗