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Higher dimensional non standard eigenvalue asymptotics

Rautenberg Nils, Camus Brice
10 Mar 2014 math.SP math-ph math.MP arxiv.org/abs/1403.2284

In this article we extend B. Simon's construction and results for leading order eigenvalue asymptotics to $n$-dimensional Schr\"odinger operators with non-confining potentials given by: $H^\alpha_n=-\Delta +\prod\limits_{i=1}^n |x_i|^{\alpha_i}$ on $\mathbb{R}^n$ ($n>2$), $\alpha:=(\alpha_1,\cdots,\alpha_n)\in (\mathbb{R}{+}^*)^n$. We apply the results to also derive the leading order spectral asymptotics in the case of the Dirchlet Laplacian $-\Delta^D$ on domains $\Omega^\alpha_n={x\in\mathbb{R}^n: \prod\limits{j=1}^n |x_j|^{\frac{\alpha_j}{\alpha_n}}<1 }$. keywords : Trace formulae; Schr\"odinger operators; Singular asymptotics.

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

Camus Brice
07 Mar 2013 math.AP arxiv.org/abs/1303.1866

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; Schr\"odinger operators; Inverse spectral problems.

Inverse spectral problems for Schr\"odinger and pseudo-differential operators

Camus Brice
20 Feb 2013 math.AP math.SP arxiv.org/abs/1302.5665

Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (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; Schr\"odinger operators; Inverse spectral problems.

Fundamental solutions for a class of non-elliptic homogeneous differential operators

Camus Brice
05 Apr 2007 math.AP math.GM arxiv.org/abs/0704.0801

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be constructed in terms of radial averages and invariant distributions on the unit sphere.

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

Camus Brice
09 Oct 2006 math.AP math.SP arxiv.org/abs/math/0610308

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.

Equilibrium and eigenfunctions estimates in the semi-classical regime

Camus Brice
24 May 2006 math.AP math-ph math.MP arxiv.org/abs/math/0605637

We establish eigenfunctions estimates, in the semi-classical regime, for critical energy levels associated to an isolated singularity. For Schr\"odinger 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.

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

Camus Brice
18 Mar 2006 math.AP math-ph math.MP arxiv.org/abs/math/0603456

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.

Fundamental solutions of homogeneous elliptic differential operators

Camus Brice
31 Oct 2005 math.AP math.GM arxiv.org/abs/math/0510672

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.

Spectral fluctuations of Schr\"odinger operators generated by critical points of the potential

Camus Brice
25 Apr 2005 math-ph math.MP math.SP arxiv.org/abs/math-ph/0504073

Starting from the spectrum of Schr\"odinger 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; Schr\"odinger operators; Equilibriums in classical mechanics.

Spectral estimates for degenerate critical levels

Camus Brice
30 Mar 2005 math.AP math.SP arxiv.org/abs/math/0503719

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.