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## Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

### Annales Henri Poincaré (2008-04-01) 9: 275-327 , April 01, 2008

### Abstract.

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem
(P)
$$ \left\{\begin{aligned} -\Delta_p u &= \lambda\vert u \vert^{p-2}u + h\left(x,u(x);\lambda\right)\,\,\hbox{ in }\,\,\Omega;\\ u&= 0\,\,\hbox{on}\,\,\partial\Omega.\\ \end{aligned}\right. $$
Here, Ω is a bounded domain in
$${\mathbb{R}}^N (N \geq 1), \Delta_p u\,\, {\mathop = \limits^{\rm def} }\,\, {\rm div}(\mid \nabla u\mid^{p-2}\nabla u)$$
denotes the Dirichlet *p*-Laplacian on
$$W^{1,p}_0(\Omega), 1 < p < \infty$$
, and
$$\lambda \in {\mathbb{R}}$$
is a spectral parameter. Let μ_{1} denote the first (smallest) eigenvalue of −Δ_{p}. Under some natural hypotheses on the perturbation function
$$h : \Omega \times {\mathbb{R}}\times
{\mathbb{R}} \rightarrow {\mathbb{R}}$$
, we show that the trivial solution
$$(0, \mu_1) \in E = W^{1,p}_0 (\Omega)\times {\mathbb{R}}$$
is a bifurcation point for problem (P) and, moreover, there are *two distinct continua*,
$$\mathcal{Z}^+_{\mu_1}$$
and
$$\mathcal{Z}^-_{\mu_1}$$
, consisting of nontrivial solutions
$$(u,\lambda) \in E$$
to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ_{1}). The continua
$$\mathcal{Z}^+_{\mu_1}$$
and
$$\mathcal{Z}^-_{\mu_1}$$
are *either* both unbounded in *E*, *or else* their intersection
$$\mathcal{Z}^+_{\mu_1} \cap \mathcal{Z}^-_{\mu_1}$$
contains also a point other than (0, μ_{1}). For the semilinear problem (P) (i.e., for *p* = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union
$$\mathcal{Z}^+_{\mu_1} \cap
\mathcal{Z}^-_{\mu_1}$$
looks like (for *p* > 2) in an interesting particular case.

Our proofs are based on very precise, local asymptotic analysis for λ near μ_{1} (for any 1 < *p* < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer’s original work.

## Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

### Annales Henri Poincaré (2013-07-01) 14: 1135-1168 , July 01, 2013

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $${\mathbb{H}^n}$$ . The graphs are considered as unbounded hypersurfaces of $${\mathbb{H}^{n+1}}$$ which carry the induced metric and have an interior boundary. For such manifolds, the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence, we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition, this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam’s article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256 , 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061 , 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.

## Construction of Real-Valued Localized Composite Wannier Functions for Insulators

### Annales Henri Poincaré (2016-01-01) 17: 63-97 , January 01, 2016

We consider a real periodic Schrödinger operator and a physically relevant family of $${m \geq 1}$$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $${d \leq 3}$$ , there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.

## Poisson and Diffusion Approximation of Stochastic Master Equations with Control

### Annales Henri Poincaré (2009-08-01) 10: 995-1025 , August 01, 2009

*Quantum Trajectories* are solutions of stochastic differential equations. Such equations are called *Stochastic Master Equations* and describe random phenomena in the continuous measurement theory of Open Quantum System. Many recent developments deal with the control of such models, i.e. optimization, monitoring and engineering. In this article, stochastic models with control are mathematically and physically justified as limits of concrete discrete procedures called *Quantum Repeated Measurements*. In particular, this gives a rigorous justification of the Poisson and diffusion approximations in quantum measurement theory with control.

## Complex Ashtekar Variables and Reality Conditions for Holst’s Action

### Annales Henri Poincaré (2012-04-01) 13: 425-448 , April 01, 2012

From the Holst action in terms of complex valued Ashtekar variables, additional reality conditions mimicking the linear simplicity constraints of spin-foam gravity are found. In quantum theory with the results of Ding and Rovelli, we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our results perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.

## Pseudo-Differential Calculus on Homogeneous Trees

### Annales Henri Poincaré (2014-09-01) 15: 1697-1732 , September 01, 2014

To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on *L*^{2} and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.

## Semiclassical Propagation of Coherent States for the Hartree Equation

### Annales Henri Poincaré (2011-12-01) 12: 1613-1634 , December 01, 2011

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in *L*^{2} by
$${C \sqrt {\varepsilon}}$$
,
$${\varepsilon}$$
being the Planck constant. Finally we present a full formal asymptotic expansion.

## Perturbative Test of Single Parameter Scaling for 1D Random Media

### Annales Henri Poincaré (2004-12-01) 5: 1159-1180 , December 01, 2004

### Abstract.

Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitly exhibited for the Anderson model.

## Poincaré Theory for Decomposable Cofrontiers

### Annales Henri Poincaré (2017-01-01) 18: 85-112 , January 01, 2017

We extend Poincaré’s theory of orientation-preserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid *A* of a surface homeomorphism is irrational and the boundary of *A* is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements classical results by Barge and Gillette on the equivalence between rational rotation numbers and the existence of periodic orbits and yields a direct analogue to the Poincaré classification theorem for circle homeomorphisms. Moreover, we show that the semiconjugacy can be obtained as the composition of a monotone circle map with a ‘universal factor map’, only depending on the topological structure of the circloid. This implies, in particular, that the monotone semiconjugacy is unique up to post-composition with a rotation. If, in addition, *A* is a minimal set, then the semiconjugacy is almost one-to-one if and only if there exists a biaccessible point. In this case, the dynamics on *A* are almost automorphic. Conversely, we use the Anosov–Katok method to build a
$$C^\infty $$
-example where all fibres of the semiconjugacy are non-trivial.

## Widths of Highly Excited Shape Resonances

### Annales Henri Poincaré (2017-04-01) 18: 1289-1304 , April 01, 2017

We study the widths of shape resonances for the semiclassical multidimensional Schrödinger operator, in the case where the frequency remains close to some value strictly larger than the bottom of the well. Under a condition on the behavior of the resonant state inside the well, we obtain an optimal lower bound for the widths.