Abstract.

We prove the convergence of the perturbative expansion, based on Renormalization Group, of the two point Schwinger function of a system of weakly interacting fermions in d = 2, with symmetric Fermi surface and up to exponentially small temperatures, close to the expected onset of superconductivity.

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 μ₁ 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, μ₁). 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, μ₁). 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 μ₁ (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer’s original work.

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.

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.

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.

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.

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 L² and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.

We consider the systematic force on a heavy probe induced by interaction with an overdamped diffusive medium where particles undergo a rotating force around a fixed center. The stiffness matrix summarizes the stability of the probe around that center, where the induced force vanishes. We prove that the introduction of the rotational force in general enhances the stability of that point (and may turn it from unstable to stable), starting at second order in the nonequilibrium amplitude. When the driving is further enhanced the stabilization occurs for a wide range of rotation profiles and the induced stiffness converges to a universal expression proportional to the average mechanical stiffness. The model thus provides a rigorous example of stabilization of a fixed point due to contact with a nonequilibrium medium and beyond linear order around equilibrium.

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 L² by $${C \sqrt {\varepsilon}}$$ , $${\varepsilon}$$ being the Planck constant. Finally we present a full formal asymptotic expansion.

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.