We prove semiclassical resolvent estimates for the Schrödinger operator with a real-valued $$L^\infty $$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the manifold at infinity. In particular, we show that for compactly supported real-valued $$L^\infty $$ potentials and asymptotically Euclidean manifolds the resolvent bound is of the form $$\exp (Ch^{-4/3}\log (h^{-1}))$$, while for asymptotically hyperbolic manifolds it is of the form $$\exp (Ch^{-4/3})$$, where $$C>0$$ is some constant.

We consider the difference $$f(H_1)-f(H_0)$$ , where $$H_0=-\Delta $$ and $$H_1=-\Delta +V$$ are the free and the perturbed Schrödinger operators in $$L^2({{\mathbb {R}}}^d)$$ , respectively, in which V is a real-valued short range potential. We give a sufficient condition for this difference to belong to a given Schatten class $${\mathbf {S}}_p$$ , depending on the rate of decay of the potential and on the smoothness of f (stated in terms of the membership in a Besov class). In particular, for $$p>1$$ we allow for some unbounded functions f.

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.

Obtaining HOMFLY-PT polynomials $$H_{R_1,\ldots ,R_l}$$ for arbitrary links with l components colored by arbitrary SU(N) representations $$R_1,\ldots ,R_l$$ is a very complicated problem. For a class of rank r symmetric representations, the [r]-colored HOMFLY-PT polynomial $$H_{[r_1],\ldots ,[r_l]}$$ evaluation becomes simpler, but the general answer lies far beyond our current capabilities. To simplify the situation even more, one can consider links that can be realized as a 3-strand closed braid. Recently (Itoyama et al. in Int J Mod Phys A28:1340009, 2013. arXiv:1209.6304 ), it was shown that $$H_{[r]}$$ for knots realized by 3-strand braids can be constructed using the quantum Racah coefficients (6j-symbols) of $$U_q(sl_2)$$ , which makes easy not only to evaluate such invariants, but also to construct analytical formulas for $$H_{[r]}$$ of various families of 3-strand knots. In this paper, we generalize this approach to links whose components carry arbitrary symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $$H_{[r_1],[r_2]}$$ for the two-component link L7a3 whose components carry $$[r_1]$$ and $$[r_2]$$ colors. Using our results for exclusive Racah matrices, it is possible to calculate symmetric-colored HOMFLY-PT polynomials of links for the so-called one-looped links, which are obtained from arborescent links by adding a loop. This is a huge class of links that contains the entire Rolfsen table, all 3-strand links, all arborescent links, and, for example, all mutant knots with 11 intersections.

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.