Authors:Rupert L. Frank; Marius Lemm Pages: 2285 - 2340 Abstract: Abstract
This paper consists of three parts. In part I, we microscopically derive Ginzburg–Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator
\({K_{T_c}+V}\)
to be n-fold degenerate and the resulting GL theory then couples n order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure d-wave order parameters and (b) mixed (s + d)-wave order parameters, in two and three-dimensions. In part III, we present explicit choices of spherically symmetric interactions V which produce the examples in part II. In fact, we find interactions V which produce ground state sectors of
\({K_{T_c}+V}\)
of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schrödinger operators
\({-\nabla^2+V}\)
, for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0473-x Issue No:Vol. 17, No. 9 (2016)

Authors:Denis Borisov; Anastasia Golovina; Ivan Veselić Pages: 2341 - 2377 Abstract: Abstract
We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e., the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0465-x Issue No:Vol. 17, No. 9 (2016)

Authors:Ram Band; David Fajman Pages: 2379 - 2407 Abstract: Abstract
A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction—those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0468-7 Issue No:Vol. 17, No. 9 (2016)

Authors:Tadayoshi Adachi; Masaki Kawamoto Pages: 2409 - 2438 Abstract: Abstract
In this paper, we study the quantum dynamics of a charged particle in the plane in the presence of a periodically pulsed magnetic field perpendicular to the plane. We show that by controlling the cycle when the magnetic field is switched on and off appropriately, the result of the asymptotic completeness of wave operators can be obtained under the assumption that the potential V satisfies the decaying condition
\({ V(x) \le C(1 + x )^{-\rho}}\)
for some
\({\rho > 0}\)
. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0457-x Issue No:Vol. 17, No. 9 (2016)

Authors:James B. Kennedy; Pavel Kurasov; Gabriela Malenová; Delio Mugnolo Pages: 2439 - 2473 Abstract: Abstract
We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0460-2 Issue No:Vol. 17, No. 9 (2016)

Authors:Yan-Hui Qu Pages: 2475 - 2511 Abstract: Abstract
We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let s be the Hausdorff dimension of the spectrum. For V > 20, we show that the restriction of the s-dimensional Hausdorff measure to the spectrum is a Gibbs type measure; the density of states measure is a Markov measure. Based on the fine structures of these measures, we show that both measures are exact dimensional; we obtain exact asymptotic behaviors for the optimal Hölder exponent and the Hausdorff dimension of the density of states measure and for the Hausdorff dimension of the spectrum. As a consequence, if the frequency is not silver number type, then for V big enough, we establish strict inequalities between these three spectral characteristics. We achieve them by introducing an auxiliary symbolic dynamical system and applying the thermodynamical and multifractal formalisms of almost additive potentials. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0462-0 Issue No:Vol. 17, No. 9 (2016)

Authors:Joachim Asch; Olivier Bourget; Victor Cortés; Claudio Fernandez Pages: 2513 - 2527 Abstract: Abstract
One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavine’s time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturbed embedded eigenstates the bound is explicit and involves Fermi’s Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time-dependent systems including the AC Stark effect as well as multistate systems. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0467-8 Issue No:Vol. 17, No. 9 (2016)

Authors:Mohamed Lazhar Tayeb Pages: 2529 - 2553 Abstract: Abstract
The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L1-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0484-7 Issue No:Vol. 17, No. 9 (2016)

Authors:Varga Kalantarov; Anton Savostianov; Sergey Zelik Pages: 2555 - 2584 Abstract: Abstract
The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown. PubDate: 2016-09-01 DOI: 10.1007/s00023-016-0480-y Issue No:Vol. 17, No. 9 (2016)

Authors:Johannes Kautzsch; Marc Kesseböhmer; Tony Samuel Pages: 2585 - 2621 Abstract: Abstract
We consider a family
\({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\)
of Markov interval maps interpolating between the tent map
\({T_{0}}\)
and the Farey map
\({T_{1}}\)
. Letting
\({\mathcal{P}_{r}}\)
denote the Perron–Frobenius operator of
\({T_{r}}\)
, we show, for
\({\beta \in [0, 1]}\)
and
\({\alpha \in (0, 1)}\)
, that the asymptotic behaviour of the iterates of
\({\mathcal{P}_{r}}\)
applied to observables with a singularity at
\({\beta}\)
of order
\({\alpha}\)
is dependent on the structure of the
\({\omega}\)
-limit set of
\({\beta}\)
with respect to
\({T_{r}}\)
. The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities. PubDate: 2016-09-01 DOI: 10.1007/s00023-015-0451-8 Issue No:Vol. 17, No. 9 (2016)

Authors:Jonas Lampart; Mathieu Lewin Pages: 1937 - 1954 Abstract: Abstract
We study vacuum polarisation effects of a Dirac field coupled to an external scalar field and derive a semi-classical expansion of the regularised vacuum energy. The leading order of this expansion is given by a classical formula due to Chin, Lee-Wick and Walecka, for which our result provides the first rigorous proof. We then discuss applications to the non-relativistic large-coupling limit of an interacting system, and to the stability of homogeneous systems. PubDate: 2016-08-01 DOI: 10.1007/s00023-016-0472-y Issue No:Vol. 17, No. 8 (2016)

Authors:Gabriel Rivière Pages: 1955 - 1999 Abstract: Abstract
We consider perturbations of the semiclassical Schrödinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window. PubDate: 2016-08-01 DOI: 10.1007/s00023-016-0464-y Issue No:Vol. 17, No. 8 (2016)

Authors:Hal M. Haggard; Muxin Han; Aldo Riello Pages: 2001 - 2048 Abstract: Abstract
We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi–Civita holonomies around each of the tetrahedron’s faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group-valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. This becomes manifest in light of their relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in covariant loop quantum gravity in the presence of a cosmological constant. PubDate: 2016-08-01 DOI: 10.1007/s00023-015-0455-4 Issue No:Vol. 17, No. 8 (2016)

Authors:Marcus Carlsson; Humberto Prado; Enrique G. Reyes Pages: 2049 - 2074 Abstract: Abstract
Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory. PubDate: 2016-08-01 DOI: 10.1007/s00023-015-0447-4 Issue No:Vol. 17, No. 8 (2016)

Authors:Vladimir Georgescu; Manuel Larenas; Avy Soffer Pages: 2075 - 2101 Abstract: We prove pointwise in time decay estimates via an abstract conjugate operator method. This is then applied to a large class of dispersive equations. PubDate: 2016-08-01 DOI: 10.1007/s00023-016-0459-8 Issue No:Vol. 17, No. 8 (2016)

Authors:Kazunori Ando; Hiroshi Isozaki; Hisashi Morioka Pages: 2103 - 2171 Abstract: Abstract
We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice. PubDate: 2016-08-01 DOI: 10.1007/s00023-015-0430-0 Issue No:Vol. 17, No. 8 (2016)

Authors:Sohei Ashida Pages: 2173 - 2197 Abstract: Abstract
We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. In Martinez and Sordoni (Mem AMS 936, 2009) such a case is also studied but their reduced Hamiltonian includes the vector potential terms. In this paper, using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms of the nucleus. Using the reduced evolution, we also obtain the asymptotic expansion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constant magnetic fields. PubDate: 2016-08-01 DOI: 10.1007/s00023-016-0458-9 Issue No:Vol. 17, No. 8 (2016)

Authors:Hajime Moriya Pages: 2199 - 2236 Abstract: Abstract
This note provides a C
*-algebraic framework for supersymmetry. Particularly, we consider fermion lattice models satisfying the simplest supersymmetry relation. Namely, we discuss a restricted sense of supersymmetry without a boson field involved. We construct general supersymmetric C
*-dynamics in terms of a superderivation and a one-parameter group of automorphisms on the CAR algebra. (We do not introduce Grassmann numbers into our formalism.) We show several basic properties of superderivations on the fermion lattice system. Among others, we establish that superderivations defined on the strictly local algebra are norm-closable. We show a criterion of superderivations on the fermion lattice system for being nilpotent. This criterion can be easily checked and hence yields new supersymmetric fermion lattice models. PubDate: 2016-08-01 DOI: 10.1007/s00023-016-0461-1 Issue No:Vol. 17, No. 8 (2016)

Authors:Jie Xiao Pages: 2265 - 2283 Abstract: Abstract
As an inclusive
\({(1,3)\ni p}\)
—extension of Bray–Miao’s Theorem 1 and Corollary 1 (Invent Math 172:459–475, 2008) for p = 2, this note presents a sharp isoperimetric inequality for the p-harmonic capacity of a surface in the complete, smooth, asymptotically flat 3-manifold with non-negative scalar curvature, and then an optimal Riemannian Penrose type inequality linking the ADM/total mass and the p-harmonic capacity by means of the deficit of Willmore’s energy. Even in the Euclidean 3-space, the discovered result for
\({p \not =2}\)
is new and non-trivial. PubDate: 2016-08-01 DOI: 10.1007/s00023-016-0475-8 Issue No:Vol. 17, No. 8 (2016)

Authors:Lu Xu; Yong Li; Yingfei Yi Abstract: Abstract
We consider a multi-scale, nearly integrable Hamiltonian system. With proper degeneracy involved, such a Hamiltonian system arises naturally in problems of celestial mechanics such as Kepler problems. Under suitable non-degenerate conditions of Bruno–Rüssmann type, the persistence of the majority of non-resonant, quasi-periodic invariant tori has been shown in Han et al. (Ann. Henri Poincaré 10(8):1419–1436, 2010). This paper is devoted to the study of splitting of resonant invariant tori and the persistence of certain class of lower-dimensional tori in the resonance zone. Similar to the case of standard nearly integrable Hamiltonian systems (Li and Yi in Math. Ann. 326:649–690, 2003, Proceedings of Equadiff 2003, World Scientific, 2005, pp 136–151, 2005), we show the persistence of the majority of Poincaré–Treschev non-degenerate, lower-dimensional invariant tori on a the given resonant surface corresponding to the highest order of scale. The proof uses normal form reductions and KAM method in a non-standard way. More precisely, due to the involvement of multi-scales, finite steps of KAM iterations need to be firstly performed to the normal form to raise the non-integrable perturbation to a sufficiently high order for the standard KAM scheme to carry over. PubDate: 2016-08-17 DOI: 10.1007/s00023-016-0516-3