Monodromy in the quantum spherical pendulum

In this article we show that monodromy in the quantum spherical pendulum can be interpreted as a Maslov effect: i.e. as multi-valuedness of a certain generating function of the quantum energy levels.     

A Necklace of Wulff Shapes

In a probabilistic model of a film over a disordered substrate, Monte-Carlo simulations show that the film hangs from peaks of the substrate. The film profile is well approximated by a necklace of Wulff shapes. Such a necklace can be obtained as the infimum of a collection of Wulff shapes resting on the substrate. When the random substrate is given by iid heights with exponential distribution, we prove estimates on the probability density of the resulting peaks, at small density.AMS subject classification: 60K35, 60K37, 82B24, 82B41     

Spectra of Sol-Manifolds: Arithmetic and Quantum Monodromy

The spectral problem of three-dimensional manifolds M³_A admitting Sol-geometry in Thurston's sense is investigated. Topologically M³_A are torus bundles over a circle with a unimodular hyperbolic gluing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of modified Mathieu functions. It is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form. As a result the spectral statistics is shown to disagree with the Berry-Tabor conjecture. The topological nature of the monodromy for both classical and quantum systems on Sol-manifolds is demonstrated.     

Stokes Matrices and Monodromy of the Quantum Cohomology of Projective Spaces

In this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a “classical” framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Received: 24 October 1998 / Accepted: 27 April 1999     

Monodromy properties of energy momentum tensor on general algebraic curves

A new approach to analyze the properties of the energy momentum tensor T (z) of conformal field theories on generic Riemann surfaces (RS) is proposed. T (z) is decomposed into N components with different monodromy properties, where N is the number of branches in the realization of RS as branch covering over the complex sphere. This decomposition gives rise to new infinite dimensional Lie algebra which can be viewed as a generalization of Virasoro algebra containing information about the global properties of the underlying RS. In the simplest case of hyperelliptic curves the structure of the algebra is calculated in two ways and its central extension is explicitly given. The algebra possess an interesting symmetry with a clear interpretation in the framework of the radial quantization of CFTs with multivalued fields on the complex sphere.     

The Feynman Identity for Planar Graphs

The Feynman identity (FI) of a planar graph relates the Euler polynomial of the graph to an infinite product over the equivalence classes of closed nonperiodic signed cycles in the graph. The main objectives of this paper are to compute the number of equivalence classes of nonperiodic cycles of given length and sign in a planar graph and to interpret the data encoded by the FI in the context of free Lie superalgebras. This solves in the case of planar graphs a problem first raised by Sherman and sets the FI as the denominator identity of a free Lie superalgebra generated from a graph. Other results are obtained. For instance, in connection with zeta functions of graphs.     

The Periodicity of Persistent Currents in Mesoscopic Rings

In a tight-binding model, we find that the equation of the Green's function in a onedimensional mesoscopic ring threaded by a Aharonov-Bohm (AB) flux can be exactly solved. From the poles of the Green's function, the eigenvalues are obtained and are periodic in the AB flux with period h/e. However, the periodicity of the persistent currents is determined by the number of electrons in the ring.     

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III

A unified treatment of monodromy and spectrum preserving deformations is presented. In particular a general procedure is described to reduce the latter into the former consistently. The concept of the τ-function, previously introduced for the former [2], is extended to the isospectral context. It is shown that the general monodromy and spectrum preserving deformation equations can be written as Hirota's bilinear differential equations by using the τ-functions as dependent variables.     

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Let G be a connected graph in which almost all vertices have linear degrees and let \(\mathcal {T}\) be a uniform spanning tree of G. For any fixed rooted tree F of height r we compute the asymptotic density of vertices v for which the r-ball around v in \(\mathcal {T}\) is isomorphic to F. We deduce from this that if \(\{G_n\}\) is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of \(G_n\) locally converges to a multi-type branching process defined in terms of W. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least \(e^{-1}-\mathsf {o}(1)\), the density of vertices of degree 2 is at most \(e^{-1}+\mathsf {o}(1)\) and the density of vertices of degree \(k\geqslant 3\) is at most \({(k-2)^{k-2} \over (k-1)! e^{k-2}} + \mathsf {o}(1)\). These bounds are sharp.     

Scattering the Geometry of Weighted Graphs

Given two weighted graphs (X, b_k, m_k), k =?1,2 with b₁ ～ b₂ and m₁ ～ m₂, we prove a weighted L¹-criterion for the existence and completeness of the wave operators W_±(H₂, H₁, I_1,2), where H_k denotes the natural Laplacian in ?²(X, m_k) w.r.t. (X, b_k, m_k) and I_1,2 the trivial identification of ?²(X, m₁) with ?²(X, m₂). In particular, this entails a general criterion for the absolutely continuous spectra of H₁ and H₂ to be equal.     

The Formation of Non-Keplerian Rings of Matter About Compact Stars

The formation of energetic rings of matter in a Kerr spacetime with an outward pointing acceleration field does not appear to have previously been noted as a relativistic effect. In this paper we show that such rings are a gravimagneto effect with no Newtonian analog, and that they do not occur in the static limit. The energy efficiency of these rings can (depending of the strength of the acceleration field) be much greater than that of Keplerian disks. Counter-intuitively these rings rotate in a direction opposite to that of compact star about which they form. The size and energy efficiency of the rings depend on the fundamental parameters of the spacetime as well as the strength the acceleration field.     

Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

We continue our investigation on the newly introduced concept of the τ function [1], associated with monodromy preserving deformations of a linear differential equation $\text{dY}\text{dx}\text{= A(x)Y}$. In this paper a series of τ functions parametrized by integers are introduced. Their ratios to the original τ function are then shown to be explicit rational expressions in terms of the coefficients of A(x). We point out, and follow where possible, a close analogy between the theory of τ functions and that of classical theta functions.     

双光子过程的能级图和时序图

从辐射与分子相互作用的跃迁几率幅出发,说明各种双光子过程的能级图和时序图与相互作用的微扰矩阵元的对应关系。     

The Einstein Relation for Random Walks on Graphs

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs.     

The Monodromy Matrix Method of Solving an Exterior Boundary Value Problem for a Given Stationary Axisymmetric Perfect Fluid Solution

A procedure is described for matching a given stationary axisymmetric perfect fluid solution to a not necessarily asymptotically flat vacuum exterior. Using data on the zero pressure surface, the procedure yields the Ernst potential of the matching vacuum metric on the symmetry axis. From this the full metric can be constructed by established procedures.