508 research outputs found
Period functions for Maass wave forms. I
Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a
smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to
C which is small as y \to \infty and satisfies Delta u = lambda u for some
lambda \in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic
Laplacian. These functions give a basis for L_2 on the modular surface Gamma\H,
with the usual trigonometric waveforms on the torus R^2/Z^2, which are also
(for this surface) both the Fourier building blocks for L_2 and eigenfunctions
of the Laplacian. Although therefore very basic objects, Maass forms
nevertheless still remain mysteriously elusive fifty years after their
discovery; in particular, no explicit construction exists for any of these
functions for the full modular group. The basic information about them (e.g.
their existence and the density of the eigenvalues) comes mostly from the
Selberg trace formula: the rest is conjectural with support from extensive
numerical computations.Comment: 68 pages, published versio
Quasimodularity and large genus limits of Siegel-Veech constants
Quasimodular forms were first studied in the context of counting torus
coverings. Here we show that a weighted version of these coverings with
Siegel-Veech weights also provides quasimodular forms. We apply this to prove
conjectures of Eskin and Zorich on the large genus limits of Masur-Veech
volumes and of Siegel-Veech constants.
In Part I we connect the geometric definition of Siegel-Veech constants both
with a combinatorial counting problem and with intersection numbers on Hurwitz
spaces. We introduce modified Siegel-Veech weights whose generating functions
will later be shown to be quasimodular.
Parts II and III are devoted to the study of the quasimodularity of the
generating functions arising from weighted counting of torus coverings. The
starting point is the theorem of Bloch and Okounkov saying that q-brackets of
shifted symmetric functions are quasimodular forms. In Part II we give an
expression for their growth polynomials in terms of Gaussian integrals and use
this to obtain a closed formula for the generating series of cumulants that is
the basis for studying large genus asymptotics. In Part III we show that the
even hook-length moments of partitions are shifted symmetric polynomials and
prove a formula for the q-bracket of the product of such a hook-length moment
with an arbitrary shifted symmetric polynomial. This formula proves
quasimodularity also for the (-2)-nd hook-length moments by extrapolation, and
implies the quasimodularity of the Siegel-Veech weighted counting functions.
Finally, in Part IV these results are used to give explicit generating
functions for the volumes and Siegel-Veech constants in the case of the
principal stratum of abelian differentials. To apply these exact formulas to
the Eskin-Zorich conjectures we provide a general framework for computing the
asymptotics of rapidly divergent power series.Comment: 107 pages, final version, to appear in J. of the AM
The energy operator for infinite statistics
We construct the energy operator for particles obeying infinite statistics
defined by a q-deformation of the Heisenberg algebra.
(This paper appeared published in CMP in 1992, but was not archived at the
time.)Comment: 6 page
Finite Size XXZ Spin Chain with Anisotropy Parameter
We find an analytic solution of the Bethe Ansatz equations (BAE) for the
special case of a finite XXZ spin chain with free boundary conditions and with
a complex surface field which provides for symmetry of the
Hamiltonian. More precisely, we find one nontrivial solution, corresponding to
the ground state of the system with anisotropy parameter
corresponding to . With a view to establishing an exact
representation of the ground state of the finite size XXZ spin chain in terms
of elementary functions, we concentrate on the crossing-parameter
dependence around for which there is a known solution. The
approach taken involves the use of a physical solution of Baxter's t-Q
equation, corresponding to the ground state, as well as a non-physical solution
of the same equation. The calculation of and then of the ground state
derivative is covered. Possible applications of this derivative to the theory
of percolation have yet to be investigated. As far as the finite XXZ spin chain
with periodic boundary conditions is concerned, we find a similar solution for
an assymetric case which corresponds to the 6-vertex model with a special
magnetic field. For this case we find the analytic value of the ``magnetic
moment'' of the system in the corresponding state.Comment: 12 pages, latex, no figure
Power partitions and a generalized eta transformation property
In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number ps(n) of partitions of a positive integer~n into s-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function Ps(q)=âân=1(1âqns)â1 near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case s=1. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for ps(n) of the same general type of the one that they gave for~s=1. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of p(200) the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for p2(100000) these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying size
Strichartz estimates with broken symmetries
In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schrödinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe an alternative method that is applicable to a wider class of matrix problems
Asymptotics of Nahm sums at roots of unity
We give a formula for the radial asymptotics to all orders of the special -hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecturerelating the modularity of Nahm sums to the vanishing of a certain invariant in -theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connectionbetween asymptotics of quantum knot invariants and asymptotics of Nahm sumsthat will be discussed further in a subsequent publication.<br
Dynamics of geodesics, and Maass cusp forms
The correspondence principle in physics between quantum mechanics and
classical mechanics suggests deep relations between spectral and geometric
entities of Riemannian manifolds. We survey---in a way intended to be
accessible to a wide audience of mathematicians---a mathematically rigorous
instance of such a relation that emerged in recent years, showing a dynamical
interpretation of certain Laplace eigenfunctions of hyperbolic surfaces.Comment: 30 pages, 17 figure
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