Jacobi Theta Functions Research Articles

Probabilistic aspects of Jacobi theta functions

In this note we deduce well-known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

On quantum states for angular position and angular momentum of light

In the present paper we construct a properly defined quantum state ψ(θ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L=−id/dθ in units of ℏ=1 . The quantum uncertainties Δθ and ΔL for the state are well-defined and are shown to give a lower value of the uncertainty product ΔθΔL in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨L⟩ of the state ψ(θ) is not required to be an integer. In the case of any half-integer mean value ⟨L⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δθ=π2/3−2 and ΔL=1/2 .

Impulsive impact of a twin hull

An impulsively starting motion of two cylindrical bodies floating on a free liquid surface is considered. The shape of the cross-section of each body and the distance between them are arbitrary. The integral hodograph method is advanced to derive the complex velocity potential defined in a rectangular parameter region in terms of the elliptic quasi-doubly periodic Jacobi theta functions. A system of singular integral equations in the velocity magnitude on the free surface and in the slope of the wetted part of each body is derived using the kinematic boundary condition, which is then solved numerically. The velocity field, the pressure impulse on the bodies and the added mass coefficients of each body immediately after the impact are determined in a wide range of distances between the bodies and for cross-sectional shapes such as the flat plate and half-circle.

Apollonian packings and Kac-Moody root systems

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system Φ \Phi . We introduce the generating function Z ( s ) Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for Φ \Phi . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ \Phi , with automorphic Weyl denominators, we express Z ( s ) Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form Δ 5 \Delta _5 . We also show that the domain of convergence of Z ( s ) Z(\mathbf {s}) is the Tits cone of Φ \Phi , and discover that this domain inherits the intricate geometric structure of Apollonian packings.

Gluon scattering in AdS at finite string coupling from localization

We consider gluons scattering in Type IIB string theory on AdS5 × S5/ℤ2 in the presence of D7 branes, which is dual to the flavor multiplet correlator in a certain 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 USp(2N) gauge theory with SO(8) flavor symmetry and complexified coupling τ. We compute this holographic correlator in the large N and finite τ expansion using constraints from derivatives of the mass deformed sphere free energy, which we compute to all orders in 1/N and finite τ using supersymmetric localization. In particular, we fix the F4 higher derivative correction to gluon scattering on AdS at finite string coupling τs = τ in terms of Jacobi theta functions, which feature the expected relations between the SL(2, ℤ) duality and the SO(8) triality of the CFT, and match it to the known flat space term. We also use the flat space limit to compute D2F4 corrections of the correlator at finite τ in terms of a non-holomorphic Eisenstein series. At weak string coupling, we find that the AdS correlator takes a form which is remarkably similar to that of the flat space Veneziano amplitude.

Integrals Involving Jacobi Theta Functions and Applications in Quantum Systems

We study four expressions involving the integrals of Jacobi’s theta functions. From Poisson’s summation formula, we write the integrals of the functions θi, (i = 1,2,3,4) in terms of modified Bessel functions of the second kind. For the integrals of θ1 , θ2 and θ3, we get expressions with real arguments, but for the integral of θ4 , we find an expression with imaginary argument. In addition, we apply our results to the description of two kinds of interacting quantum systems: boson gas and fermion gas both under a thermal bath and an external magnetic field. PACS numbers:

Analytical study of the pantograph equation using Jacobi theta functions

The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y′(x)=ay(qx)+by(x), where a and b are two non-zero real or complex numbers. When 0<q<1 and y(0)=1, the associated Cauchy problem admits a unique power series solution, ∑n≥0(−a/b;q)nn!(bx)n, that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.

Three types of boundary conditions in a four-fermion interacting model for quarks

In this note, we study the phase transition of an effective four-fermion model describing interactions between quarks, taking into account finite size and chemical potential effects on the system, with periodic, quasi-periodic, and anti-periodic boundary conditions in D - Euclidean dimensions through Jacobi theta functions. Strictly speaking, this is performed by the generalized Matsubara formalism and the Schwinger proper-time representation. We consider the system at temperature T and the spatial coordinates compacted along the direction j with length Lj. The gap equation of the system is solved numerically and we investigate its behavior when the thermodynamic parameters of the model are changed.

Recurrence relations and Jacobi theta functions

We consider two recurrence relations for the number of representations of an integer as sums of squares, and we show that the corresponding coefficients into such recurrences can be written in terms of theta functions.

Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N-solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

Large gap asymptotics on annuli in the random normal matrix model

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form exp(C1n2+C2nlogn+C3n+C4n+C5logn+C6+Fn+O(n-112)),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\exp \\Bigg ( C_{1} n^{2} + C_{2} n \\log n + C_{3} n + C_{4} \\sqrt{n} + C_{5}\\log n + C_{6} + {\\mathcal {F}}_{n} + \\mathcal {O}\\Big ( n^{-\\frac{1}{12}}\\Big )\\Bigg ), \\end{aligned}$$\\end{document}where n is the number of points of the process. We determine the constants C1,…,C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{6}$$\\end{document} explicitly, as well as the oscillatory term Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only C1,…,C4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{4}$$\\end{document} were previously known, (ii) when the hole region is an unbounded annulus, only C1,C2,C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},C_{2},C_{3}$$\\end{document} were previously known, and (iii) when the hole region is a regular annulus in the bulk, only C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} was previously known. For general values of our parameters, even C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} is new. A main discovery of this work is that Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

Quadratic superlattices: A type of nonperiodic lattice with extended states

Deterministic nonperiodic lattices often exhibit properties intermediate between those of random and periodic crystals. Properties that have attracted interest include complex heirarchical structure factors, the breaking up of the band into a dense set of critical states and minigaps, and the interplay between extended and localized states. In one dimension, considerable attention has focused on nonperiodic lattices showing properties intermediate between periodic and random. We propose a one-dimensional nonperiodic lattice with lattice sites at ${0}^{2}d, {1}^{2}d, {2}^{2}d$, ... with length $d$ playing the r\^ole of the lattice constant. Here we show that the structure factor ${S}_{0}(k)$, which governs how waves are scattered and is reflected in many physical properties, is simply related to the Jacobi theta function ${\ensuremath{\vartheta}}_{3}(q)$. ${S}_{0}(k)$ is periodic in wave vector $k$ and is singular continuous, consisting of a dense set of peaks with $k$ given by rational-fraction multiples of $\frac{\ensuremath{\pi}}{d}$, though the scaling properties of ${S}_{0}(k)$ are the same as for periodic crystals. The electronic structure considered in a tight-binding model shows a bandlike spectrum, but broken by a dense set of minigaps near peak locations in ${S}_{0}(k)$. These lattices show novel properties such as the coexistence of a singular-continuous spectrum with extended states. Quadratic lattices may enable the physical realization of special functions of mathematical and physical importance as well as for the design of nonperiodic diffraction gratings, antenna arrays, ad coupled optical resonators to produce complex quasirandom behavior and are of interest to study quantum interactions in nonperiodic optical lattices.

Heat coefficients for magnetic Laplacians on the complex projective space P n(ℂ)

We denote by Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of Δ ν . Using a suitable polynomial decomposition of the multiplicity of each β m , we write down a trace formula for the heat operator associated with Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with Δ ν .

The lambda extensions of the Ising correlation functions

We revisit, with a pedagogical heuristic motivation, the lambda extension of the low-temperature row correlation functions of the two-dimensional Ising model. In particular, using these one-parameter series to understand the deformation theory around selected values of , namely with m and n integers, we show that these series yield perturbation coefficients, generalizing form factors, that are D-finite functions. As a by-product these exact results provide an infinite number of highly non-trivial identities on the complete elliptic integrals of the first and second kind. These results underline the fundamental role of Jacobi theta functions and Jacobi forms, the previous D-finite functions being (relatively simple) rational functions of Jacobi theta functions, when rewritten in terms of the nome of elliptic functions.

A Generalization of Iseki's Formula and The Jacobi Theta Function

In this paper we give a generalization of Iseki's formula and use it to prove the transformation law of $\theta_1(z,\tau)$.

On Some Expansion Formulas for Products of Jacobi’s Theta Functions

In this paper, we establish several expansion formulas for products of the Jacobi theta functions. As applications, we derive some expressions of the powers of (q;q)∞ by using these expansion formulas.

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

For a prime $p\equiv 3$ $(\text{mod }4)$ and $m\ge 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\Gamma_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\mathrm{SL}_2(\mathbb{Z})$.

A domain free of the zeros of the partial theta function

The partial theta function is the sum of the series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|&lt;1$). Its name is due to similaritieswith the formula for the Jacobi theta function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. The function $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently in the study of section-hyperbolic polynomials,i.~e. real polynomials with all coefficients positive,with all roots real negative and all whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$\theta$ is an entire function of order $0$ in the variable~$x$. When$q$ is real and $q\in (0,0.3092\ldots )$, $\theta (q,.)$ is a function of theLaguerre-P\'olyaclass $\mathcal{L-P}I$. More generally, for $q \in (0,1)$, the function $\theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $\mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric progression with ratio $1/q$. A similar result is true for$q\in (-1,0)$ when there are also infinitely-many positive zeros.We consider thequestion how close to the origin the zeros of the function $\theta$ can be.In the generalcase when $q$ is complex it is truethat their moduli are always larger than $1/2|q|$. We consider the case when $q$ is real and prove that for any $q\in (0,1)$,the function $\theta (q,.)$ has no zeros on the set $$\displaystyle \{x\in\mathbb{C}\colon |x|\leq 3\} \cap \{x\in\mathbb{C}\colon {\rm Re} x\leq 0\}\cap \{x\in\mathbb{C}\colon |{\rm Im} x|\leq 3/\sqrt{2}\}$$which containsthe closure left unit half-disk and is more than $7$ times larger than it.It is unlikely this result to hold true for the whole of the lefthalf-disk of radius~$3$. Similar domains do not exist for $q\in (0,1)$, Re$x\geq 0$, for$q\in (-1,0)$, Re$x\geq 0$ and for $q\in (-1,0)$, Re$x\leq 0$. We show alsothat for $q\in (0,1)$, the function $\theta (q,.)$ has no real zeros $\geq -5$ (but one can find zeros larger than $-7.51$).

Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits

We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.

Superpotentials from singular divisors

We study Euclidean D3-branes wrapping divisors D in Calabi-Yau orientifold compactifications of type IIB string theory. Witten’s counting of fermion zero modes in terms of the cohomology of the structure sheaf {mathcal{O}}_D applies when D is smooth, but we argue that effective divisors of Calabi-Yau threefolds typically have singularities along rational curves. We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf {mathcal{O}}_{overline{D}} of the normalization overline{D} of D. We establish this by detailing compactifications in which the singularities can be unwound by passing through flop transitions, giving a physical incarnation of the normalization process. Analytically continuing the superpotential through the flops, we find that singular divisors whose normalizations are rigid can contribute to the superpotential: specifically, {h}_{+}^{bullet}left({mathcal{O}}_{overline{D}}right)=left(1,0,0right) and {h}_{-}^{bullet}left({mathcal{O}}_{overline{D}}right)=left(0,0,0right) give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups Γ. We use the action of Γ on effective divisors to determine the exact effective cones, which have infinitely many generators. The resulting nonperturbative superpotentials are Jacobi theta functions, whose modular symmetries suggest the existence of strong-weak coupling dualities involving inversion of divisor volumes.