Jacobi Theta Research Articles

On kq representations in quantum mechanics

The Bloch-type functions for linear harmonic oscillators introduced recently by Krivoshlykov et al. (1977) are shown to be identical to the kq representation functions of Zak (1972). As a by-product of this analysis an interesting linear partial differential equation is obtained for Jacobi's theta 3 function with certain periodic boundary conditions. Finally the authors point out briefly the relation between the so called coherent angular momentum states and kq-type representations.

Dual Amplitudes with Two Off-Shell Lines

An integral-representation is explicitly established for the dual-resonance amplitudes with two off-the-mass-shell states which interact with several unexcited states within the framework of Schwarz's off-shell dual scalar (orbital) theory. The procedure to be employed is based on a functional-integral method and shares its most steps in common with those of the functional method used in the on-shell theory: One crucial difference is the presence of the Carrigan-Fairlie boundary condition. The main ingredients of the formula are compactly written in terms of Jacobi's theta functions which also play a judicious role in other various stages of the present formulation. The representation established is shown to be valid only when the intercept of the leading trajectory is one and the transverse degree of space-time dimensions is twenty-four. It is also demonstrated how our amplitudes reduce to the totally on-shell Bardak'ii-Ruegg amplitudes if both of the two-off-shell particles are put on the ground state shells: We also obtain the amplitudes with one off-shell line in the course.

Half-space electrostatic lattice sums

A new inversion formula analogous to the Jacobi theta function imaginary transformation for sums of the type l=0 Sigma infinity exp(-t(l+x)2) is derived and used to find computable forms for the electrostatic energy of point charges and dipoles above the surface of an electrostatically neutral simple cubic lattice of charges and a simple cubic lattice of point dipoles.

Modular Group and Non-Cyclic Symmetry of the Veneziano Formula

Transformation properties of the four-point Veneziano formula as well as its Koba­ Nielsen ·variables are discussed in the language of modular-group theory. The modular group is here associated with Jacobi's theta functions which naturally appear in this area as a conse­ quence of the functional-integral consideration on a rectangular world sheet. Invariance of Veneziano's differential form under the action of the modular group determines the value of the Regge intercept at which the Virasoro condition (also invariant) holds.

Explicit expressions for geometrical dimensions of symmetrical coupled striplines

It is shown how the inverse solution of certain wave-impedance problems can be found explicitly by using Jacobi's theta functions. It is possible to find directly the geometrical dimensions, given the transmission-line wave-impedance parameters. Inverse formulas for a shielded single stripline and coupled symmetrical striplines are presented. The numerical computations are simplified by the excellent convergence properties of Jacobi's theta functions.

The Doppler analog of the Elsasser band model

An expression has been obtained for the spectral absorption coefficient of a distribution of 2 N + 1 equally intense, equally spaced Doppler broadened spectral lines of equal half-width in the limit of large N, i.e., the Doppler analog of an Elsasser band. The expression is P ω=P 0 exp α 2 4β 2 θ 3 πα 2β 2 πi β 2 = S d θ 3 α 2i − β 2 πi where P o¬ 1 n2 π 1 2 S γ , α¬ 2πi(ω−ω 0) d , β= πγ d 1 n2 π 1 2 and the function θ 3( u| τ) is the third Jacobi theta function. Several interesting properties of the distribution have been obtained, namely, (1) P ω±d=P ω , (2) P ω0±nd=P 0θ 3 0 πi β 2 is a maximum, (3) P ω0±(n+ 1 2 )d =P 0θ 2 0 πi β 2 is a minimum, and (4) the ratio of the minimum to maximum values of the absorption coefficient is k 1 2 , where k is the modulus of the complete elliptic integral of the first kind, K( k). This last result may be used to establish conditions for the validity of limiting cases. For example, for values of k 1 2 < 0·01 (corresponding to γ ⧸ d < 0·18), the band may be considered to consist of isolated lines; conversely, for values of >k 1 2 > 0·99 (corresponding to γ ⧸ d > 0·65) the lines are so highly overlapped that the absorption coefficient may be considered constant with a value P ω=P 0 K(k) 2π 1 2 (1+k 1 2 )= S d K′(k) 2π 1 2 (1+k 1 2 )

A note on the Jacobi theta formula

A note on the Jacobi theta formula

On certain identities in theta functions

1. The purpose of this paper is to obtain certain identities in Jacobi's theta functions by a new method. This method may be outlined as follows. We consider two sets of elements S and Si, conceptually distinct, such that the elements of S are in one-to-one correspondence with the elements of S1. In certain cases these elements are algebraic numbers and in other cases they are quaternions; in every case, corresponding elements have the same norms. We construct two functions (I(x, y, ) and T(xi, yi, ), the terms of which are in one-to-one correspondence with the elements of S and Si respectively. Hence there is a one-to-one correspondence between the terms of 1 and those of T. We then show that if xi, yi, * are properly defined linear functions of x, y, , the corresponding terms are equal and therefore (I(x, y, ) =T(xi, yi, ). We shall prove Jacobi's well known fundamental identity,t i.e.

O isogonálném zobrazení čtyř thetafunkcí Jacobiho, jichž modul jest číslo ryze imaginární

O isogonálném zobrazení čtyř thetafunkcí Jacobiho, jichž modul jest číslo ryze imaginární

$E_n$ Jacobi forms and Seiberg–Witten curves

We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.