scholarly journals Solution of Eight-vertex Lattice Model Without Elliptic Functions

1974 ◽  
Vol 27 (4) ◽  
pp. 433 ◽  
Author(s):  
Kailash Kumar
Keyword(s):  
Eigenvalue Problem ◽  
Transfer Matrix ◽  
Elliptic Functions ◽  
Point Of View ◽  
Vertex Model ◽  
Type Problem ◽  
Matrix Eigenvalue Problem ◽  
Lattice Statistical Mechanics ◽  
Algebraic Operations ◽  
Functional Matrix

Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.

A HIERARCHY OF HAMILTONIAN EQUATIONS ASSOCIATED WITH THE MODIFIED JAULENT–MIODEK HIERARCHY

2010 ◽  
Vol 24 (02) ◽  
pp. 183-193
Author(s):  
HAI-YONG DING ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN ◽  
LI-LI ZHU
Keyword(s):  
Eigenvalue Problem ◽  
Evolution Equations ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Matrix Eigenvalue ◽  
Integrable Evolution

By considering a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is presented.

scholarly journals Field Analysis of Ridged Waveguides Using Transfer Matrix

2012 ◽  
Vol 433-440 ◽  
pp. 3863-3869
Author(s):  
Wei Hua Zong ◽  
Ming Xin Shao ◽  
Xiao Yun Qu
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Previous Analysis ◽  
Linear Equations ◽  
Field Analysis ◽  
Weight Functions ◽  
Mode Matching ◽  
Matching Method ◽  
Ridged Waveguide ◽  
Functional Matrix

The mode matching method is applied to analyze generalized ridged waveguides. The tangential fields in each region are expressed in terms of the product of several matrices, i.e., a functional matrix about x-F(x), a functional matrix about y-G(y) and a column vector of amplitudes. The boundary conditions are transformed into a set of linear equations by taking the inner products of each element of G(y) with weight functions. Two types of ridged waveguide are calculated to validate the theory. Several new modes not reported in previous analysis are presented.

A Matrix Eigenvalue Problem (G. Efroymson, A. Steger and S. Steinberg)

SIAM Review ◽  
1980 ◽  
Vol 22 (1) ◽  
pp. 99-100
Author(s):  
T. Sekiguchi ◽  
N. Kimura
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Eigenvalue Problem ◽  
Matrix Eigenvalue

Quantum Tavis–Cummings Model

2019 ◽  
pp. 474-488
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Transfer Matrix ◽  
Strong Interaction ◽  
Building Blocks ◽  
Vertex Model ◽  
Algebraic Bethe Ansatz ◽  
Quantum Matter ◽  
Ansatz Approach

This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.

scholarly journals Instability Through Anisotropy in Spherical Stellar Systems

1987 ◽  
Vol 127 ◽  
pp. 515-516
Author(s):  
P.L. Palmer ◽  
J. Papaloizou
Keyword(s):  
Growth Rate ◽  
Eigenvalue Problem ◽  
Growth Rates ◽  
Stellar Systems ◽  
Matrix Eigenvalue Problem ◽  
Simple Method ◽  
Anisotropic Models ◽  
Poisson Equations ◽  
Degree Of Anisotropy ◽  
The Stability

We consider the linear stability of spherical stellar systems by solving the Vlasov and Poisson equations which yield a matrix eigenvalue problem to determine the growth rate. We consider this for purely growing modes in the limit of vanishing growth rate. We show that a large class of anisotropic models are unstable and derive growth rates for the particular example of generalized polytropic models. We present a simple method for testing the stability of general anisotropic models. Our anlysis shows that instability occurs even when the degree of anisotropy is very slight.

A FOUR-BY-FOUR MATRIX EIGENVALUE PROBLEM, RELATED HIERARCHY OF LAX INTEGRABLE EQUATIONS AND THEIR HAMILTONIAN STRUCTURES

2008 ◽  
Vol 22 (23) ◽  
pp. 4027-4040 ◽  
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG ◽  
WEI-LI CAO
Keyword(s):  
Eigenvalue Problem ◽  
Evolution Equations ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Integrable Equations ◽  
Matrix Eigenvalue ◽  
Integrable Evolution

Starting from a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is proved.

scholarly journals NEW EXACT SOLUTIONS OF SPATIALLY AND TEMPORALLY VARYING REACTION-DIFFUSION EQUATIONS

2008 ◽  
Vol 06 (04) ◽  
pp. 371-381 ◽  
Author(s):  
NALINI JOSHI ◽  
TEGAN MORRISON
Keyword(s):  
Reaction Diffusion ◽  
Elliptic Functions ◽  
Point Of View ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Source Terms ◽  
Symmetry Approach ◽  
New Exact Solutions ◽  
Special Cases ◽  
New Feature

This paper considers reaction-diffusion equations from a new point of view, by including spatiotemporal dependence in the source terms. We show for the first time that solutions are given in terms of the classical Painlevé transcendents. We consider reaction-diffusion equations with cubic and quadratic source terms. A new feature of our analysis is that the coefficient functions are also solutions of differential equations, including the Painlevé equations. Special cases arise with elliptic functions as solutions. Additional solutions given in terms of equations that are not integrable are also considered. Solutions are constructed using a Lie symmetry approach.

scholarly journals THE 8V CSOS MODEL AND THE sl2 LOOP ALGEBRA SYMMETRY OF THE SIX-VERTEX MODEL AT ROOTS OF UNITY

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1899-1905 ◽  
Author(s):  
TETSUO DEGUCHI
Keyword(s):  
Transfer Matrix ◽  
Lattice Model ◽  
Algebraic Method ◽  
System Size ◽  
Spin Operator ◽  
Total Spin ◽  
Coupling Constants ◽  
Roots Of Unity ◽  
Vertex Model ◽  
Elliptic Quantum

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2N η = m1 + im2τ, where N, m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 ( mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 ( mod N) of the six-vertex model at Nth roots of unity is given by [Formula: see text], where [Formula: see text] is the maximal value of the total spin operator SZ in the degenerate eigenspace.

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