Monodromy matrices for Harper equation

2018 ◽  
Author(s):  
A. Fedotov ◽  
E. Shchetka
Keyword(s):  
Monodromy Matrices ◽  
Harper Equation

APPLYING THE 1/N APPROXIMATION TO THE DERIVATION OF THE BAND ENERGY OF THE HARPER EQUATION

2000 ◽  
Vol 14 (11) ◽  
pp. 373-376
Author(s):  
E. PAPP
Keyword(s):  
Magnetic Field ◽  
Reflection Symmetry ◽  
Band Energy ◽  
First Order ◽  
The Magnetic Field ◽  
Harper Equation

The 1/N approach to the Harper equation proposed previously is generalized towards performing the pertinent band-energy description. Leading forms and corrections proceeding to first order in the magnetic field are written down. The energy reflection symmetry has also been discussed.

RENORMALIZATION IN QUASIPERIODICALLY FORCED SYSTEMS

2003 ◽  
Vol 03 (02) ◽  
pp. L251-L258 ◽  
Author(s):  
A. H. OSBALDESTIN ◽  
B. D. MESTEL
Keyword(s):  
Localized States ◽  
Rigorous Results ◽  
Two Level Systems ◽  
Self Similar ◽  
Strange Nonchaotic Attractors ◽  
Harper Equation

We review our recent rigorous results on renormalization in a variety of quasiperiodically forced systems. Our results include a description of (i) self-similar fluctuations of localized states in the Harper equation, including the renormalization strange set (known as the orchid) in the generalized Harper equation; and (ii) self-similarities in the correlations of strange nonchaotic attractors, barrier billiards, and quantum two-level systems.

scholarly journals On the Ultralocal and Non-Ultralocal Poisson Brackets

1997 ◽  
Vol 12 (13) ◽  
pp. 905-911
Author(s):  
G. P. Pronko
Keyword(s):  
Mechanical System ◽  
Gauge Transformation ◽  
Poisson Brackets ◽  
Monodromy Matrices

We consider the classical KdV theory as an example of non-ultralocal mechanical system (i.e. where fundamental Poisson brackets contain the derivative of δ-function). We show that by an appropriate gauge transformation, the Poisson brackets of the associated monodromy matrices may be transformed into those of the classical Heisenberg magnetic.

RECTANGULAR YANG–BAXTER ALGEBRAS AND ALTERNATING A-TYPE INTEGRABLE VERTEX MODELS

2005 ◽  
Vol 02 (06) ◽  
pp. 1063-1080
Author(s):  
S. GRILLO ◽  
H. MONTANI
Keyword(s):  
Algebraic Geometry ◽  
Bethe Ansatz ◽  
Transfer Matrix ◽  
Lattice Models ◽  
Natural Generalization ◽  
Vertex Models ◽  
Algebraic Bethe Ansatz ◽  
Monodromy Matrices

Given a couple of Yang–Baxter operators 𝖱[k] and 𝖱[l] corresponding to integrable anisotropic vertex models of Ak-1 and Al-1 type, respectively, we construct and study a class of related lattice models whose monodromy matrices alternate between the mentioned operators. In order to do that, we use a natural generalization of the idea of coproduct in a bialgebra, that appears in the scenario of non-commutative algebraic geometry, related to the notion of internal homomorphisms of quantum spaces. We build up all eigenstates and eigenvalues of the transfer matrix by means of algebraic Bethe ansatz technics, where not only one vector, but a pseudovacuum subspace is needed for the process of diagonalization.

scholarly journals Interplays between Mesoscopic Josephson Junctions and the Harper Equation

2009 ◽  
Vol 116 (4) ◽  
pp. 486-488
Author(s):  
E. Papp ◽  
C. Micu ◽  
I. Bica
Keyword(s):  
Josephson Junctions ◽  
Harper Equation

scholarly journals Construction of the Fuchs equation with four given finite critical points and a given reducible monodromy group in the resonance case

2019 ◽  
Vol 55 (2) ◽  
pp. 199-206
Author(s):  
V. V. Amel’kin ◽  
M. N. Vasilevich
Keyword(s):  
Critical Points ◽  
Monodromy Group ◽  
Projective Line ◽  
Resonance Case ◽  
Nonsingular Transformation ◽  
Triangular Form ◽  
Rank 2 ◽  
Linear Differential ◽  
Complex Projective ◽  
Monodromy Matrices

One inverse problem of the analytic theory of linear differential equations is considered. Namely, the completely integrable Fuchs equation with four given finite critical points and a given reducible monodromy group of rank 2 on the complex projective line is constructed. Reducibility of the monodromy group of rank 2 means that 2×2-monodromy matrices (the generators of the monodromy group) can be simultaneously reduced by a linear nonsingular transformation to an upper triangular form. In so doing we study the case when the eigenvalue ξj of the diagonal matrix of the monodromy formal exponent at a corresponding Fuchs critical point is equal to an integer different from zero (resonance takes place).

Explicit Polynomial Solutions to the q- Difference Form of the Harper-Equation

1997 ◽  
Vol 11 (18) ◽  
pp. 773-778 ◽  
Author(s):  
C. Micu ◽  
E. Papp
Keyword(s):  
Recurrence Relations ◽  
Polynomial Solutions ◽  
Difference Form ◽  
Harper Equation

This paper deals with the derivation of concrete polynomial solutions to the q-difference form of the Harper equation proposed recently [P. B. Wiegmann and A. V. Zabrodin, Phys. Rev. Lett.72, 1890 (1994)], now by using the q-calculus in conjunction with quite directly solvable recurrence relations. Regularity patterns and the onset of level-structures with increasing complexity are discussed.

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