We discuss the integrable boundary conditions for the one-dimensional (1D) Hubbard Model in the framework of the Quantum Inverse Scattering Method (QISM). We use the fermionic R-matrix proposed by Olmedilla et al. to treat the twisted periodic boundary condition and the open boundary condition. We determine the most general form of the integrable twisted periodic boundary condition by consideri...

Over the past decade, the intimate relationships between matrix integrals and nonlinear integrable systems have been clarified, particularly in the context of string theory. In such cases, nonperturbative properties of physical quantities can be evaluated by the use of the integrable structures of the models. (For review, see refs. 1-4.) Here, we consider a matrix integral over an ensemble of H...

We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT)-a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M family of integrable matrices consists of exactly N-M independent commuting N×N matrices linear in a real parameter. We first develop a rotationally invariant param...

For the Hamiltonian integrable systems governed by SL(2)-invariant r-matrix (such as Heisenberg magnet, Toda lattice, nonlinear Schrödinger equation) a general procedure for constructing Bäcklund transformation is proposed. The corresponding BT is shown to preserve the Poisson bracket. The proof is given by a direct calculation using the r-matrix expression for the Poisson bracket.

Glossary Dilation matrix A 2× 2 matrix M is called a dilation matrix if all the entries of M are integers and all the eigenvalues of M are greater than one in modulus. Isotropic dilation matrix A dilation matrix M is said to be isotropic if M is similar to a diagonal matrix and all its eigenvalues have the same modulus. Wavelet system A wavelet system is a collection of square integrable functi...

We show that the most general two–matrix model with bilinear coupling underlies c = 1 string theory. More precisely we prove that W 1+∞ constraints, a subset of the correlation functions and the integrable hierarchy characterizing such two– matrix model, correspond exactly to the W 1+∞ constraints, to the discrete tachyon correlation functions and to the integrable hierarchy of the c = 1 string...

In this paper, the relations between three special forms of matrix integrals and their associated integrable differential-difference systems are considered. It turns out that these matrix integrals with special β = 2 and 1, 4 satisfy the differential-difference KP equation, the two-dimensional Toda lattice, the semi-discrete Toda equation and their Pfaffianized systems, respectively.

We show that the one dimensional unitary matrix model with potential of the form aU + bU2 + h.c. is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space dimension in an external potential of the form a cos(x+α) + b cos(2x+ β) and interacting through two-body potentials of the inverse sine square type. This system const...

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian...