Addition chains and solutions of l(2n) = l(n) and l(2n − 1) = n + l(n) − 1

AbstractAn addition chain for a positive integer n is a set 1 = a0 < al < … < ar = n of integers such that for each i ⩾ 1. a1 = a1 + a k for some k ⩽ j < i. The smallest length r for which an addition chain for n exists is denoted by l(n). This paper introduces the function h(x) which denotes the number of integers n less than or equal to x for which l(2n) = l(n) and proves that h(x) > (logx)2. A necessary theorem for establishing this result is that there exist infinitely many infinite classes of integers for which l(2n) = l(n). The proof of this theorem is outlined. Also, this paper establishes seven new cases for which l(2n − 1) = n + l(n) − 1. These are cases n=15, 16, 17, 18, 20, 24 and 32

From Toda to KdV

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty$ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm$. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$

Auxiliary matrices on both sides of the equator

The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter's TQ-equation corresponding to the Bethe ansatz equations above and below the equator. In contrast, one finds for even spin-chains only one linear independent solution and complete strings. The other main result is the proof of a previous conjecture on the degeneracies of the six-vertex model at roots of unity. The proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight-vertex model conjectured by Fabricius and McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some typos correcte

The integrable quantum group invariant A_{2n-1}^(2) and D_{n+1}^(2) open spin chains

A family of A_{2n}^(2) integrable open spin chains with U_q(C_n) symmetry was recently identified in arXiv:1702.01482. We identify here in a similar way a family of A_{2n-1}^(2) integrable open spin chains with U_q(D_n) symmetry, and two families of D_{n+1}^(2) integrable open spin chains with U_q(B_n) symmetry. We discuss the consequences of these symmetries for the degeneracies and multiplicities of the spectrum. We propose Bethe ansatz solutions for two of these models, whose completeness we check numerically for small values of n and chain length N. We find formulas for the Dynkin labels in terms of the numbers of Bethe roots of each type, which are useful for determining the corresponding degeneracies. In an appendix, we briefly consider D_{n+1}^(2) chains with other integrable boundary conditions, which do not have quantum group symmetry.Comment: 47 pages; v2: two references added and minor change

Exact Solutions and Degenerate Properties of Spin Chains with Reducible Hamiltonians

The Jordan--Wigner transformation plays an important role in spin models. However, the non-locality of the transformation implies that a periodic chain of $N$ spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the $N-1$ bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an anti-periodic chain of lattice fermions without redundancy when the Jordan--Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model ($\sigma_x\sigma_y-\sigma_x\sigma_y$) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.Comment: 9 pages, 3 figure

Analytical Bethe Ansatz for A2n−1(2),Bn(1),Cn(1),Dn(1)A^{(2)}_{2n-1}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n quantum-algebra-invariant open spin chains

We determine the eigenvalues of the transfer matrices for integrable open quantum spin chains which are associated with the affine Lie algebras $A^{(2)}_{2n-1}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n$, and which have the quantum-algebra invariance U_q(C_n), U_q(B_n), U_q(C_n), U_q(D_n)$, respectively.Comment: 14 pages, latex, no figures (a character causing latex problem is removed

On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf-Cole transformation

We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the Hopf-Cole transformation. More concretely, 1) we show that the integrability properties (Lax pair, infinitely-many commuting symmetries, large classes of analytic solutions) of (2+1)-dimensional PDEs integrable by the Inverse Scattering Transform method ($S$-integrable) can be generated by the integrability properties of the (1+1)-dimensional matrix B\"urgers hierarchy, integrable by the matrix Hopf-Cole transformation ($C$-integrable). 2) We show that the integrability properties i) of $S$-integrable PDEs in (1+1)-dimensions, ii) of the multidimensional generalizations of the GL(M,\CC) self-dual Yang Mills equations, and iii) of the multidimensional Calogero equations can be generated by the integrability properties of a recently introduced multidimensional matrix equation solvable by the method of characteristics. To establish the above links, we consider a block Frobenius matrix reduction of the relevant matrix fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius matrix, followed by a systematic elimination procedure of some of these blocks. The construction of large classes of solutions of the soliton equations from solutions of the matrix B\"urgers hierarchy turns out to be intimately related to the construction of solutions in Sato theory. 3) We finally show that suitable generalizations of the block Frobenius matrix reduction of the matrix B\"urgers hierarchy generates PDEs exhibiting integrability properties in common with both $S$- and $C$- integrable equations.Comment: 30 page