States with v 1 = λ , v 2 = − λ and reciprocal equations in the six - vertex model

States with v 1 = λ, v 2 = −λ and reciprocal equations in the six-vertex model Abstract The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v 1 ,. .. , v n , the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v 1 , v 2 are equal to λ, −λ , with ∆ = − cosh λ and ∆ related to the Boltzmann weights of the six-vertex model by the usual combination ∆ = (a 2 + b 2 − c 2)/2ab. The eigenvectors associated to these eigenvalues are Bethe states (although they seem not). We count the number of such states (eigenvectors) for n = 2, 3, 4, 5 when N , the columns in a row of a square lattice , is arbitrary. The number obtained is independent of the value of ∆ , but depends on N. We give the explicit expression of the eigenvalues in terms of a, b, c (when possible) or in terms of the roots of a certain reciprocal polynomial , being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered. For real a, b, c such eigenvalues are real. 1 The problem Some time ago the author of this note read in the paper Completeness of the Bethe Ansatz for the Six and Eight-Vertex Models by R.J Baxter [1, Sect. 4] the following sentence concerning certain proper states of the transfer matrix in the six-vertex model at zero-field: