In this paper we show that a system of three fermions is exactly solvable for the case of a single-j in the presence of an angular momentum-J pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-j symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin J = J max = 2j − 1, the " non-integer " eigenvalues ...

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap pote...

In this paper we show that a system of three fermions is exactly solvable for the case of a single-j in the presence of an angular momentum-J pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-j symbols. It is also found that the " non-integer " eigenvalues of three fermions with angular momentum I around the maximum appear as " non-integer " eig...

I show that the potential V (x,m) = [b 4 −m(1−m)a(a+ 1) ] sn(x,m) dn(x,m) − b(a+ 1 2 ) cn(x,m) dn(x,m) constitutes an unusual QES band-structure problem in one dimension. In particular, I show that for any positive integral or half-integral a, 2a+ 1 band edge eigenvalues and eigenfunctions can be obtained analytically. However, the associated orthogonal polynomials satisfy fourterm (rather than...

It is known that the trigonometric Calogero-Sutherland model is obtained by the trigonometric limit (τ → √ −1∞) of the elliptic CalogeroMoser model, where (1, τ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero-Sutherland model, if exp(2π √ −1τ) is small enough then there exist square-integrable eig...

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

We study random walks on the integers mod Gn that are determined by an integer sequence {Gn}n?1 generated a linear recurrence relation. Fourier analysis provides explicit formulas to compute eigenvalues of transition matrices and we use this bound mixing time walks.

The definition of the standard derivative operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new fractional operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charmon...

Some new properties of the eigenvalues of the subdirect sums are presented for the particular case of 1-subdirect sums. In particular, it is shown that if an eigenvalue λ is associated with certain blocks of matrix A or matrix B then λ is also an eigenvalue associated with the 1-subdirect sum A ⊕1 B. Some results concerning eigenvectors of the k-subdirect sum A⊕k B for an arbitrary positive int...

The notion of the Laplacian of weighted graphs will be introduced, the eigenvectors belonging to k consecutive eigen-values of which define optimal k-dimensional Euclidean representation of the vertices. By means of these spectral techniques some combinatorial problems concerning minimal (k+ 1)-cuts of weighted graphs can be handled easily with linear algebraic tools. (Here k is an arbitrary in...