Matrix-valued bispectral operators and quasideterminats

C. BOYALLIAN Y J. LIBERATI

JOURNAL OF PHYSICS. A - MATHEMATICAL AND GENERAL

iop science

Año: 2008 vol. 41 p. 1 - 11

0305-4470

Abstract: We consider a matrix-valued version of the bispectral problem, that is, find differential operators L(x, d/dx) and B(z, d/dz) with matrix coefficients such thatthere exists a family of matrix-valued common eigenfunctions ψ(x, z): L(x,d/dx)ψ(x, z) = f (z)ψ(x, z) and ψ(x, z)B(z,d/dz)= T(x)ψ(x, z), where f and T are matrix-valued functions. Using quasideterminants, we prove that the operators L obtained by non-degenerated rational matrix Darboux transformations from g(d/dx)D are bispectral operators, where g(y) ∈ C[y] and D is a diagonal matrix. We also give a procedure to find an explicit formula for the operator B extending previous results in the scalar case.