Operadores biespectrales matriciales

LIBERATI, JOSE I.

Cordoba, Argentina

Congreso; LVII Reunión Anual de la Unión Matemática Argentina; 2007

We consider a matrix-valued version of the  bispectralproblem, that is,  find  differential  operators$L(x, rac{d}{dx})$ and $B(z, rac{d}{dz})$ with matrixcoefficients such that there exists a family of matrix-valued common eigenfunctions $psi(x,z)$: % $$ Lleft(x, rac{d}{dx} ight)psi(x,z)= f(z)psi(x,z),qquad psi(x,z)Bleft(z,  rac{d}{dz} ight) =Theta(x)psi(x,z), $$ %where $f$ and $Theta$ are matrix-valued functions. Usingquasideterminants, we prove that the operators $L$ obtained bynon-degenerated rational matrix Darboux transformations from$gleft( rac{d}{dx} ight)I$ ($g(y)inCC[y]$) are bispectraloperators. We also give an explicit formula for the operator $B$ extendingprevious results in the scalar case.