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