Display Abstract

Title On Criticality of higher order difference operators

Name Petr Hasil
Country Czech Rep
Email hasil@math.muni.cz
Submit Time 2010-02-21 11:58:06
Special Session 48: Differential, Difference, and Dynamic Equations
This is a joint work with Prof. Ond\v{r}ej Do\v{s}l\'y. We study the so-called $p$-critical difference operators, i.e., the non-negative operators that can be turned to negative ones by small (in a certain sense) negative perturbations. This research is motivated by [2], where the concept of critical operators is introduced for difference operators of the second order. We have generalized this concept in [1] for $2n$-order Sturm-Liouville difference operators, and we have suggested a criterium of $p$-criticality for one term $2n$-order difference operators $l(y)_k = \Delta^n (r_k \Delta^n y_k)$. This criterium turned out to be valid and we have proved it using a structure of the solution space of the equation $l(y)_k = 0$, see [3]. [1] O.~Do\v{s}l\'{y}, P.~Hasil, \emph{Critical higher order Sturm-Liouville difference operators}, J. Differ. Equ. Appl., to appear. [2] F.~Gesztesy, Z.~Zhao, \emph{Critical and subcritical Jacobi operators defined as Friedrichs extensions}, J. Differential Equations \textbf{103} (1993), 68--93. [3] P.~Hasil, \emph{Criterium of p-criticality for one term 2n-order difference operators}, submitted.