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  Volume 4, Issue 3, Article 56
 
Separation and Disconjugacy

    Authors: Richard C. Brown,  
    Keywords: Separation, Symmetric second order differential operator, Disconjugacy, Limit-point.  
    Date Received: 21/11/02  
    Date Accepted: 25/03/03  
    Subject Codes:

26D10, 34C10,34L99, 47E05.

  
    Editors: A. M. Fink,  
 
    Abstract:

We show that certain properties of positive solutions of disconjugate second order differential expressions $ M[y]=-(py^{prime})^{prime}+qy$ imply the separation of the minimal and maximal operators determined by $ M$ in $ % L^2(I_a)$ where $ I_a=[a,infty)$, $ a>-infty$, i.e., the property that $ % M[y]in L^2(I_a)Rightarrow qyin L^2(I_a)$. This result will allow the development of several new sufficient conditions for separation and various inequalities associated with separation. Some of these allow for rapidly oscillating $ q$. It is shown in particular that expressions $ M$ with $ WKB$ solutions are separated, a property leading to a new proof and generalization of a 1971 separation criterion due to Everitt and Giertz. A final result shows that the disconjugacy of $ M-lambda q^2$ for some $ % lambda>0$ implies the separation of $ M$.

         
       
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