Continuous spectra of an even order differential operator
Don B Hinton · Illinois Journal of Mathematics · 1974
We consider here a differential operator of order 2n defined by (1) l(y)(1/w){(--1)(ry(")) () qy}.The coefficients w, r, and q are real continuous functions defined on a ray [a, and w and r are positive.Associated with is the Hilbert space H of all complex-valued, measurable functions f satisfying f.o wJ/I dx <We recall that determines a certain minimal closed operator L0 in H in the following way.Let g be the set of all y e H such that (i) y, y', ..., (ry(,)),, ", (ry()) (,-1) are absolutely continuous on compact subintervals of [a, o ) and (ii) l(y) e H. Define 0 as the set of all y e g which have compact support interior to (a, o ), and let L and L be the restrictions of to g and ), respectively.Then L is a densely defined Symmetric operator in H;hence admits a closure L0 with domain 0. As in [8, Section 17], it may be shown that L L.