The spectrum of differential operators with almost constant coefficients II

摘要

The absolutely continuous spectrum of differential operators of the formLy=w−1∑k=0n(−1)k(pky(k))(k)onL2([0,∞),w)is determined. With pn(x),w(x)>0 the coefficients pk are assumed to satisfy p̃k(x)=(pkγ2kw−1)(x)→ck,γ=(w·pn−1)1/2n. If the coefficients satisfy some additional smoothness and decay conditions, the absolutely continuous part Hac of any self-adjoint extension of L is unitarily equivalent to the operator of multiplication by P(x)=∑0nckx2k on L2([0,∞)). Several extensions of this result as well as examples are shown.