The Spectrum of Differential Operators with Almost Constant Coefficients

Horst Behncke University of Osnabruck

The absolutely continuous spectrum of differential operators of order 2n of the form

Ly = w^{−1} ∑_{k=0}^{n}(−1)^{n}(p_{k}y^{(k)})^{(k)}

on L^{2}((0, ∞), w) is determined. We assume w, p_{n} > 0 and that the real valued coefficients admit a decomposition

p_{k} = p_{k1} + p_{k2} + p_{k3} + p_{k4}  k = 0, …, n
w = w_{1} + w_{2} p_{n3} = p_{n4} = 0

with

[p_{k3}, q_{k}, p′_{k,2}, p′′_{k1}γ^{−1}, p_{k,1}^{′2}w^{−1}γ^{2k−1}], γ^{2k}w^{−1}w′_{2}w^{−1}, w′′_{1}w^{−1}, w_{1}^{′2}w^{−1}γ^{−1} ∈ L_{1}

q_{k}γ^{2k}w^{−1}, p′_{k,1}γ^{2k−1}w^{−1}, w′_{1}w^{−1}γ^{−1} → 0 x → ∞

where q_{k} = ∫_{∞}^{t} p_{k4} and γ = (wp_{n}^{−1})^{1/2n}. Moreover assume (p_{k,1}+p_{k,2})γ^{2k}w^{−1} → c_{k}.

Then the absolutely continuous part of the spectrum of any selfadjoint extension of L is unitarily equivalent to that of the operator multiplication with ∑_{k=0}^{n} c_{k}x^{2k} on L^{2}(0, ∞).

This theorem extends all known results for such operators. The proof is based on a variant of the Liouville-Kummer transformation and asymptotic integration.

Various extensions of this result are also considered.