On the $C*$-comparison algebra of a class of singular Sturm-Liouville expressions on the real line

Abstract

In this article we study a $C^*$-comparison algebra in the sense of [C2] with generators related to the ordinary differential expression H on the full real line R where,with constants $𝛼≥ 0, 𝛽∈ R$, (1.1) $$H = -\partial_x (1+x²)^𝛽 \partial_x+(1+x²)^𝛼, \;x∈ R.$$ More precisely, the algebra, called $\mathbf A$, is generated by the multiplications $a( M) : u( x) → a(x)u(x)$, by functions $a(x)∈ C([-∈fty,+∈fty])$ and the (singular integral) operators $S₀ = ( 1 + x²)^𝛼/2Λ, iS₁ = (1+ x²)^𝛼/2 \partial_x Λ$,and their adjoints. Here $Λ = H^-1/2$, the inverse positive square root of the unique self-adjoint realization H of the expression (1.1), in the Hilbert space $\mathbf H = L² ( R )$ .(We use the same notation for both, (1.1) and its realization.) The case of $𝛽 < 𝛼+ 1 $ was discussed earlier in [Tg1], even for all n-dimensional problem. The commutators are compact and the Fredholm properties of operators in $\mathbf A$ are determined by a complex-valued symbol on a symbol space homeomorphic to that of the usual Laplace comparison algebra on $Rⁿ$, although the symbol itself is calculated by different formulas.