, Volume 34, Issue 1, pp 45-70

The C*-algebra of the N-dimensional harmonic oscillator

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Abstract

In this paper we study the Fredholm thoery of a C*-algebraOl of o-order pseudo-differential operators on L2(ℝn). IfK denotes the ideal of all compact operators of L2, the algebraOl will be generated by (i) the idealK, (ii) a function algebra CS(ℝn) and (iii) by the bounded operators xjΛ, DjΛ, j=1,...,n, Λ= H−1/2, H=1+¦x¦2−Δ. We show thatOl/K is a commutative C*-algebra with identity and obtain its Gelfany space M. This provides Fredholm criterion and index formula for a graded algebra of partial differential operators including all oeprators with polynomial coefficients. We also give Fredholm criterion and index formula for systems of such operators.