On the Spectrum of Periodic Elliptic Operators
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چکیده
It was observed in [Su5] that the spectrum of a periodic Schrodinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators. Let X be a Riemannian manifold of dimension n on which a discrete group F acts isometrically, effectively, and properly discontinuously. We assume that the quotient space F \ X (which may have singularities) is compact. Let E be a F-equivariant hermitian vector bundle over X, and D : C°°(E) • C°°(E) a formally self-adjoint elliptic operator which commutes with the F-action. For short, we call such a D a F-periodic operator. It is easy to show (see Section 1) that the symmetric operator D with the domain C^{E) is essentially self-adjoint, so that D has a unique self-adjoint extension in the Hilbert space L(E) of square integrable section of E, which we denote also by D by a slight abuse of notation. Let Cfed(F, # ) denote the tensor product of the reduced group C*-algebra of F with the algebra # of compact operators on a separable Hilbert space of infinite dimension, and by Xir the canonical trace on C*ed(F, # ) . We define the Kadison constant C (F) by C(F) =inf {trrP ; P is a non-zero projection in C*ed(F, # ) } . By definition, F is said to satisfy the Kadison property if C(F) > 0. It is a conjecture proposed by Kadison that, if F is torsion free, then C(F) — 1. A
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تاریخ انتشار 1992