A theorem about two-parameter families of Schrödinger operators in proved; the potential is parameter dependent.

This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {(x, y) : 0 < y < εh(x)}, where h(x) is a positive periodic function. The main assumption is that h(x) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0.

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Let Omega(0) be a domain in the cube (0, 2pi)(n), and let chi(tau)(x) be a function that equals I inside Omega(0), equals tau in (0, 2pi)(n)\Omega(0), and that is extended periodically to R-n. It is known that, in the limit tau --> infinity, the spectrum of the operator -delchi(tau)(x)del exhibits the band-gap structure. We establish the asymptotic behavior of the density of states function in the bands.

Under an additional symmetry condition, we prove that the spectrum of a second order self-adjoint elliptic differential operator with periodic coefficients is purely absolutely continuous.

This paper achieves, among other things, the following: • It frees the main result of [BFKM] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [BZ] from finite dimensional representations of Γ to representations on an A−Hilbert module of finite type (A a finite von Neumann algebra). The result of [BZ] corresponds to A = I C. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion R, first introduced by Carey, Mathai and Mishchenko [CMM] in special cases. The main result of this paper calculates explicitly this relative torsion (cf Theorem 0.1).

We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$ into disjoint components. In particular we extend the $L-2$ analytic and Reidemeister torsions to compact manifolds with boundary. If the system $(M,\partial_-M, \partial_+M, \Cal F)$ is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove glueing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]

For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative torsion. The relative torsion is a positive real number and unlike the analytic torsion or the Reidemeister torsion, which are defined only when the pair manifold- representation is of determinant class, is always defined. When the pair is of determinant class the relative torsionis equal to the quotient of the analytic and the Reidemeister torsion.We calculate the relative torsion.

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.

Given a compact Riemannian manifold (Md, g), a finite dimensional representationρ:π1(M)→GL(V) of the fundamental groupπ1(M) on a vector spaceVof dimensionland a Hermitian structureμon the flat vector bundle[formula]associated toρ, Ray–Singer [RS] have introduced the analytic torsionT=T(M, ρ, g, μ)>0. Witten's deformationdq(t) of the exterior derivativedq,dq(t)=e−htdqeht, withh:M→Ra smooth Morse function, can be used to define a deformationT(h, t)>0 of the analytic torsionTwithT(h, 0)=T. The main results of this paper are to provide, assuming that gradghis Morse Smale, an asymptotic expansion for logT(h, t) fort→∞ of the form[formula]and to present two different formulae fora0. As an application we obtain a shorter derivation of results due to Ray–Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation T=(h, g) and the dual triangulation TD=(d−h, g).

In this paper we present a formula for the determinant of a matrix-valued elliptic differential operator of even order on a line segment [0, T] with boundary conditions.

For any two-dimensional Riemannian manifold (M, g) we introduce a new functional, hg, on the space of closed simple nonparametrized curves on M. This functional associates to any simple curve Γ the regularize determinant of the Laplace operator on the manifold obtained by cutting M along Γ and imposing Dirichlet boundary conditions. When M is of genus zero we derive a formula for the variation of hg, we prove that the critical points are conformal circles (i.e., the curves which, with respect to the unique metric of constant curvature 1 in the conformal class {e2αg:α ∈ C∞(S2, R)} of g, have constant geodesic curvature), and that the hessian of the functional at a critical point is nondegenerate in directions normal the critical submanifold (Theorem 1.1). We also construct smooth flows on the space of nonparametrized curves retracting the space onto the critical sub-manifold and show that they are gradient-like for our function. These flows deform a given closed embedded curve on S2 to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3).

We express the ζ-regularized determinant of an elliptic pseudodifferential operatorA overS1 with strongly invertible principal symbol in terms of the Fredholm determinant of an operator of determinant class, canonically associated toA, and local invariants. These invariants are given by explicit formulae involving the principal and subprincipal symbol of the operator. We remark that,generically, elliptic pseudodifferential operators have a strongly invertible principal symbol.

The multiplicity of the second eigenvalue of the Dirichlet Laplacian on smooth Riemannian surfaces with boundary that satisfy certain convexity condition is at most two. The proof is based on variational formulas for eigenvalues under the change of the domain.

For a closed codimension one submanifold Γ of a compact manifold M, let MΓ be the manifold with boundary obtained by cutting M along Γ. Let A be an elliptic differential operator on M and B and C be two complementary boundary conditions on Γ. If (A, B) is an elliptic boundary valued problem on MΓ, then one defines an elliptic pseudodifferential operator R of Neumann type on Γ and prove the following factorization formula for the ζ-regularized determinants: DetADet(A, B) = KDetR, with K a local quantity depending only on the jets of the symbols of A, B and C along Γ. The particular case when M has dimension 2, A is the Laplace-Beltrami operator, and B resp. C is the Dirichlet resp. Neumann boundary condition is considered.

For an elliptic differential operatorA overS1,\(A = \sum\limits_{k = 0}^n {A_k (x)D^k } \), withA k (x) in END(ℂr) and θ as a principal angle, the ζ-regularized determinant DetθA is computed in terms of the monodromy mapP A , associated toA and some invariant expressed in terms ofA n andA n−1 . A similar formula holds for finite difference operators. A number of applications and implications are given. In particular we present a formula for the signature ofA whenA is self adjoint and show that the determinant ofA is the limit of a sequence of computable expressions involving determinants of difference approximation ofA.