By Vincent Rivasseau (Chief Editor)

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Since Op(b)u ∈ S(Rd ) for any b ∈ Sα and u ∈ S(Rd ), the product Op(b)Op(g), b ∈ Sα ,g ∈ Sβ , is well defined on S(Rd ). A straightforward calculation leads to the following formula for the symbol b ◦ g of the product Op(b) Op(g): (b ◦ g)(x, ξ) = 1 d(Γ) ˆb(θ, ξ + φ)ˆ g (φ, ξ)ei(θ+φ)x , θ,φ and hence 1 (b ◦ g)(χ, ξ) = d(Γ) θ+φ=χ ˆb(θ, ξ + φ)ˆ g (φ, ξ), χ ∈ Γ† , ξ ∈ Rd . 1) Here and below θ, φ ∈ Γ† . 3 Let b ∈ Sα , g ∈ Sβ . Then b ◦ g ∈ Sα+β and b◦g (α+β) p,s ≤ Cl,s b (α) l,s g (β) l,s , for some l = l(p).

T. the lattice Λ ⊂ Rn . Let a(ω) = |Rω|2 , with a non-degenerate matrix R. Then the operator H = I ⊗ Op(a) + Op(b) ⊗ I is self-adjoint on H2 (Rd ). 6 Let the operator H be as above and let C ⊂ Rl , D ⊂ Rn be subsets of Rl and Rn . Then D(λ, H; D × C) = 1 (2π)l C D(λ − a(k), B; D)dk. Proof. t. the lattice Λ × (2πZ)l with the fundamental domain M = O × [0, 2π)l . The Floquet representative of the operator H is the operator H(K) = I ⊗ A(k) + B(µ) ⊗ I, K = (µ, k), k ∈ [0, 1)l , µ ∈ O† . Since the symbol a does not depend on x, after separating variables we have N (λ, a(m + k) + B(µ); D), ∀µ ∈ O† , N (λ, H(µ, k); D × C) = m∈Zl , m+k∈C and thus D(λ, H; D × C) = 1 (2π)l+n C O† N (λ, a(k) + B(µ); D)dµdk, which leads to the required formula.

6, 2005 Density of States 37 and a smooth real-valued periodic potential V . However, some general definitions are more natural to give for more general operators. For these purposes it is not even necessary to assume that H0 = −∆, but it would be sufficient to suppose that H0 = Op(h0 ) with h0 (ξ) = |Fξ|m , m > 0, where F is a non-degenerate d × dmatrix. Also, the perturbation is allowed to be an arbitrary PDO of order lower than H0 . Another reason of considering more general operators is methodological: a number of intermediate results requires the use of such PDO’s.

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