By Ramis E., Deschamps C., Odoux J.

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Then α(l + 1, d + t0 + 1) = α(l + k, d + t0 ) + 1 (t0 − 1)k + k + 1 = t0 k + 1 hold. Next α(l + 2, d + t0 + 1) = max(α(l + 1, d + t0 + 1), α(l + k + 1, d + t0 )) + 1 t0 k + 2. By use of the estimates α(N + l, d + t0 + 1) max(α(N − 1 + l, d + t0 + 1), α(N − 1 + l + k, d + t0 )) + 1, one can obtain the bounds α(N + l, d + t0 + 1) t0 k + N. This completes the proof. 1 Formal Taylor Expansion and ODE Let us consider a Cα+1 function u : (0, ∞) → (0, ∞). Below we proceed to approximate u very roughly by discrete dynamics defined by relative elementary functions of n variables.

1, we see that this operator is unitarily equivalent to the Schrödinger operator H = 1,a + W, with an−1,n ∼ nγ +2β and the potential Wn ∼ −β(β + γ − 1)n2β+γ −2 , which is therefore also not essentially self-adjoint. 2. 1, such an operator must verify the 1 condition an−1,n ∈ l 1 (N), which is indeed the case. Following the terminology of the previous sections, it means the non completeness of (N, d p ) with the −1/2 weights pn−1,n = an−1,n . 4 Example 4 Let us consider the Laplacian H = ω,c on a spherically homogeneous rooted tree G = (V, E) (see [3] and references within).

The only non trivial point is to prove that the arrow p : E → C N is surjective. Let us consider P˜ a self-adjoint extension of P which exists because n+ = n− . Let us consider the map ρ : C N → E defined by ρ(x) = ( P˜ − i)−1 (x, 0, 0, · · · ). Then p ◦ ρ = IdC N . e. ((H − i)u)n = 0 for n large enough) which is not in l 2 (N). 3 Let us consider the following linear dynamical system on Cd : ∀n 0, U n+1 = AU n + R(n)U n where 1. 2. A is hyperbolic: all eigenvalues λ j of A satisfy |λ j| = 1 R(n) → 0 as n → ∞.