By Walter E Thirring
Mathematical Physics, Nat. Sciences, Physics, arithmetic
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Extra resources for A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems
24) Hf + φ|∇hτ ψ = d φ|ψ Hf Hf (τ ) and ∇hτ1 ∇hτ2 − ∇hτ2 ∇hτ1 − ∇h[τ1 ,τ2 ] ψ = −∇vR⊥ (τ1 ,τ2 )ν ψ, Rh (τ1 , τ2 )ψ := where R⊥ is the normal curvature mapping (deﬁned in the appendix). The proof of this result can be found at the beginning of Section 4. In order to deduce the formula for the eﬀective Hamiltonian we need that Hε can be expanded with respect to the normal directions when operating on functions that decay fast enough. g. chapter XVI, §4 of ). 7. Let m ∈ N0 . 25) v ∗ h 2 3 R ∇ φ , ν, ε∇ ψ, ν φ∗ ( 21 ∇vν,ν W + Vgeom )ψ dν dμ, where II is the second fundamental form, W is the Weingarten mapping, and R is the Riemann tensor (see the appendix for the deﬁnitions).
Therefore we expect Pε to have an expansion in ε starting with P0 : Pε = P0 + εP1 + ε2 P2 + O(ε3 ). We ﬁrst construct Pε in a formal way ignoring problems of boundedness. Afterwards we will show how to obtain a well-deﬁned projector and the associated unitary Uε . We make the ansatz P1 := T1∗ P0 + P0 T1 with T1 : H → H to be determined. Assuming that [P1 , −ε2 Δh + Ef ] = O(ε) we have [Hε , Pε ]/ε = = = = [H0 /ε + H1 , P0 + εP1 ] + O(ε) [H0 /ε + H1 , P0 ] + [H0 , P1 ] + O(ε) [−εΔh + H1 , P0 ] + [Hf − Ef , P1 ] + O(ε) [−εΔh + H1 , P0 ] + (Hf − Ef )T1∗ P0 − P0 T1 (Hf − Ef ) + O(ε) We have to choose T1 such that the ﬁrst term vanishes.
Although the curvature of the connection ipε always vanishes, it may have a non-trivial holonomy over the circle, which we will discuss next. e. with constant |η|. Let x be a 2π-periodic coordinate for it. The eigenfunction ϕf (x) can be chosen realvalued for each ﬁxed x because Hf is real. This associates a real line bundle to Ef . From the topological point of view, there are two real line bundles over the circle: the trivial one and the non-trivializable M¨obius band. In the former case the global section ϕf can be chosen real everywhere.
A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems by Walter E Thirring