i) The unitary transform is defined by : H H, ψ
ii) The dilation operator is defined by (Dεψ)(q, ν) := ε−k/2 ψ(q, ν/ε).
iii) The dilated Hamiltonian and potential are defined by
∗Mρ ∗HεMρDε,
∗Mρ ∗V εMρDε
= Vc +
The index ε will consistently be placed down to denote dilated objects, while it will
placed up to denote objects in the original scale.
The leading order of will turn out to be the sum of −Δv + Vc(q, ·) + W (q, 0)
and −ε2Δh (for details on and the expansion of see Lemmas 3.1 & 3.7
below). When
acts on functions that are constant on each fibre, it is simply
the Laplace-Beltrami operator on C carrying an
Hereby the analogy with the
Born-Oppenheimer setting is revealed where the kinetic energy of the nuclei carries
the small parameter given by the ratio of the electron mass and the nucleon mass
(see e.g. [36,46]).
We need that the family of q-dependent operators −Δv + Vc(q, ·) + W (q, 0) has
a family of exponentially decaying bound states in order to construct a subspace of
states that are localized close to the constraint manifold. The following definition
makes this precise. We note that the conditions are simpler to verify than one might
have thought in the manifold setting, since the space and the operators involved
are euclidean.
Definition 1.3. Let Hf (q) :=
and V0(q, ν) := Vc(q, ν) + W (q, 0).
i) The selfadjoint operator (Hf (q),H2(NqC,dν)) defined by
(1.15) Hf (q) := −Δv + V0(q, .)
is called the fiber Hamiltonian. Its spectrum is denoted by σ
Hf (q)
ii) A function Ef : C R is called an energy band, if Ef (q) σ
Hf (q)
all q C. Ef is called simple, if Ef (q) is a simple eigenvalue for all q C.
iii) An energy band Ef : C R is called separated, if there are a constant
cgap 0 and two bounded continuous functions : C R defining an
interval I(q) := [f−(q),f+(q)] such that
(1.16) {Ef (q)} = I(q) σ(Hf (q)) , inf
Hf (q)
\{Ef (q)}, {Ef (q)}
= cgap.
iv) Set ν := 1 + |ν|2 = 1 + g(q,0)(ν, ν). A separated energy band Ef is
called a constraint energy band, if there is Λ0 0 such that the family of
spectral projections {P0(q)}q∈C with P0(q) L
Hf (q)
and Hf (q)P0(q) =
Ef (q)P0(q) satisfies
eΛ0 ν P0(q)eΛ0 ν
L(Hf (q))
Remark 1.4. Condition iii) is known to imply condition iv) in lots of cases
(see [21] for a review of known results), in particular for eigenvalues below the
continuous spectrum, which is the most important case in the applications. Besides,
condition iii) is a uniform but local condition (see Figure 3).
The family of spectral projections {P0(q)}q∈C associated with a simple energy
band Ef defines a smooth line bundle EP0 := {(q, ϕ) | q C, ϕ P0(q)Hf (q)}
over C. Smoothness of EP0 follows e.g. from Lemma 4.10. If this bundle has
a normalized section ϕf Γ(EP0 ), it holds for all q C that (P0ψ)(q, ν) =
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