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Quantum waveguides with magnetic fields

We study generalised quantum waveguides in the presence of moderate and strong external magnetic fields. Applying recent results on the adiabatic limit of the connection Laplacian we show how to construct and compute effective Hamiltonians that allow, in particular, for a detailed spectral analysis of magnetic waveguide Hamiltonians. Using our general construction, we discuss a number of explicit examples and thereby recover known results as special cases.

In this work we study generalised quantum waveguides as introduced in [HLT15] in the presence of moderate and strong external magnetic fields. In a nutshell, a generalised quantum waveguide is an ε-thin neighbourhood of a submanifold, or a boundary thereof. The corresponding Hamiltonian is the magnetic Laplacian in the interior with Dirichlet boundary conditions, or the Laplacian on the surface. The strength of the magnetic field is either held fixed or of the same order as the constraining forces, which increase in the asymptotic limit ε 1. Our results concern the construction of effective Hamiltonians, defined only on the submanifold, that provide an approximation of the spectrum of the full waveguide Hamiltonian for small ε. Our construction is based on recent results on the adiabatic limit of the connection Laplacian [HL17] that hold for very general geometries. Here we analyse the effective Hamiltonians in detail for magnetic quantum waveguides and dicuss the relevant terms for different energy scales, geometries, and strengths of the magnetic field.
More specifically, for us a quantum waveguide is an ε-thin neighbourhood T ε of a smoothly embedded b-dimensional submanifold B c − → R b+f of codimension f . The size and shape of the cross-section, i.e. the intersection of T ε with a plane normal to the submanifold, may vary along the submanifold. Common examples are quantum strips (b = 1 and f = 1), quantum tubes (b = 1 and f = 2) and quantum layers (b = 2 and f = 1). Additionally, we consider the case of so called hollow waveguides, which are modelled on the boundary of a massive waveguide as just described. In the absence of any magnetic field, the properties of freely moving quantum particles are modelled by the Laplacian on L 2 (T ε ) with Dirichlet boundary conditions for massive waveguides, or by the Laplace-Beltrami operator of the induced metric on the boundary of T ε for hollow waveguides. Since localised and normalised elements of these domains (apart from the constant transversal ground state for hollow waveguides) necessarily have derivatives of order ε −1 in the transversal directions, we will introduce the pre-factor ε 2 in front of the Laplacian in order to rescale the energies properly. The effects of an external magnetic field -represented by a magnetic potential A ∈ C ∞ (T * R b+f ) -are incorporated by minimal coupling, replacing the flat connection d by the magnetic connection ∇ ε −σ A := d + iε −σ A. The parameter σ ∈ {0, 1} controls the coupling strength. We refer to σ = 0 as moderate magnetic fields and to σ = 1 as strong magnetic fields. Thus, on the macroscopic scale where the waveguide is of width ε, a moderate magnetic field B is of order one, while a strong magnetic field B is of order ε −1 .
There exists a vast literature on specific types of quantum waveguides without magnetic fields, see e.g. [EK15,HLT15] and references therein. There are significantly less results on the magnetic case. For example, Ekholm and Kovařík [EK05] consider quantum strips (B = R, f = b = 1) and Grushin [Gru08] special tubes (B = S 1 , f = 2), in both cases with magnetic fields of moderate strength. The more recent work [KR14] of Krejčiřík and Raymond deals with quantum strips and quantum tubes (B = R, f = 1, 2) of constant cross-section and moderate as well as strong magnetic fields. Finally, in [KRT15] the same authors with Tušek treat ε-tubular neighborhoods of hypersurfaces in R b+1 in the presence of moderate magnetic fields.
In our approach we are able to treat all these geometries, and actually much more general ones, in a unified framework. For the case of moderate fields, our main result, Theorem 4.4, states that for any generalised waveguide the effective Hamiltonian can be obtained by minimal coupling of the effective non-magnetic Hamiltonian from [HLT15] to an effective vector potential. For strong magnetic fields the more complicated effective Hamiltonian is given in Theorem 5.1. Note that in both cases the quality of the approximation depends on the relevant energy scale of the problem, see Corollary 3.2 and Theorem 3.3. In Section 6 we explicitly compute and discuss the effective operators for different types of massive and hollow quantum tubes (b = 1, f = 2) and for moderate and strong magnetic fields. In particular, we recover as a special case the effective Hamiltonian of [KR14], cf. Eq. (32). The general effective Hamiltonians of Sections 4 and 5 can be easily evaluated also for all other special cases considered previously in the literature.
As a completely new example, treated in detail at the end of Section 6.5, let us mention the case of a hollow waveguide T ε modelled on the surface of a cylindrical tube of varying radius ε (x) along a smooth curve c : R → R 3 in the presence of a strong magnetic field ε −1 B. The resulting effective Hamiltonian is at leading order where B and B ⊥ are the projections of B| c onto the tangential resp. normal directions of the curve. The spectrum of H eff is ε-close to the spectrum of the Laplace-Beltrami operator −∆ ε −1 A T ε on the surface T ε ⊂ R 3 in the sense that for all E ∈ R (see Theorem 3.3 and mind the rescaling of energies by a factor ε 2 ).
Let us give a slightly more detailed plan of the paper. In Chapter 2, we identify the family {T ε } 0<ε≤1 (the "tube") with an ε-independent manifold M (the "waveguide") that has the additional structure of a fibre bundle M π M − − → B, with typical fibre ("typical cross section") F given by a compact subset of R f . The corresponding diffeomorphism Ψ ε : M → T ε was constructed in [HLT15, Chapter 2] and lifts to a unitary operator , with an induced Riemannian metric G ε on M and an induced magnetic potential A ε ∈ C ∞ (T * M ). The precise form of the two last-mentioned objects will be discussed in Section 2.1 and Section 2.2, respectively. In Section 2.3 we finally show that, after changing the volume measure, −ε 2 ∆ ε −σ A is unitarily equivalent to the operator on an ε-independent Hilbert space of functions on M . It is the sum of a horizontal operator −ε 2 L(G ε hor , ε −σ A ε hor ) that acts "tangentially" to B, a vertical operator H V,σ (depending on the vertical contribution A ε V of the induced magnetic potential) that acts transversally to B, and the so-called bending potential V bend . The operator H σ is of the general form to which the results of [HL17] can be applied. The statements resulting from [HL17] concerning the effective Hamiltonian, its approximation properties, and its relation to the so called adiabatic Hamiltonian are stated and discussed in Chapter 3.
In Chapter 4 and Chapter 5, we compute the relevant terms in an asymptotic expansion of the adiabatic Hamiltonian for moderate and strong magnetic fields, respectively. The form of these expansions and the approximation errors depend, among other things, on the energy scale under consideration. We consider energies at distance of order ε α above the infimum Λ 0 of the spectrum of the non-magnetic vertical operator for α ∈ [0, 2]. So α = 0 corresponds to energies of order one in units of the level spacing of transversal eigenvalues and α = 2 to energies ε 2 -close to the infimum of the spectrum. Roughly speaking, the spectrum of the adiabatic Hamiltonian turns out to be ε 2+α -close to the spectrum of the magnetic waveguide Laplacian. Therefore, in the asymptotic expansion of the adiabatic Hamiltonian terms of order ε 2+α and smaller can be discarded. Whether the interval [Λ 0 , Λ 0 + Cε α ] for α > 0 contains any spectrum of the magnetic waveguide Hamiltonian at all depends on the geometry of the waveguide.
The main result of Chapter 4, Theorem 4.4, states that for moderate magnetic fields the adiabatic Hamiltonian can be obtained by minimal coupling of the non-magnetic adiabatic Hamiltonian as computed in [HLT15] to an effective magnetic potential, for all values of α ∈ [0, 2]. For strong magnetic fields the adiabatic Hamiltonian is not merely altered by minimal coupling to an effective magnetic potential, but the effects of the field are more subtle, see Theorem 5.1 and Corollary 5.2.
In Section 6 we finally consider the special case of quantum tubes, i.e., the case b = 1 and f = 2, where the expansions of the adiabatic Hamiltonian become even more explicit.

Generalised Quantum Waveguides
Let c : B → R b+f be the smooth embedding of a smooth, complete bdimensional manifold B into the Euclidean space R b+f with the standard metric δ. We have an orthogonal decomposition into vectors tangent and normal to c(B), that is (1) The Riemannian metric δ and its Levi-Civita connection ∇ δ give rise to • a Riemannian metric g B = c * δ on B and its associated Levi-Civita connection ∇ g B , • a bundle metric δ NB = δ| NB and a metric connection ∇ NB on NB, where the metric is obtained by restriction of the Euclidean metric and the connection is the projection of ∇ δ to the subbundle NB.
Suppose that there exists a tubular neighbourhood T r ⊂ R b+f of c(B) with globally fixed radius, i.e., there is r > 0 such that normals to c(B) of length r do not intersect. This is equivalent to the requirement that the map restricted to NB r := ν ∈ NB such that ν δ NB < r be a diffeomorphism onto T r . If this holds, the entire analysis can be carried out on (a subset of) the normal bundle. This suggests that we view the initial tube as an ε-independent, fibrewise embedded submanifold : M → NB r , which is then mapped into R b+f by means of a rescaled diffeomorphism Φ ε : ν → c(π NB (ν)) + εν, 0 < ε ≤ 1.
Definition 2.1. Let B c − → R b+f be a smooth, complete, embedded b-dimensional submanifold with tubular neighbourhood T r ⊂ R b+f of fixed radius r > 0, and F be a compact manifold of dimension dim(F ) ≤ f with smooth boundary. Let M − → NB r be a connected submanifold that has the structure of a fibre bundle π M : M → B with typical fibre F such that the diagram commutes. We then call M a generalised quantum waveguide. f ≥ 2 and the waveguide M is the (fibrewise) boundary of a massive waveguide. We impose the following uniformity properties for the geometry the generalised quantum waveguide and the magnetic potential: Definition 2.2. Let M be a generalised quantum waveguide as in Definition 2.1 and A ∈ C ∞ (T * R b+f ) be a magnetic potential. These conditions on M are trivially satisfied for compact manifolds and in many concrete examples, such as "asymptotically straight" or periodic waveguides.
We will now discuss the geometry of such quantum waveguides in some more detail, in order to get an explicit expression for the operator H σ on L 2 (M, vol g ), obtained from the Laplacian on T ε with magnetic potential ε −σ A via Ψ ε .
2.1. The Pullback of the Riemannian Metric. Let VM := ker(Tπ M ) be the vertical subbundle of TM . Its elements are vectors that are tangent to the fibres of M . For a massive waveguide, these vectors are tangent to the fibres of NB, so they point along the normals to B. They are thus naturally identified with NB by "forgetting the base point", mapping T ν (N x B) → N x B in the obvious way. Thus, in the massive case, VM ∼ = NB. This identification is formalised by introducing the map K NB : T(NB) → NB, which projects to vertical vectors, orthogonally with respect to δ NB , and then uses the identification above.
The orthogonal complement of the vertical subbundle VM (with respect to G = G ε=1 ) is called the horizontal subbundle HM ∼ = π * M TB. It turns out that the choice of horizontal bundle does not depend on ε, that is, the decomposition TM = HM ⊕ VM, is G ε -orthogonal for every ε > 0 (see [HLT15,Equation (4.5)] for the massive case and [HLT15, Lemma 5.2] for the hollow case). The pullback metric G ε can then be written as the sum of a horizontal and a vertical part. More precisely, it is of the form Here, g V := G ε | VM is the ε-independent restriction of G ε to the vertical subbundle. The restriction g Mx := g V | Mx defines a Riemannian metric on each fibre M x of M . Moreover, h ε ∈ C ∞ (T * M ⊗2 ) is a symmetric (but not necessarily non-degenerate) tensor that satisfies h ε (V, ·) = 0 for all V ∈ C ∞ (VM ). Thus, if we define for any vector field X ∈ C ∞ (TB) its horizontal lift X H ∈ C ∞ (HM ) as the unique horizontal vector field satisfying Tπ M (X H ) = X, we have for all X, Y ∈ C ∞ (TB) and V, W ∈ C ∞ (VM ). The explicit form (3) shows that the pullback metric G ε may be considered as a small perturbation of the metric We will refer to this metric as the (rescaled) submersion metric, as it makes Tπ M an isometry from HM to TB (up to a factor ε −1 ), so g := g ε=1 is a Riemannian submersion. We finally mention that the horizontal deviation h ε can be computed explicitly from the quantities associated with the embeddings B → R b+f and M → NB. We have appears only in the hollow case for all X, Y ∈ C ∞ (TB), where • W ∈ C ∞ (N * B ⊗End(TB)) is the Weingarten map of the embedding B → R b+f and II ∈ C ∞ (N * B ⊗ T * B ⊗2 ) is the associated second fundamental form II(·)(X, Y ) := g B (W(·)X, Y ), • and ‫(ג‬X) is a vertical vector that measures the difference between the horizontal lifts with respect to the hollow waveguide and its underlying massive waveguide (cf. [HLT15, Section 5.1] and Eq. (6) below).
2.2. The Pullback of the Magnetic Potential. In this section we give explicit expressions for the pulled-back magnetic potential Ψ * ε A, in a specific gauge. The role of this gauge is to make the effect of the ε-scaling of the fibres on the magnetic potential more explicit. The idea is the following: On T ε perform a Taylor expansion of A with respect to the normal directions By a gauge transformation, we can make the one-form A c(B) vanish on normal vectors, turning it into a one-form on the tangent space to B. When A is written in this way, one sees easily that the leading term is intrinsic to B. It can thus naturally be integrated into the effective operator and also commutes with any differential operators acting in the normal directions. Lemma 2.3. Let r > 0 and A ∈ C ∞ (T * T r ) be a one-form with globally bounded derivatives on T r . Define the function and let A B := c * A. Then H vanishes on the vertical vectors VM , A ε V vanishes on HM and both have globally bounded derivatives with respect to the metric g = g ε=1 , uniformly in ε.
By the same reasoning, TΦ ε (v) = εK NB v, so we have This is clearly of order ε 2 , and * is a partial isometry. To calculate the horizontal component, take w ∈ T x B and denote by w # its horizontal lift to T ν N B for ν ∈ N x B. Note that w # = T (w H ) for massive waveguides, but not in general. We have TΦ ε (w # ) = w − εW(ν)w, see e.g. [HLT15, Section 4.1]. This gives The first term is A B , while the remaining ones are clearly of order ε. As dΩ ε = εdΩ 1 this proves the claim.
The explicit expressions for A ε H and A ε V are easily obtained from this proof. To leading order we have, for a vertical field V ∈ C ∞ (VM ), Here, we have used that the vertical vector has length of order one, see [HLT15, Section 5.1].

The Induced
Operator. The operator we want to study is the Laplacian with magnetic potential ε −σ A of the waveguide. For massive waveguides this means the Laplacian on the tube T ε , with Dirichlet boundary boundary conditions. For hollow waveguides this is the Laplace-Beltrami operator on the submanifold Ψ ε (M ) ⊂ R b+f , with the induced metric. In both cases, the operator is, up to a global factor ε 2 , equivalent to the Laplacian on (5)), that is, the operator whose quadratic form is given by . It is convenient to work on the ε-independent Hilbert space H := L 2 (M, vol g ), where g = g ε=1 is the unscaled submersion metric. This will also make many effects of the scaling and the geometry appear more explicitly in the operator. The correspondence is given by the unitary map with , is the relative density of the two measures. The factor ε −b is due to the scaling of the total volume and does not play any role in the operator.
The transformed operator can be split into an horizontal differential operator, a vertical operator (with Dirichlet conditions), and an ε-dependent potential.
The factor ε 2 is due to the fact that, on the co-vectors T * M , G ε = ε 2 G ε hor + g V . The horizontal and vertical operators are defined by substituting their arguments in the quadratic form The restriction of the vertical operator to the fibre of M x over x acts as (minus) the Laplacian associated to the metric g V | Mx = g Mx and the magnetic potential ε 2−σ A ε V , that is: note our sign convention. The geometric potential V ρε arises from the change of the volume measure and captures effects of the extrinsic curvature of the embedding B → R b+f . It takes the form , which is the same as for A = 0, see [HLT15, Equation (2.5)]. The operators in the second line are defined by inserting their arguments in (9). This potential is of order ε 2 for massive waveguides and of order ε for hollow waveguides.
The operator H σ is self adjoint on the Dirichlet domain Remark 2.4. Due to the fact that the magnetic potential in the vertical operator is of order ε 2−σ , we will be able to treat this perturbatively, even for σ = 1. We have and the error will be uniform in x under our hypothesis. Physically speaking, this effect occurs because the shrinking of the initial tube T ε in the transversal directions leaves the magnitude of the magnetic field unchanged, while the influence of the vertical differential operator increases. The perturbation by A ε V will play a role in the effective operator, if we aim for high precision when σ = 0, and in general when σ = 1.
Remark 2.5. We will exclusively work with L 2 -Sobolev spaces, so we will drop the corresponding superscript from now on and write W k,2 = W k . When these spaces are associated with the Riemannian manifolds (M, g), (B, g B ) and (M x , g Mx ), for x ∈ B, we will omit the dependence on the metric in the notation. In our setting, these manifolds (with boundary) are of bounded geometry in the sense of [Sch96], and the Sobolev norms are defined by summing local norms [Sch96, Defintion 3.23]: where {U ν } ν∈Z is an atlas of normal coordinate charts κ ν : U ν → R n , adapted to the metric, with subordinate partition of unity χ ν .

Adiabatic perturbation theory
In this section we discuss the approximation of the spectrum of H σ by that of the adiabatic operator H σ a , which is an operator on L 2 (B). Our results rely on the general construction of (super-)adiabatic approximations in [HL17], and we will refer to that paper for most of the proofs. Before going into the technical details, let us briefly sketch the basic ideas of our approach. We will consider a fibrewise vertical operator be the ground-state energy of its restriction to L 2 (M x ). Let P A (x) be the projection to the associated eigenspace. Since, under our assumptions, the ground-state is a simple eigenvalue, the ranges of P A (x) define a line bundle over B. This line bundle is in fact trivial, because the range of P A (x) is close to the groud-state eigenspace of L(g V , 0)| Mx . The projection P A defines an operator on H by setting, for The adiabatic operator is then given by By the triviality of the line bundle associated with P A , we may consider this as an operator on L 2 (B). Consider the difference If we can show that this is small, the adiabatic operator provides an approximation of H σ on the range of P A . For σ = 0 one sees rather easily that as iεA ε hor is of order ε by itself and [d, P A ] can be bounded uniformly in ε, as the eigenfunctions of H V A (x) vary smoothly with x. To obtain the same result for σ = 1 it is important to recall that ε −1 A ε hor = ε −1 π * M A B +A ε H , and then note that π * M A B commutes with the projection P A , because it does not depend on the fibre-coordinates. This implies (11), since L (G ε hor , ε −σ A ε hor ) is a quadratic expression in d + iε −σ A ε hor . So in both cases (11) is of order ε, as an operator from D(H σ ) to H (it is not of order ε 2 , as the graph-norm of H σ is also ε-dependent). Consequently, the adiabatic operator provides an approximation of H σ , with errors of order ε.
However, since the magnetic and geometric terms in the operator H σ are of order ε 1−σ and ε 2 , respectively, it is desirable to construct a slightly tilted super-adiabatic projection P ε , in a suitable sense, and an intertwining unitary operator U σ ε on H with P ε A U ε = U ε P A , such that the effective operator provides a better approximation of H σ than H σ a does. The details of the construction of P ε A can be found in [HL17].
More specifically, we gather parts of the geometric potential with the vertical Laplace operator and rewrite the Schrödinger operator (8) as and accordingly for the remainder -the so-called "bending potential". We consider H V A as a fibrewise perturbation of the non-magnetic operator ) solely consists of discrete eigenvalues of finite multiplicity accumulating at infinity. Let λ 0 be the ground-state energy of H V 0 with a spectral projection P 0 . That is, for x ∈ B, (cf. [Haa16, Section 5.2]): • massive quantum waveguides: The smallest eigenvalue λ 0 > 0 of L(g V , 0) with Dirichlet boundary conditions on M x . The associated eigenfunction φ 0 (x) can be chosen real and positive, and P 0 (x) projects to the space spanned by this function. • hollow quantum waveguides: It holds that λ 0 ≡ 0 with (ε-dependent) ground state The latter is the reason why, in the hollow case, we added additional terms to H V A . These terms are of order ε, but they have a trivial effect on λ 0 , which would be somewhat obscured if we treat them as a part of the bending potential.
To first order in perturbation theory, λ A − λ 0 is given by This equals zero because φ 0 is real-valued. Thus, the asymptotic expansion of the magnetic ground state eigenband reads hol.
with x-uniform errors due to Definition 2.2.
The bounded geometry of the waveguide M implies that λ 0 (x) is separated from the rest of σ(H V 0 (x)) by an x-uniform gap [LT17, Proposition 4.1]. Thus λ A (x) also has a gap, of size δ > 0 uniformly in x and ε, for ε small enough (see [Haa16,Lemma 5.13]). Moreover This gives a natural choice for the isomorphism of the range of P A with In general, H σ a and H σ eff approximate H σ only on the range of P A . If we restrict ourselves to energies below this restriction is not necessary. This is due to the fact that, for a function ψ ∈ L 2 (M ) in the space 1 (−∞,Λ] (H σ )H, the vertical mode ψ| Mx has to concentrate in the ground state of H V A . These considerations lead to the following theorem: Theorem 3.1. Let (M, A) be a magnetic quantum waveguide of bounded geometry and set Λ : Then for all N ∈ N there exist a projection P ε A and a unitary operator U σ ε ∈ L(H) ∩ L(D(H σ )), intertwining P A and P ε A , such that for every cut-off function In particular, the Hausdorff distance between the spectra of H σ and H σ eff is small, i.e., it holds for every δ > 0 that Proof. For σ = 0 this is an immediate consequence of [HL17, Corollary 2.3]. More presicely (in the notation of the mentioned work), one takes the trivial line bundle E = M × C endowed with the connection ∇ M ×C = d + iπ * M A B and treats the corrections of the metric and the connection as a perturbation.
For σ = 1 we use the connection ∇ M ×C = d + iA ε=0 H and treat the higher order corrections again as a perturbation. The contribution of the remaining strong part ε −1 π * M A B of the magnetic potential may be added to the horizontal Laplacian, since it does not depend on the fibre coordinates (see [Haa16,Lemma 5.19] for the details).
For N = 1 we can choose P ε A = P A , so at first sight the approximation of H σ by H σ a (which is much simpler compared to H σ eff since it does neither incorporate the complicated super-adiabatic projection P ε A nor the unitary U σ ε ) yields errors of order ε. More careful inspection shows that for N > 1 we have H σ a − H σ eff = O(ε 2 ) as an operator from W 2 (B) to L 2 (B), so the statement on the spectrum holds for H σ a with an error of order ε 2 . Corollary 3.2. Let (M, A) be a magnetic quantum waveguide of bounded geometry and set Λ := inf x∈B min(σ(H V 0 (x))\{λ 0 (x)}). Then there exists a a unitary operator U σ ε ∈ L(H) ∩ L(D(H σ )) such that for every cut-off function and, in particular, it holds for every δ > 0 that . We may further improve the adiabatic approximation if we focus on the bottom on the low-lying part of the spectrum "near" inf σ(H σ ). The behaviour in this energy regime clearly corresponds to that of the low-lying part of the adiabatic operator H σ a and therefore is dominated by the characteristics of the unperturbed ground state band λ 0 = min σ(H V 0 ): on B, the results obtained in [Sim83] suggest that the leading part of the adiabatic operator behaves like an harmonic oscillator close to x 0 , schematically with eigenvalue spacing of order ε, so the interesting scale for small energies is α = 1. In this case there is no spectrum of H σ a in the (−∞, Λ 0 + Cε 2 ] for ε small enough which will imply σ(H σ ) ∩ (−∞, Λ 0 + Cε 2 ] = ∅. • If λ 0 (x) = Λ 0 is constant, the adiabatic operator is given by . . ) and its eigenvalues, if they exist, scale as ε 2 . We will see that the latter approximate those of H σ up to errors of order ε 4 . More generally we will investigate energies of order ε α above the bottom Λ 0 := inf x∈B λ 0 (x) of the vertical operator for 0 < α ≤ 2. The following theorem states that the mutual approximation in this regime is better, compared to Corollary 3.2, by a factor ε α/2 and even by a factor ε α if one merely considers eigenvalues. For moderate magnetic fields (σ = 0) this can be done without further assumptions, for strong magnetic fields (σ = 1) the additional hypothesis that A B = c * A may be gauged away is needed.
Theorem 3.3. Let (M, A) be a magnetic quantum waveguide of bounded geometry and 0 < α ≤ 2. If σ = 1, assume additionally that A B is an exact one-form on B. Then, for every C > 0, it holds that is of order ε 2+α/2 . If Λ 0 +Cε α is strictly below the essential spectrum of H σ a , in the sense that for some δ > 0 the spectral projection 1 (−∞,Λ 0 +(C+δ)ε α ] (H σ a ) has finite rank, for ε > 0 small enough. Then, if µ σ 0 < µ σ 1 ≤ · · · ≤ µ σ K are all the eigenvalues of H σ a below Λ 0 +Cε α , H σ has at least K +1 eigenvalues ν σ 0 < ν σ 1 ≤ · · · ≤ ν σ K below its essential spectrum and Proof. The statement for the moderate case (σ = 0) is an immediate application of [Haa16,Proposition 4.14] and [Haa16,Theorem 4.15], where again E = M × C and ∇ M ×C = d + iπ * M A B (plus higher order corrections that are again treated as a perturbation). The basic idea of this improvement to work is the observation that the super-adiabatic corrections to H σ=0 a (which must also be included in order to obtain a better approximation) essentially consist of horizontal differential operators. But such derivatives ∇ are of order ε α/2 and are therefore small on this ε-dependent energy scale, i.e., on the image of 1 (−∞,Λ 0 +Cε α ] (H σ=0 ) (see [Haa16,Lemma B.1] for the precise statement).
For the strong case (σ = 1) we mention that -if A B is exact and may be gauged away -the relevant connection on M × C is given by ∇ M ×C = d + iA ε=0 H plus higher order corrections and thus has basically the same form as in the moderate case.
We remark that results can be obtained also for energies higher than Λ and projections to other eigenbands than the ground state. The relevant condition is that they are separated from the rest of the spectrum of H V A by a local gap δ, which guarantees regularity of the corresponding spectral projection. As pointed out above, the approximation of spectra is not mutual as for low energies, but there is always spectrum of H σ near that of H σ eff (see [HL17, Corollary 1.2]).

Moderate Magnetic Fields (σ = 0)
This section is dedicated to the calculation of the adiabatic operator (10) for the case of moderate magnetic fields. To simplify the notation, we will frequently drop the superscript σ. Since we may apply both Corollary 3.2 and Theorem 3.3 without further restrictions, we treat both cases at once and aim for the approximation of the spectrum of H by that of H a for energies of the order ε α above Λ 0 = inf λ 0 , for α ∈ [0, 2].
We need to compute H a up to errors of order ε 2+α on the appropriate energy scale. This is implemented by estimating the errors in L(K α (B), L 2 (B)), with the ε and α-dependent spaces equipped with the restriction of the L 2 -norm, where C is an arbitrary constant for α > 0 and C < Λ − Λ 0 for α = 0. Note that K α 1 (B) ⊂ K α 2 (B) is a closed subspace for α 1 > α 2 .
We will see be below (see also [WT13, Theorem 2.5]) that the nonmagnetic adiabatic operator may be written in the form The quantities in this equation are the following: • The Laplacian L on the base B which is defined by substituting the non-magnetic effective metric, given for t 1 , t 2 ∈ T x B by , into the quadratic from The underline L is used here to distinguish the so-defined operator on the base from its analogue L on M , defined by (9). • The non-magnetic Born-Huang potential is given by Equivalently, it may be defined by its quadratic form + 2 Re φ 0 (π * M ψ)G ε hor (dπ * M ψ, dφ 0 ) vol g . We emphasise that this defines a potential, and not a derivation, due to the real part.
• The averaged bending potential is Remark 4.1. The notation · 0 is used here to denote averaging over the fibre with weight |φ 0 | 2 , we will later use the notation · A when the weight is given by |φ A | 2 .
Remark 4.2. The leading order of the non-magnetic Born-Huang potential V BH is given by and equals the adiabatic potential found in [HLT15, Equation (12)]. Here, we used the facts that φ 0 vanishes on the boundary ∂M x and that π * M g B (dφ 0 , ·) is a horizontal vector field. Moreover, it is shown in [HLT15, Section 5.2] that the second integral coincides with 1 2 η 0 , where η ∈ C ∞ (HM ) stands for the mean curvature vector of the fibres M x → M . Thus, if we deal with massive quantum waveguides (for which η vanishes), the non-magnetic Born-Huang potential has the expansion In order to simplify later calculations, we establish the following Lemma: Lemma 4.3. Let φ ∈ {φ 0 , φ A }, ∇ be a connection on the trivial line-bundle B ×C, ϕ, ψ ∈ W 1 (B) and ξ ∈ C ∞ (T * B ⊗C) and µ ∈ C ∞ (T * M ⊗C) complex one-forms. Then the quadratic form Proof. Expanding the expression, one sees that the terms containing ξ cancel each other out, and Q ξ = Q 0 for all ξ.
The adiabatic operator for moderate magnetic fields is obtained from the non-magnetic adiabatic operator H nm a by minimal coupling to an effective magnetic potential A ε eff : Theorem 4.4. The adiabatic operator for moderate magnetic fields is given by the non-magnetic adiabatic operator, minimally coupled to the effective magnetic potential A ε eff := A B + εA 1 , where for X ∈ C ∞ (TB), up to errors of order ε 2+α in L(K α (B), L 2 (B)) for α ∈ [0, 2]. That is, Proof. We proceed by calculating the quadratic form of H σ a on L 2 (B). By definition, we have where we choose ϕ ∈ W 1 (B) and ψ ∈ K α (B). Since λ A = λ 0 + O(ε 4 ) and φ A = φ 0 + O(ε 2 ) for weak magnetic fields, we immediately see that It now remains to analyse the quadratic form of the projected horizontal Laplacian. We need to estimate the errors by ε 2+α ϕ L 2 (B) ψ Kα(B) . To achieve this, we will make use of the fact (see Equation (18) below) that εdψ L 2 (T * B) = O(ε α/2 ) for ψ ∈ K α (B). When derivatives act on ϕ, we integrate by parts, so that they act on ψ and we can use the same estimate. We will not perform this step in detail, but just remark that integration by parts is always justified due to the factors of φ 0 or φ A in the integrals, which vanish on the boundary of a massive waveguide.
We split the quadratic form using the projection P A : where we used the abbreviation ∇ A ε hor = d + iA ε hor . We start by simplifying this expression using Lemma 4.3. We have and, because [A B (X), P A ] = 0 for any vector field, We can thus apply Lemma 4.
which means that we can continue as if ξ = 0. In particular, this proves the formula (14) when A = 0.
We now show that, in Equation (17a) with ξ = 0, A ε hor A can be replaced by A ε eff and G ε hor A with G ε hor 0 , up to errors of order ε 3+α/2 . To achieve this, observe that Now expand where the error is measured by g B ). Together with Equation (18) and integration by parts this shows that . For the remaining terms (17b)-(17d), we use the expansion , which yields, by the same reasoning as above and neglecting ξ, The terms involving iεP ⊥ 0 A ε=0 H φ 0 are also of order ε 3+α/2 , because Furthermore, the remaining terms in Equation (19) because φ 0 and A ε eff are real. The relevant contribution of (17b), (17c) thus reduces to The contribution of the last term, (17d), is just as the imaginary part of G ε hor (dφ 0 , P ⊥ 0 A ε=0 H φ 0 ) vanishes. Combining this with (21) gives exactly the quadratic form of V BH .
We remark that this proof shows that the error is actually of order ε 3+α/2 , which is slightly better than ε 2+α . But this is not very relevant, as the error of the original approximation of H by H a is ε 2+α .

Strong Magnetic Fields (σ = 1)
In this section we compute the adiabatic operator (10). We will again drop the superscript σ, most of the time.
Theorem 5.1. The adiabatic operator for σ = 1 is given by where the magnetic Born-Huang potential is given by , and the first-order differential operator This statement is exact, with no further errors, but the terms in the adiabatic operator may of course be expanded using perturbation theory for λ A and φ A , as well as an expansion of A ε H . In particular, when A B is an exact form, Theorem 3.3 applies and one can obtain a more explicit form of the effective Hamiltonian on ε α -energy scales, cf. Section 6.5.2.
Proof. The potentials λ A and ε 2 V bend A are the exact projections of H A V and the bending term, so there is nothing to prove here and we are again left with calculation the projection of L G ε hor , ε −1 A ε hor with P A . As in the proof of Theorem 4.4, we start by applying Lemma 4.3, for which we need and . Writing out the quadratic form of ε 2 P A L(G ε hor , ε −1 A ε hor )P A as in (17a) -(17d), we can thus neglect ξ = φ A , dφ A L 2 (Mx) by Lemma 4.3. The term with the projection P A applied twice then yields exactly the quadratic form of ε 2 L( G ε hor A , ε −1 A B + A P ).
Using that A P is real and integration by parts, one sees that the terms with exactly one P ⊥ A yield the quadratic form of the operator Note that this operator is self adjoint but D mag and the complex potential separately are not. The first term in this expression is completed by the one with two occurrences of P ⊥ A to the potential V A BH . In view of Corollary 3.2 there is no additional restriction on A and we need to calculate H a up to errors of order ε 2 : Corollary 5.2 (α = 0). The adiabatic operator for strong magnetic fields is given by , where the effective magnetic potential A ε eff is the same as in Theorem 4.4. Proof. We have λ A = λ 0 + O(ε 2 ) and ε 2 V bend A = O(ε 2 ), so it remains to show that ε 2 D mag = O(ε 2 ), and that A P can be replaced by A ε=0 H 0 , with appropriate errors. The latter is a consequence of Equation (22), as as a one-from on (B, g B ) (cf. Equation (18)). This holds because φ A = φ 0 + O(ε) with real φ 0 and Equation (20).

Application to Quantum Tubes
This chapter is devoted to the explicit computation of the magnetic adiabatic operators of massive or hollow quantum tubes, that is waveguides modelled along closed or infinite open curves in R 3 . We will discuss the leading contribution of the magnetic effects for moderate and strong magnetic fields in Section 6.4 and Section 6.5, respectively. 6.1. The Geometry. Let c : B → R 3 be either an infinite curve, i.e., B = R, or a closed curve of length 2L, i.e., B = R/2LZ ∼ = [−L, L), that is smoothly embedded in (R 3 , δ), bounded with all its derivatives and parametrised by arc length, so g B = dx ⊗ dx. In both cases the normal bundle NB is diffeomorphic to the trivial bundle B ×R 2 . For the following explicit computations we choose coordinates induced by parallel orthonormal frames in NB. While for B = R such frames exist globally, for B = R/2LZ they might be discontinuous at one point. Hence, in this case, we restrict the computations to B \{x 0 } ∼ = (−L, L). In order to not overburden the notation, we do not make this explicit in every step, but keep in mind that B needs to be replaced by B \ {x 0 } whenever necessary. The relevant geometric objects will always be well defined on all of B, see for example Remark 6.1.
Explicitly, a parallel orthonormal frame can be constructed as follows: At 0 ∈ B we pick an orthonormal basis {τ 0 , e 1,0 , e 2,0 } of T c(0) R 3 ∼ = R 3 such that τ 0 = c (0) is tangent and (e 1,0 , e 2,0 ) are normal to the curve at c(0). One then obtains a frame (τ, e 1 , e 2 ) of TR 3 c(B) as the solution of the coupled system of differential equations with initial data (τ (0), e 1 (0), e 2 (0)) = (τ 0 , e 1,0 , e 2,0 ), where are the mean curvatures of the curve, see for instance [Bis75]. This is an orthonormal frame of ( TR 3 c(B) , δ) with τ (x) = c (x) for all x ∈ B which we identify with an orthonormal frame of TB ⊕ NB by means of (1). Put differently, this frame is obtained by the parallel transport of the orthonormal basis (τ 0 , e 1,0 , e 2,0 ) along the curve B with respect to the induced connection c * ∇ δ . Consequently, x → τ (x) = ∂ x is a trivialisation of the tangent bundle TB, whereas x → (e 1 (x), e 2 (x)) yields bundle coordinates Ξ : B × R 2 → NB, (x, n 1 , n 2 ) → n 1 e 1 (x) + n 2 e 2 (x) that come along with coordinate vector fields ∂ # x , ∂ n 1 and ∂ n 2 , where ∂ # x is the horizontal lift (which coincides with the product lift) of ∂ x to NB.
It is given by (see [Haa16,Example 5.6]) where we used the notation dx # := π * NB dx. Remark 6.1. Note that even in the case, where our coordinates are defined only on (B \ {x 0 }) × R 2 , the metric G ε int is well defined on all of NB r . This is because the bases (e 1 (−L), e 2 (−L)) and (e 1 (L), e 2 (L)) differ at most by a transformation in SO(2), while the appearing expressions are invariant under rotations of the fibres.
The final form of the family {G ε := * G ε int } 0<ε≤1 now depends on the concrete realisation of the waveguide induced by the embedding : M → NB r . Note that for B = R we can without loss of generality pick M = R×F , while for B = R/2LZ the bundle M π M − − → B need not be trivial. As we will see in the following example, the global structure of M is then encoded in the map .
Example 6.2. Let us consider two illustrative choices of the embedding (see [Haa16,Example 5.7] and [Haa16, Example 5.9]): (i) massive quantum tube, dim(F ) = 2: Let F ⊂ R 2 be a connected (in general not rotationally invariant) bounded domain and R > 0 such that F ⊂ B R (0). Then for y = (y 1 , y 2 ) ∈ F the mapping : (x, y 1 , y 2 ) → (x)r(x)y 1 e 1 (x) + (x)r(x)y 2 e 2 (x), cos ϕ(x) models a waveguide whose cross-section is the fixed set F that twists around the curve with respect to an x-dependent angle ϕ : B → R and scales with an L)) such that the cross-sections as subsets of NB r match smoothly, i.e., (r(−L)y) j e j (−L) : y ∈ F = (r(L)y) j e j (L) : y ∈ F and ϕ ∈ C ∞ b ([−L, L)). While the coordinates (x, y 1 , y 2 ) again provide coordinate vector fields ∂ pr x , ∂ y 1 and ∂ y 2 , it is more convenient to the horizontal lift ∂ H x of ∂ x to M instead of ∂ pr x . This lift satifies T (∂ H x ) = ∂ # x and is given by with the intuitive notation for the cross product in R 2 . It incorporates both the effect induced by the variation of the scaling factor (in terms of ) and the effect of the twist (in terms of ϕ ) along the curve. Finally, the associated family of pullback metrics on M reads We take the cross-section to be a circle S 1 = R/2πZ with an xdependent radius ∈ C ∞ b (B, [l − , l + ]) such that 0 < l − < l + < r. The associated embedding is given by Similarly as above we have coordinate vector fields (∂ pr x , ∂ y ) but we use the horizontal lift x for the calculation. This eventually leads to the following family of pullback metric on M : where κ(x, y) := κ 1 (x) cos y + κ 2 (x) sin y is the radial projection of the mean curvature vector.
These examples illustrate the great generality of our approach, i.e., even more complex realisations of quantum waveguides may be represented by rather simple embeddings.
We finally remark that in general the horizontal part of the pullback metric takes the form as a pointwise mapping From this we immediately infer for the effective metric on covector fields.
6.3. The Pullback of the Magnetic Potential. When dealing with quantum tubes, an appropriate modification of the gauge function Ω ε introduced in Lemma 2.3 leads to a very convenient gauge for the intermediate magnetic potential A ε int (pulled back via Φ ε ) on NB r in the case of quantum tubes. Although we will use (local) coordinates (x, n 1 , n 2 ) -and therefore a (local) orthonormal frame {τ, e 1 , e 2 } of TB ⊕ NB -as introduced in Subsection 6.1 for its construction, it will turn out that the resulting magnetic potential A ε int is actually independent of the chosen frame. Let for j ∈ {1, 2} be the components of the magnetic potential, where τ (x) and e j (x) are considered as vectors in R 3 . Then the pulled-back magnetic potential with respect to these coordinates reads We set A Taylor expansion of the coefficients A x (x, εn) and A n j (x, εn) up to errors of order ε 2 eventually leads to Here, we introduced the physically relevant magnetic field B = dA ∈ C ∞ b ( Λ 2 TR 3 T r ) evaluated on the curve, which consists of one parallel component and two perpendicular components with respect to the curve c(B) ⊂ R 3 (with respect to the local orthonormal frame {τ, e 1 , e 2 }). While the component B parallel to the curve clearly is unaffected by the choice of transversal directions, both B ⊥ ×n and n×dn are invariant under a change of a (local) orthonormal frame. Thus, the above expression for A ε int is well-defined. Example 6.3. Let us examine the induced magnetic potential A ε for the geometric situations illustrated in Example 6.2. Detailed calculations of A ε in these cases can be found in [Haa16,Example 5.10].
(i) The horizontal part of A ε reduces to π * whereas the vertical part ε 2 A ε V is given at leading order by A ε=0 V (∂ y 1 ) = − 1 2 B 2 y 2 , A ε=0 V (∂ y 2 ) = 1 2 B 2 y 1 . Consequently, the magnetic potential of this massive quantum waveguide reads (ii) Although ‫∂(ג‬ x ) does not vanish in the hollow case, its contribution to the horizontal part A ε H is of lower order and may thus be neglected, i.e., A ε=0 H (∂ H x ) = B ⊥ × cos y sin y . As far as the vertical part is concerned, we obtain cos y sin y × − sin y cos y = 1 2 B 2 and therefore Remark 6.4. The intermediate magnetic potential A ε int -and therefore the resulting pulled-back magnetic potential A ε -may be further simplified if one deals with infinite curves (B = R) with a trivial topology. In this case, we may additionally gauge away the leading part π * NB A B by subtracting the differential of the gauge transformation Here, we set A| c(0) = 0 as reference point without loss of generality. Put differently, the magnetic potential A restricted to the infinite curve c(R) ⊂ R 3 can be gauged away completely in both the horizontal and vertical directions. 6.4. Moderate Magnetic Fields (σ = 0). We now come to the examination of the adiabatic operator that is associated with quantum tubes in the presence of moderate magnetic fields. Therefore, Theorem 4.4 states that the only possible influence of the magnetic potential arises both from A B and from averaged against the non-magnetic ground state φ 0 , i.e., A 1 (∂ x ) = A ε=0 H (∂ H x ) 0 . While the first part A B only depends on the topology of the submanifold B (A B = 0 for B = R due to Remark 6.4, but A B = 0 in general for B = S 1 ), the second part A 1 is affected by the concrete modelling of the waveguide (fibres) in terms of the embedding . The following corollary yields easily verifiable conditions conditions on the geometry which cause a vanishing of the latter potential A 1 : Corollary 6.5. Assume that (i) massive quantum tubes, dim(F ) = 2: the map | (x,y) is a vector space isomorphism for all (x, y) ∈ M and that the non-negative ground state φ 0 (x) is centred for all x ∈ B, i.e., y 1 0 = 0 = y 2 0 , (ii) hollow quantum tubes, dim(F ) = 1: the centre of mass of each cross-section lies in the origin, i.e., the integrals vanish for all x ∈ B. Then the adiabatic operator is given by In particular, if B = R is an infinite curve, there are no magnetic effects within the adiabatic operator, i.e., H σ=0 While this trivially reduces to (i) if is a pointwise vector space isomorphism, we note for (ii) that φ 0 (x) is y-independent at leading order by (13) and may thus be extracted from the integral.
Applying this corollary to the two geometrical situations introduced in Example 6.2, we find the following: (i) massive example: We have M x = F and φ 0 (x) = Φ 0 for all x ∈ B, where Φ 0 is the normalised ground state of the Dirichlet Laplacian −∆ y on L 2 (F, dy). Consequently, the condition of Corollary 6.5(i) reduces to the xindependent claim on Φ 0 to be centred, i.e., where g(y) Φ 0 := Φ 0 , gΦ 0 L 2 (F ) for any integrable function g on F . One possible realisation thereof is a circular disc F = D 2 or more generally an ellipse F = E 2 . (ii) hollow example: We have M x = S 1 = R/2πZ for all x ∈ B, and the condition of Corollary 6.5(ii) is satisfied since and similarly for the other integral. Consequently, if we assume B = R, the resulting adiabatic operator is independent of any magnetic effects up to errors of order ε 2+α in L(K α (R), L 2 (R)). It is discussed in great detail within [HLT15, Section 5.2]. Let us finally compare our adiabatic operator with the effective operator in [KR14, Definition 2.5 with δ = 0]. The geometric situation considered there corresponds to that of Example 6.2(i) with ≡ 1. We refer to this as rigid massive quantum tubes. In particular, this implies that λ 0 (x) = λ 0 is constant and we retrieve the case α = 2, where dψ(∂ x ) L 2 (B) = O(1) for ψ ∈ W 2 (B). Therefore, the operator H σ=0 a − λ 0 is of order ε 2 , hence we may divide it by the same factor and get by Theorem 4.4: . Its eigenvalues eventually approximate those of the initial magnetic Laplacian −∆ A δ of the conventional quantum tube T ε ⊂ R 3 up to errors of order ε 2 by Theorem 3.3. We now compute explicit (local, frame invariant) formulas for the individual terms appearing in H σ=0 rig which itself is a local operator.
The Laplacian. Using the expressions (24) and (25) for the horizontal metric and magnetic potential, respectively, the expansion of the quadratic form of the Laplacian within the above Hamiltonian equals The Born-Huang Potential. As far as the non-magnetic Born-Huang potential is concerned, we observe that by means of (23) with ≡ 1. In view of Remark 4.2 the first term doesn't contribute to V BH and we have where L y := y×∇ y stands for the (skew-symmetric) vertical angular momentum operator. Again, all introduced objects do not depend on the explicitly chosen orthonormal frame.
The Bending Potential. Using the fact that ρ ε = 1 − ε ry, κ R 2 , the formulas of [HLT15, Section 4.3] lead to ε 2 L(π * M g B , 0) ln ρ ε = ε 3 ry, κ + O(ε 4 ) and . Thus, the bending potential reads . and hence Conclusion. We conclude that the leading contribution of H σ=0 rig equals which is in accordance with [KR14, Theorem 2.6 with δ = 0] for infinite curves. The subsequent order of H σ=0 rig incorporates magnetic effects in terms of A B and B ⊥ . If Φ 0 is centred (26), this O(ε)-contribution to the Hamiltonian reduces to the potential Here, in particular, we observe that magnetic effects induced by B ⊥ are not occurrent at this order as stated by Corollary 6.5(i). Moreover, if in addition the cross-section F (and, therefore, the non-magnetic ground state Φ 0 ) is rotationally invariant, it holds that L y Φ 0 = 0 and this potential vanishes. 6.5. Strong Magnetic Fields (σ = 1). We continue with the analysis of the adiabatic operator for the framework of quantum tubes with strong magnetic fields. 6.5.1. The Case α = 0. While hollow quantum waveguides always possess spectrum below this energy regime (more precisely, the constant nonmagnetic ground state band in the hollow cases induces spectrum of order ε 2 above the bottom), the regime α = 0 corresponds to the low-lying part of the spectrum of generic massive quantum waveguides. Therefore, we expand H σ=1 a from Corollary 5.2 for massive quantum tubes as an operator in L(K 0 (B), L 2 (B)), i.e., on states ψ ∈ W 2 (B) with εdψ(∂ x ) L 2 (B) = O(1).
The effective magnetic potential A ε eff = A B + εA 1 is (locally) given by Moreover, the incorporated metric (evaluated on one-forms) reads (24) and the comment after Corollary 5.2. Thus, we get the following expansion for the adiabatic operator: The relevant quantities herein, that correspond to the geometric setting of Example 6.2(i) and Example 6.3(i), are given by and thus vanish for a centred non-magnetic ground state φ 0 , eventually leading to H σ=1 is this case. 6.5.2. The Case α = 2. We intend to discuss the main magnetic effects within the adiabatic operator if the unperturbed ground state band λ 0 (x) = λ 0 is constant. Therefore, in order to apply Theorem 3.3, we restrict ourselves to infinite curves (B = R), where A B = 0. It turns out that the resulting operators are of order ε 2 in L(K 2 (R), L 2 (R)), where dψ(∂ x ) L 2 (R) = O(1) for ψ ∈ W 2 (R). In view of Theorem 5.1, we obtain is the second-order correction of the magnetic ground state band to λ 0 . By Theorem 3.3 we could achieve an accuracy of order ε 2 for the approximation of the initial Laplacian −∆ ε −1 A δ of the conventional quantum tube T ε ⊂ R 3 . We, however, focus on the leading contribution of these adiabatic operators and hence on the main magnetic effects.
• The leading order of the Laplacian is given by −∆ A 1 x . In view of (12), the expansion of the averaged bending potential reads hol. .
• Standard perturbation theory reveals that (see for example [Kat80, Section II - § 2]) where We mention that λ 0,2 is non-negative due to the diamagnetic inequality [FH10] and may be expressed more concretely in the following sense: Let {ζ k (x)} k≥0 be an orthonormal basis of the vertical non-magnetic Laplacian −∆ V (x) = −∆ g Mx on L 2 (M x ) with eigenvalues {λ k (x)} k≥0 , then one has We note that, in the hollow case, the corrections to λ 0,2 that arise from H V 0 = −∆ V + O(ε) (inducing an O(ε)-deviation between the corresponding eigenfunctions) are of higher order and may thus be neglected.
• In view of Remark 4.2, the expansion of the magnetic Born-Huang potential equals where A 0 P ⊥ := P ⊥ 0 A ε=0 H φ 0 ∈ C ∞ (H * M ) ⊗ P ⊥ 0 H andη ∈ C ∞ (M ) is the projected mean curvature vector along ∂ H x . Herein, we used the fact that φ 0 , A 0 P ⊥ (∂ H x ) L 2 (Mx) = 0. Furthermore, the first term of this leading order contribution can be rewritten as follows: • The first-order differential operator vanishes at leading order.
Consequently, we conclude that with errors in L(W 2 (R), L 2 (R)), where the κ-potential is not present for hollow quantum waveguides.
Massive Waveguides. Let us evaluate the resulting operator (31) in more detail for rigid massive quantum tubes (η = 0) as discussed at the end of the previous section, corresponding to the geometric framework of Example 6.2(i) with ≡ 1. We recall that A ε=0 H (∂ H x ) = B ⊥ × ry in this case and thus A 1 (∂ x ) = B ⊥ × r y Φ 0 by (28), where Φ 0 stands for the normalised ground state of the Dirichlet Laplacian −∆ y on L 2 (F, dy). Consequently, we retrieve with ∂ H x = ∂ pr x − ϕ L y : The last transformation follows from a straightforward calculation, using the fact that This operator exactly coincides with the effective operator in [KR14, Definition 2.5 with δ = 1]. We also find the same representation for the correction λ 0,2 of the ground state band: = B 2 1 4 y 2 We eventually note that this operator reduces to if Φ 0 is centred (26). We finally consider the case where F is rotationally invariant and without loss of generality ϕ ≡ 0 (i.e., r = 1 2×2 ). This yields that Φ 0 is rotationally invariant and, in particular, centred. The rotational invariance implies L y Φ 0 = 0 and thus Therefore, using the fact that Φ 0 , −iL y Φ 0,1 L 2 (F ) = −iL y Φ 0 , Φ 0,1 L 2 (F ) = 0 due to integration by parts and Φ 0 | ∂F = 0, the corresponding adiabatic operator furthermore reduces to . Example 6.6. If we assume that F is the circular disc with radius , a small calculation shows that where j 01 ≈ 2.405 denotes the first zero of the zeroth Bessel function J 0 of first kind.