Spectral properties of relativistic quantum waveguides
William Borrelli, Philippe Briet, David Krejcirik, Thomas Ourmieres-Bonafos
aa r X i v : . [ m a t h . SP ] J a n SPECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES
WILLIAM BORRELLI, PHILIPPE BRIET, DAVID KREJˇCIˇR´IK, AND THOMAS OURMI`ERES-BONAFOS
Abstract.
We make a spectral analysis of the massive Dirac operator in a tubular neighborhoodof an unbounded planar curve, subject to infinite mass boundary conditions. Under general as-sumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonianon the base curve which approximates the original operator in the thin-strip limit. We also inves-tigate the existence of bound states in the non-relativistic limit and give a geometric quantitativecondition for the bound states to exist.
Keywords : quantum waveguides, Dirac operator, infinite mass boundary conditions, non-relativistic limit, thin-waveguide limit, norm-resolvent convergence. : 35P05, 81Q10, 81Q15, 81Q37, 82D77. Introduction
Motivations and state of the art.
Consider a massive particle in a guide modelled by auniform tubular neighbourhood of an infinite planar curve. A classical particle, moving according toNewton’s laws of motion with regular reflections on the boundary, will eventually leave any boundedset in a finite time, except for initial conditions of measure zero in the phase space correspondingto transverse oscillations. It came as a surprise in 1989 that the situation changes drastically forquantum particles modelled by the Schr¨odinger equation. In the pioneering paper [15] and furtherimprovements [13, 16, 19], it was demonstrated that the quantum Hamiltonian identified with theDirichlet Laplacian possesses discrete eigenvalues unless the base curve is a straight line. Roughly,and with a sharp contrast with the classical setting, the particle gets trapped in any non-triviallycurved quantum waveguide. The existence and properties of the geometrically induced bound stateshave attracted a lot of attention in the last decades and the research field is still very active. Werefer to the monograph [14] and the latest developments in [22] with further references.The goal of the present paper is to consider relativistic counterparts of the quantum waveguides.Here we model the relativistic quantum Hamiltonian by the Dirac operator in the same tubularneighbourhood as above, subject to infinite mass boundary conditions. The latter is probably thereason why the relativistic setting has escaped the attention of the community until now. Indeed,the self-adjointness of the Dirac operators on domains and the right replacement for the Dirichletboundary conditions have been understood only recently [1, 2, 3, 24].There are four motivations for the present study. First, we would like to understand the influ-ence of relativistic effects on spectral properties. Do the geometrically induced bound states existindependently of the mass of the particle? It is expected that they do exist for heavy particlesbecause the Dirac operator converges, in a suitable sense involving an energy renormalization, tothe Dirichlet Laplacian in the limit of large masses. For light particles, however, the answer is farfrom being obvious because it is well known that relativistic systems are less stable [25]. In thispaper we confirm the expectation by justifying the non-relativistic limit and provide partial (bothqualitative and quantitative) answers for the whole ranges of masses.Our second motivation is related to quantisation on submanifolds. It is well known (see [21]for an overview with many references) that the non-relativistic quantum Hamiltonian converges toa one-dimensional Schr¨odinger operator on the base curve. (The convergence involving an energyrenormalization can be understood either in a resolvent sense [12, 20, 21] or as an adiabatic limit[17, 23, 32].) It is remarkable that this non-relativistic effective operator is not the free quantumHamiltonian on the submanifold but it contains an extrinsic geometric potential depending on thecurvature of the base curve. In this paper we find that the relativistic setting is very different, for the limiting operator describing the effective dynamics on the submanifold is just the free Diracoperator of the base curve.Recently, the Dirac operator on metric graphs has been considered as a model for the transportof relativistic quasi-particles in branched structures [33] and the existence and transport of Diracsolitons in networks have been studied in [29]. Previous studies deal with the quantisation ofgraphs and spectral statistics for the Dirac operator [4], and self-adjoint extensions and scatteringproperties for different graph topologies [9]. Rigorous mathematical studies on linear and nonlinearDirac equations on metric graphs recently appeared [5, 6, 7]. The result of the present paper can beunderstood as the first step toward a rigorous justification of the metric graph model as the limitof shrinking branched waveguides.The last but not least motivation of this paper is that the present model is relevant for transportof quasi-particles in graphene nanostructures [27]. This makes our results not only interesting in themathematical context of spectral geometry and in the physical concept of quantum relativity, butdirectly accessible to laboratory experiments with the modern artificial materials. We hope thatthe present results will stimulate an experimental verification of the geometrically induced boundstates in graphene waveguides.1.2.
Geometrical setting and standing hypotheses.
Before presenting our main results in moredetail, let us specify the configuration space of the quantum system we are interested in.Let Γ ⊂ R be a curve with an injective and C arc-length parametrization γ : R → R , i.e., γ ( R ) = Γ. We define ν ( s ) the normal of Γ at the point γ ( s ) chosen such that for all s ∈ R the couple (cid:0) γ ′ ( s ) , ν ( s ) (cid:1) is a positive orthonormal basis of R . The curvature of Γ at the point γ ( s ), denoted κ ( s ) is defined by the Frenet formula γ ′′ ( s ) = κ ( s ) ν ( s ) . (1)All along this paper, we make the following assumptions on the curvature κ :(A) lim s →±∞ κ ( s ) = 0,(B) κ ′ ∈ L ∞ ( R ) . Now, for 0 < ε < ( k κ k L ∞ ( R ) ) − (with the convention that the right-hand side equals + ∞ if κ = 0identically), we define the tubular neighbourhood of radius ε of Γ as the domainΩ ε := { γ ( s ) + εtν ( s ) : s ∈ R , | t | < } . (2)It is a well-known result of differential geometry that under these conditionsΦ ε : ( s, t ) ∈ Str γ ( s ) + εtν ( s ) ∈ R (3)is a local C -diffeomorphism from the strip Str := R × ( − ,
1) to Ω ε . In order to ensure that themap Φ ε becomes a global diffeomorphism we additionally assume that(C) Φ ε is injective.Finally, for technical reasons, we also assume the more restrictive range of admissible width ε :(D) 0 < ε < (2 k κ k L ∞ ( R ) ) − . Remark 1.
Despite being quite general, assumptions (A), (B) are probably not optimal. In [21] theauthors deal with three-dimensional non-relativistic waveguides under minimal technical assump-tions on the base curve (in particular, the curvature does not need to be differentiable), and thensimilar results can be expected in the present case. However, for ease of presentation we prefer notto investigate this aspect here. The assumption (D) is purely technical and allows to apply Kato’sperturbation theory.1.3.
Main results.
We are interested in the relativistic quantum Hamiltonian of a (quasi-)particleof (effective) mass m ≥ ε . Namely, we define the operator D Γ ( ε, m ) in the Hilbert space L (Ω ε , C )as dom ( D Γ ( ε, m )) := { u ∈ H (Ω ε , C ) : − iσ σ · ν ε u = u on ∂ Ω ε } , D Γ ( ε, m ) u := − iσ · ∇ u + mσ u, (4) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 3 where ν ε is the outward pointing normal on ∂ Ω ε .In (4) we use the notation σ · v := σ v + σ v , where v ∈ R , and σ k are the Pauli matrices σ := (cid:18) (cid:19) , σ := (cid:18) − ii (cid:19) , σ := (cid:18) − (cid:19) . (5)Our first result is about the self-adjointness and the structure of the spectrum of the operator D Γ ( ε, m ). Theorem 2.
The operator D Γ ( ε, m ) defined in (4) is self-adjoint. Its spectrum is symmetric withrespect to the origin and there holds: Sp ess ( D Γ ( ε, m )) = (cid:0) − ∞ , − p ε − E ( mε ) + m (cid:3) ∪ (cid:2)p ε − E ( mε ) + m , + ∞ (cid:1) , where E ( m ) is the unique root of the equation m sin(2 √ E ) + √ E cos(2 √ E ) = 0 (6) lying in the line segment [ π , π ) . In order to prove Theorem 2, a first step is to study the operator D Γ ( ε, m ) in the special caseof Ω ε being a straight strip. In this setting, a partial Fourier transform gives a fiber decompositionof the operator D Γ ( ε, m ) and we are left with the investigation of one-dimensional operators whichcan be understood explicitly.The second step is to show that in the case of general waveguides Ω ε , D Γ ( ε, m ) can be seen as aperturbation of the operator in the straight strip. To this aim, we will use the following proposition,which allows to work with the ε -independent Hilbert space L ( Str , C ). Proposition 3.
The operator D Γ ( ε, m ) defined in (4) is unitarily equivalent to the operator E Γ ( ε, m ) defined on L ( Str , C ) as: E Γ ( ε, m ) := 11 − εtκ ( − iσ ) ∂ s + 1 ε ( − iσ ) ∂ t + εtκ ′ − εtκ ) ( − iσ ) + mσ , dom ( E Γ ( ε, m )) := { u = ( u , u ) ⊤ ∈ H ( Str , C ) : u ( · , ±
1) = ∓ u ( · , ± } . The main novelty here lies in a matrix-valued gauge transform involving the geometry of the basecurve Γ in order to deal with the infinite mass boundary conditions. In particular, compared tosimilar strategies for non-relativistic waveguides, it allows to gauge out one part of the geometricinduced potential.The next two main results of this paper concern the study of the operator D Γ ( ε, m ) in the thinwaveguide asymptotic regime ε → m → + ∞ , respectively. It turns outthat up to renormalization terms, both regimes are driven by effective operators but of very distinctkind. In the thin waveguide regime ε → m → + ∞ , the operator behaves as theDirichlet Laplacian in the domain Ω ε .Finally, our last result is a quantitative result on the existence of bound states involving only thegeometry of the domain Ω ε .1.3.1. Main result in the thin waveguide regime ε → . In this paragraph, we fix m ≥ ε → D ( m ) u := − iσ ∂ s u + mσ u, u ∈ dom ( D ( m )) := H ( R , C ) . (7)It is well known that D ( m ) is a self-adjoint operator with purely absolutely continuous spectrum Sp ( D ) = ( −∞ , − m ] ∪ [ m, + ∞ ), as can be seen performing a Fourier transform (see [30, Thm. 1.1]for the analogue in dimension three).Since this operator acts in L ( R , C ), it is more convenient to work in the ε -independent Hilbertspace L ( Str , C ) and with the unitarily equivalent operator E Γ ( ε, m ) introduced in Proposition 3. PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 4
Theorem 4 (Thin width limit) . There exists a closed subspace F ⊂ L ( Str , C ) and a unitary map V such that V : L ( Str , C ) → L ( R , C ) ⊕ F and for ε → there holds V (cid:16) E Γ ( ε, m ) − π ε ( P + − P − ) − i (cid:17) − V − = ( D ( m e ) − i ) − ⊕ O ( ε ) , (8) in the operator norm, where P ± are explicit orthogonal projectors in L ( Str , C ) and where theeffective mass m e is given by m e := π m . The projectors P ± in the renormalization term of Theorem 4 are projectors on positive andnegative spectral subspaces of a one-dimensional transverse Dirac operator. It is remarkable thatthe geometry of the base curve Γ only appears at higher order terms. We do not know if for ε small enough Sp dis ( D Γ ( ε, m )) = ∅ . In particular, it would be interesting to investigate further theremainder term in Theorem 4 to understand if the geometry can play a role in the creation of boundstates.Once again, the proof of Theorem 4 is divided in two steps. We first prove Theorem 4 in thespecial case of Ω ε being a straight strip via a projection on the modes of a one-dimensional transverseDirac operator. The obtained operator can be seen as a block operator 2 × m is non-zero there are off-diagonal terms. Theyare handled using Schur’s complement theory but a special care is needed in order to control the ε -dependence of each term.In the second step, we use a perturbation argument to prove that the general waveguides Ω ε canbe seen as a perturbation of sufficiently high order of the special case of the straight strip. This steprequires a thorough control in ε of the norm of the resolvent of some operators.1.3.2. Large mass regime m → + ∞ . In order to state our results in the large mass regime m → + ∞ we need a few notation and definition. First, all along the paper N := { , , . . . } denotes the set ofpositive natural integers. We also recall the well-known definitions of the min-max values as well asthe min-max principle (see [11, Thm. 4.5.1 & 4.5.2]). Definition 5.
Let q be a closed semi-bounded below quadratic form with dense domain dom( q ) ina complex Hilbert space H . For n ∈ N , the n -th min-max value of q is defined as µ n ( q ) := inf W ⊂ dom( q )dim W = n sup u ∈ W \{ } q ( u ) k u k H . (9)We also denote by q the associated sesquilinear form. If A is the unique self-adjoint operatoracting on H associated with the sesquilinear form q via Kato’s first representation theorem (see [18,Ch. VI, Thm. 2.1])), we shall refer to (9) as the n -th min-max value of A and set µ n ( A ) := µ n ( q ). Proposition 6 (min-max principle) . Let q be a closed semi-bounded below quadratic form withdense domain in a Hilbert space H and let A be the unique self-adjoint operator associated with q .Then, for n ∈ N , we have the following alternative:(1) if µ n ( A ) < inf Sp ess ( A ) then µ n ( q ) is the n -th eigenvalue of A (counted with multiplicity),(2) if µ n ( A ) = inf Sp ess ( A ) then for all k ≥ n there holds µ k ( q ) = inf Sp ess ( A ) . Now, we fix ε > m → + ∞ . Up to an adequaterenormalization, this limit can be interpreted as a non-relativistic limit and the Dirichlet Laplacianis expected to be the effective operator in this case (see [30, Sec. 6] for general remarks on thislimit). To this aim, we introduce L Γ ( ε ), the (spinorial) Dirichlet Laplacian in the waveguide Ω ε ,defined by L Γ ( ε ) := − ∆ , dom ( L Γ ( ε )) := H (Ω ε , C ) ∩ H (Ω ε , C ) . (10)The following proposition summarises results established in [13, 19]. Proposition 7. L Γ ( ε ) is self-adjoint and there holds Sp ess ( L Γ ( ε )) = h π ε , + ∞ (cid:17) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 5
Moreover, if Γ is not a straight line, there exists N Γ ∈ N ∪ { + ∞} such that ♯Sp dis ( L Γ ( ε )) = 2 N Γ . (11)The factor 2 in (11) comes from the fact that in (10) we consider the Dirichlet Laplacian actingon C -valued functions instead of the usual scalar one. In particular, any eigenvalue of L Γ ( ε ) haseven multiplicity. Here, we use the convention that if N Γ = + ∞ then 2 N Γ = + ∞ .Our first result in the large mass regime reads as follows. Proposition 8.
Let us assume that Γ is not a straight line, fix ε > and let n ∈ { , . . . , N Γ } .There exists m > such that for all m > m Sp dis ( D Γ ( ε, m )) ≥ n. Proposition 8 is proved by comparing the quadratic forms of the renormalized operator D Γ ( ε, m ) − m to the quadratic form of L Γ ( ε ), using the min-max principle (Proposition 6), theasymptotic behavior of E ( m ) when m → + ∞ and Proposition 7.Actually, one can show that all the min-max values of the renormalized operator D Γ ( ε, m ) − m converge to those of the Dirichlet Laplacian in the regime m → + ∞ . This is the purpose of thefollowing theorem. Theorem 9 ( Large mass limit ) . Let us assume additionally that Γ is of class C and that κ ′ ( s ) → and κ ′′ ( s ) → when | s | → + ∞ . Then for all n ∈ N there holds: lim m → + ∞ (cid:0) µ n ( D ( ε, m )) − m (cid:1) = µ n ( L Γ ( ε )) . (12)In particular, consider the positive part of the operator D Γ ( ε, m ) defined by D +Γ ( ε, m ) := x> ( D Γ ( ε, m )) . Since the spectrum of D Γ ( ε, m ) is symmetric with respect to zero, under thehypothesis of Theorem 9, we obtain for all n ∈ N µ n ( D +Γ ( ε, m )) = m + 12 m µ n ( L Γ ( ε )) + o (cid:0) m (cid:1) , m → + ∞ ; (13)where we have taken into account that the spectrum of L Γ ( ε ) has even multiplicity. Asymptotics (13)illustrates the physically expected fact that in the large mass regime m → + ∞ , the positive part ofthe Dirac operator with infinite mass boundary condition converges to the scalar Dirichlet Laplacian.The main novelty in Theorem 9 with respect to the previous work [1] is that we have to deal withthe unbounded domain Ω ε . This difficulty is overcome by a standard argument, approximating themin-max values of D ( ε, m ) − m by those of similar operators in bounded waveguides using theso-called IMS localization formula (see [10, Thm. 3.2]).1.4. Outline of the paper.
Section 2 deals with the infinite mass Dirac operator in the straightstrip and with the study of a one-dimensional Dirac operator on a finite interval, obtained byseparating variables.Then, in Section 3, we show that the Hamiltonian (4) is unitarily equivalent to a Dirac operatorin a straight strip, perturbed by a term encoding the geometric properties of the waveguide. Usingoperator-theoretic methods we are able to prove the self-adjointness and to locate the essentialspectrum, as stated in Theorem 2.Section 4 is devoted to the proof of Theorem 4, which is achieved in two steps. First, we deal withthe case of the straight waveguide and second, we add the perturbation induced by the curvature. Acareful analysis of the resolvent operator allows to prove that, after a suitable renormalization, theHamiltonian (4) converges in the norm resolvent sense to that of a one dimensional Dirac operatoron the line.Section 5 contains the proof of Theorem 9, showing that in the large mass regime the min-max values of the (renormalized) squared Hamiltonian converge to those of the vectorial DirichletLaplacian L Γ ( ε ).Finally, in Section 6, we obtain a quantitative condition for the existence of at least two boundstates in the gap of the essential spectrum. Even though the existence of bound states can be PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 6 obtained as a corollary of Proposition 8, we mention this alternative proof because here the conditionis given by a simple inequality involving geometric properties of the waveguide Ω ε .2. Straight waveguides
In this section we collect results concerning some auxiliary one-dimensional operators that natu-rally appear in the study of the Hamiltonian (4) in the thin waveguide regime.2.1.
The transverse Dirac operator.
For k ∈ R , consider the one-dimensional transverse Diracoperator T ( k, m ) := − iσ ddt + kσ + mσ , dom ( T ( k, m )) := { u = ( u , u ) ⊤ ∈ H (cid:0) ( − , , C (cid:1) , u ( ±
1) = ∓ u ( ± } . (14)The following proposition holds true. Proposition 10.
Let k ∈ R , m ≥ . The operator T ( k, m ) is self-adjoint and has compact resolvent.Moreover, the following holds:(i) Sp (cid:0) T ( k, m ) (cid:1) ∩ h − √ m + k , √ m + k i = ∅ ,(ii) the spectrum of T ( k, m ) is symmetric with respect to zero and can be represented as Sp (cid:0) T ( k, m ) (cid:1) = S p ≥ {± p m + k + E p ( m ) } , with E p ( m ) > for all p ≥ ,(iii) for all p ∈ N , E p ( m ) is the only root lying in (cid:2) (2 p − π , p π (cid:1) of (6) ,(iv) there holds E ( m ) = π
16 + m + O ( m ) , when m → , (v) there holds E ( m ) = π − π m + O ( m − ) , when m → + ∞ . The proof of Proposition 10 will also yield the following corollary which is of crucial importancein the study of the regime ε →
0. It concerns the operator T := T (0 , Corollary 11.
The operator T is self-adjoint and has compact resolvent. Its spectrum is symmetricwith respect to zero and verifies Sp ( T ) = n ± k π k ∈ N o . Corresponding normalized eigenfunctions are given by u ± k ( t ) := 12 cos (cid:16) k π t + 1) (cid:17) (cid:18) (cid:19) ±
12 sin (cid:16) k π t + 1) (cid:17) (cid:18) − (cid:19) . Proof of Proposition 10 and Corollary 11.
The multiplication operators by σ and σ are boundedand self-adjoint in L (cid:0) ( − , , C (cid:1) thus T ( k, m ) is self-adjoint if and only if T is self-adjoint. Anintegration by parts easily yields that T is symmetric and by definition, one hasdom ( T ∗ ) = n u ∈ L (cid:0) ( − , , C (cid:1) : ∃ w ∈ L (cid:0) ( − , , C (cid:1) such that ∀ v ∈ dom ( T ) , h u, T v i L (( − , , C ) = h w, v i L (( − , , C ) o . For every v ∈ D := C ∞ (cid:0) ( − , , C (cid:1) and u ∈ dom ( T ∗ ), there holds hT ∗ u, v i L (( − , , C ) = h u, T v i L (( − , , C ) = h u, − iσ v ′ i L (( − , , C ) = h u, iσ v ′ i D ′ , D = h− iσ u ′ , v i D ′ , D = hT ∗ u, v i D ′ , D , PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 7 where h· , ·i D ′ , D is the duality bracket of distributions. In particular, we know that T ∗ u = − iσ u ′ ∈ L (cid:0) ( − , , C (cid:1) thus we get u ∈ H (cid:0) ( − , , C (cid:1) . Moreover, if v ∈ dom ( T ) there holds hT ∗ u, v i L (( − , , C ) = h− iσ u ′ , v i L (( − , , C ) = h u, − iσ v ′ i L (( − , , C ) + h h− iσ u, v i C i − = h u, T v i L (( − , , C ) − u (1) v (1) + u (1) v (1)+ u ( − v ( − − u ( − v ( − . Since v ∈ dom ( T ) we obtain0 = − ( u (1) + u (1)) v (1) + ( u ( − − u ( − v ( − . This holds for any v ∈ dom ( T ), so that u ( ±
1) = ∓ u ( ±
1) and v ∈ dom ( T ). In particular T ∗ = T . Observe that, by the closed graph theorem, dom ( T ( k, m )) is continuously embedded in H (cid:0) ( − , , C (cid:1) which itself is compactly embedded in L (cid:0) ( − , , C (cid:1) . Thus, T ( k, m ) has compactresolvent.Let us prove Point (i) by picking u ∈ dom ( T ( k, m )) and considering kT ( k, m ) u k = k u ′ k L (( − , , C ) + ( m + k ) k u k L (( − , , C ) + 2 mk ℜ (cid:0) h σ u, σ u i L (( − , , C ) (cid:1) + 2 m ℜ (cid:0) h− iσ u ′ , σ u i L (( − , , C ) (cid:1) + 2 k ℜ (cid:0) h− iσ u ′ , σ u i L (( − , , C ) (cid:1) . (15)We rewrite (15), arguing as follows. Using the anti-commutation rules of Pauli matrices and theboundary condition we get2 ℜ (cid:0) h σ u, σ u i L (( − , , C ) (cid:1) = 2 ℜ (cid:0) h− iσ u ′ , σ u i L (( − , , C ) (cid:1) = 0and 2 ℜ (cid:0) h− iσ u ′ , σ u i L (( − , , C ) (cid:1) = k u (1) k C + k u ( − k C . (16)In particular, we obtain kT ( k, m ) u k L (( − , , C ) = k u ′ k L (( − , , C ) + ( m + k ) k u k L (( − , , C ) + m ( k u (1) k C + k u ( − k C ) ≥ ( m + k ) k u k L (( − , , C ) . Hence, by the min-max principle (see Proposition 6), if λ ∈ Sp ( T ( k, m )), we get | λ | ≥ √ m + k .Moreover, the last inequality is strict. Indeed, if u is an eigenfunction of T ( k, m ) associated with aneigenvalue λ such that | λ | = √ m + k we necessarily get that u is a constant C -valued functionon ( − ,
1) satisfying the boundary conditions given in (14). It is a contradiction because it impliesthat u = 0 identically. Hence, Sp ( T ( k, m )) ∩ [ −√ m + k , √ m + k ] = ∅ and Point (i) is proved.Now, let λ ∈ Sp ( T ( k, m )) and pick an associated eigenfunction u = ( u , u ) ⊤ ∈ dom ( T ( k, m )).There holds (cid:26) mu + ku − u ′ = λu ,ku + u ′ − mu = λu . (17)The second equation gives ( m + λ ) u = ku + u ′ and multiplying the first line by ( λ + m ) we get − u ′′ = Eu , E := λ − ( m + k ) . Recall that m ≥ E > k ∈ R . Thus we find u ( t ) = α cos (cid:0) √ E ( t + 1) (cid:1) + β sin (cid:0) √ E ( t + 1) (cid:1) , for some constants α, β ∈ R and as m + λ = 0 we get u ( t ) = 1 λ + m cos (cid:0) √ E ( t + 1) (cid:1)(cid:0) kα + √ Eβ (cid:1) + 1 λ + m sin (cid:0) √ E ( t + 1) (cid:1)(cid:0) kβ − √ Eα (cid:1) . The boundary condition at t = − m + λ − k ) α − √ Eβ = 0 . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 8
The boundary condition at t = 1 gives (cid:0) ( m + λ + k ) cos(2 √ E ) − √ E sin(2 √ E ) (cid:1) α + (cid:0) ( m + λ + k ) sin(2 √ E ) + √ E cos(2 √ E ) (cid:1) β = 0 . To obtain a non-zero eigenfunction u , there has to hold0 = (cid:12)(cid:12)(cid:12)(cid:12) m + λ − k −√ E ( m + λ + k ) cos(2 √ E ) − √ E sin(2 √ E ) ( m + λ + k ) sin(2 √ E ) + √ E cos(2 √ E ) (cid:12)(cid:12)(cid:12)(cid:12) . Computing the determinant, we are left with the implicit equation m sin(2 √ E ) + √ E cos(2 √ E ) = 0 . (18)In particular, it yields that the spectrum of T ( k, m ) is symmetric with respect to the origin and weremark that when m = k = 0, we necessarily have √ E = | λ | = p π (with p ∈ N ) and that in thiscase, a normalized eigenfunction associated with λ = ± π is given by u ± k ( t ) = 12 cos (cid:16) k π t + 1) (cid:17) (cid:18) (cid:19) ±
12 sin (cid:16) k π t + 1) (cid:17) (cid:18) − (cid:19) , which proves Corollary 11.Remark that for m >
0, a solution E to (18) verifies cos(2 √ E ) = 0 and we obtaintan(2 √ E ) + √ Em = 0 . (19)Now, for p ∈ N = N ∪ { } define the line segments I := [0 , π ) and I p +1 = ((2 p + 1) π , (2 p + 3) π ) g p : I p → R , g p ( x ) = tan( x ) + xm . (20)Remark that g ′ p ( x ) > g ( x ) = 0 is x = 0. For all p ≥ x → (2 p − π g p ( x ) = −∞ , g p ( pπ ) = p πm > . In particular, for all p ≥ x p ∈ I p to g p ( x ) = 0. Moreover, it satisfies x p ∈ (cid:0) (2 p − π , pπ (cid:1) . Hence, for p ≥ E p ( m ) is defined as the unique solution E to g p (2 √ E ) = 0.In particular E p ( m ) ∈ ((2 p − π , p π ) which proves Points (ii) and (iii).Now, we prove (iv). Guided by (18) we define the C ∞ function F : (cid:26) R × R → R ( µ, m ) m sin( µ ) + µ cos( µ ) . One remarks that F ( π ,
0) = 0 and ∂ µ F ( π ,
0) = π . Hence, by the implicit function theorem, thereexists δ , δ > C ∞ function µ : ( − δ , δ ) → ( π − δ , π + δ ) verifying µ (0) = π and suchthat for all | m | < δ there holds F ( µ ( m ) , m ) = 0. Moreover, when m → µ ( m ) = µ (0) + µ ′ (0) m + O ( m ) = π π m + O ( m ) . Necessarily, for m > E ( m ) = µ ( m ) . Hence, when m →
0, thereholds E ( m ) = π
16 + m + O ( m ) , which is precisely Point (iv).Finally, we prove (v). Once again, guided by (18) we define the C ∞ function G : (cid:26) R × R → R ( µ, ν ) µ ) + µν cos( µ ) . One remarks that G ( π,
0) = 0 and ∂ µ G ( π,
0) = −
2. Hence, by the implicit function theorem, thereexists δ , δ > C ∞ function µ : ( − δ , δ ) → ( π − δ , π + δ ) verifying µ (0) = π and such thatfor all | ν | < δ there holds G ( µ ( ν ) , ν ) = 0. Moreover, when ν → µ ( ν ) = µ (0) + µ ′ (0) ν + O ( ν ) = π − π ν + O ( ν ) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 9
Necessarily, for m > E ( m ) = µ ( m − ) . Hence, when m → + ∞ ,there holds E ( m ) = π − π m + O ( m − ) , which gives (v). (cid:3) The Dirac operator in the straight strip.
As will be seen further on in Section 3, Theorem2 can be obtained via classical perturbation theory arguments. They rely on the fact that theoperator D Γ ( ε, m ) can be seen as a perturbation of the operator D Γ ( ε, m ) in the straight strip Str ( ε ) := R × ( − ε, ε ). Here the base curve Γ := R × { } is a straight line, which we parametrize by γ ( s ) := s (1 , Proposition 12.
Let ε > . The operator D Γ ( ε, m ) is self-adjoint on its domain. Moreover, thereholds Sp (cid:0) D Γ ( ε, m ) (cid:1) = Sp ess (cid:0) D Γ ( ε, m ) (cid:1) = (cid:0) − ∞ , − p ε − E ( mε ) + m (cid:3) ∪ (cid:2)p ε − E ( mε ) + m , + ∞ (cid:1) , where E ( m ) is defined in Theorem 2. In order to work with operators defined on a fixed geometrical domain, we recall that
Str = Str (1)and consider the unitary map U : L (cid:0) Str ( ε ) , C (cid:1) → L (cid:0) Str , C (cid:1) , ( U v )( x ) := √ εv ( x , εx ) . The operator E ( ε, m ) := U D Γ ( ε, m ) U − verifies E ( ε, m ) = − iσ ∂ s − iε − σ ∂ t + mσ (21)with domain dom ( E ( ε, m )) = U dom ( D Γ ( ε, m )) which rewrites asdom ( E ( ε, m )) = { u = ( u , u ) ⊤ ∈ H ( Str , C ) : u ( · , ±
1) = ∓ u ( · , ± } . (22)In (21), we have used the new coordinates ( s, t ) ∈ Str defined by s = x and t = ε − x .Now, we are in a position to prove Proposition 12. We work with the unitarily equivalent operator E ( ε, m ) rather than the operator D Γ ( ε, m ) and the proof relies on a direct integral decompositionof the operator E ( ε, m ) as presented, e.g., in [28, § XIII.16.].
Proof of Proposition 12.
Consider the unitary partial Fourier transform in the s -variable F : L ( Str , C ) → L ( Str , C ) , ( F u )( k, t ) := 1 √ π Z R e − isk u ( s, t ) ds . The operator E ( ε, m ) is unitarily equivalent to the direct integral E ( ε, m ) = F − b E ( ε, m ) F , b E ( ε, m ) := Z ⊕ R b E ( ε, m ; k ) dk, where dom (cid:16) b E ( ε, m ) (cid:17) is the subspace of functions u = ( u , u ) ⊤ ∈ L ( Str , C ) such that for almostall k ∈ R we have ∂ t u ( k, · ) ∈ L (( − , , C ), u ( k, ±
1) = ∓ u ( k, ±
1) and for almost all t ∈ ( − , R R k | u ( k, t ) | dk < + ∞ .One observes that b E ( ε, m ; k ) satisfies b E ( ε, m ; k ) = ε T ( kε, mε ) where the operator T ( · , · ) isdefined in (14). In particular, b E ( ε, m ; k ) is self-adjoint and so is b E ( ε, m ) by [28, Thm. XIII.85 (a)].In particular, we have proved that E ( ε, m ) is a self-adjoint operator.By [28, Thm. XIII.85 (d)] there holds Sp ( E ( ε, m )) = [ k ∈ R Sp ( b E ( ε, m ; k )) . (23)Remark that we have Sp ( b E ( ε, m ; k )) = ε − Sp ( T ( kε, mε )) = [ p ∈ N (cid:26) ± q m + k + ε − E p ( εm ) (cid:27) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 10
By (iii) Proposition 10, for all p ≥ E p ( εm ) ∈ (cid:2) (2 p − π , p π (cid:1) . In particular, there holds Sp (cid:0) E ( ε, m ) (cid:1) = (cid:0) − ∞ , − p m + ε − E ( εm ) (cid:3) ∪ (cid:2)p m + ε − E ( εm ) , + ∞ (cid:1) . It concludes the proof of Proposition 12. (cid:3) First properties in curved waveguides
The main goal of this section is to prove Theorem 2. As mentioned before, the overall strategyconsists in regarding the operator D Γ ( ε, m ) in the curved strip Ω ε as a perturbation of the operator D Γ ( ε, m ) in the straight strip Str ( ε ).In the first paragraph of this section we derive an operator in a straight waveguide, unitarilyequivalent to D Γ ( ε, m ), which is given by a Dirac-type operator in the horizontal strip Str = R × ( − ,
1) perturbed by a curvature-induced potential. The second and third paragraphs deal with theself-adjointness and the invariance of the essential spectrum, respectively. The key arguments relyon perturbation theory.3.1.
Straightening the waveguide.
This paragraph is devoted to the proof of Proposition 3. Theoverall scheme is well-known in the study of non-relativistic waveguides and numerous works havetaken advantage of such a reduction (see, e.g., [13]). However, we give a complete proof here becausethe algebraic structure of the Dirac operator allows to gauge out one part of the curvature-inducedpotential, which appears to be a new effect.
Proof of Proposition 3.
The proof is divided into three steps. In the first one, we rewrite the problemin tubular coordinates in order to work in the strip
Str . The resulting operator acts in a weighted L -space and we perform a unitary transform in order to work in a non-weighted L -space; thisis the purpose of the second step. Finally, we build a unitary map in order to recover the sameboundary condition as the one of the operator E ( ε, m ) investigated in Section 2.2. This last steppartially simplifies the curvature-induced potential. Step 1.
Consider the unitary map U : L (Ω ε , C ) −→ L ( Str , C ; gdsdt ) , ( U u )( s, t ) := u (Φ ε ( s, t )) , (24)where Φ ε is the parametrization of the waveguide given in (3) and where g ( s, t ) := ε (cid:0) − εtκ ( s ) (cid:1) . Next, we consider the operator D Γ , ( ε, m ) := U D Γ ( ε, m ) U − . One sees that its domain isdom ( D Γ , ( ε, m )) = U dom ( D Γ ( ε, m ))= n u = ( u , u ) ⊤ ∈ L ( Str , C ; gdsdt ) :(1 − εtκ ) − ∂ s u, ∂ t u ∈ L ( Str , C ; gdsdt ) , for all s ∈ R u ( s, ±
1) = ± i n ( s ) u ( s, ± o , where for s ∈ R we have set n ( s ) := ν ( s ) + iν ( s ). The operator D Γ , ( ε, m ) acts on u ∈ dom ( D Γ , ( ε, m )) as D Γ , ( ε, m ) u = − i − εtκ σ γ ′ ∂ s u − iε σ ν ∂ t u + mσ u, where for x = ( x , x ) ∈ R we have set σ x := σ · x . Step 2.
In order to flatten the metric, consider the unitary map U : L ( Str , C ; gdsdt ) −→ L ( Str , C ) , U u := √ gu . (25)Let D Γ , ( ε, m ) := U U D Γ ( ε, m ) U − U − = U D Γ , ( ε, m ) U − . The domain of D Γ , ( ε, m ) is givenby dom ( D Γ , ( ε, m )) = U dom ( D Γ , ( ε, m ))= (cid:8) u = ( u , u ) ⊤ ∈ H ( Str , C ) :for all s ∈ R u ( s, ±
1) = ± i n ( s ) u ( s, ± (cid:9) , PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 11 and for u ∈ dom ( D Γ , ( ε, m )) the operator D Γ , ( ε, m ) acts as D Γ , ( ε, m ) u = 11 − εtκ ( − iσ γ ′ ) ∂ s u + 1 ε ( − iσ ν ) ∂ t u + εtκ ′ − εtκ ) ( − iσ γ ′ ) u + κ − εtκ ) ( − iσ ν ) u + mσ u. Note that the H ( Str , C ) regularity of functions in dom ( D Γ , ( ε, m )) is a consequence of the regu-larity hypothesis on the curve Γ (see (A) and (B)). Step 3.
Recall that n = ν + iν and by the Frenet formula (1) we have n ′ = iκ n . In particular,there holds n ( s ) = exp (cid:18) i Z s κ ( ξ ) dξ (cid:19) n . where we have set n := n (0). Moreover, there exists θ ∈ R such that n := e iθ . By setting θ ( s ) := θ + Z s κ ( ξ ) dξ , (26)we get n ( s ) = exp( iθ ( s )). For any fixed s ∈ R , consider the unitary matrix U θ ( s ) := (cid:18) exp (cid:0) i ( π + θ ( s )) (cid:1) − exp (cid:0) − i ( π + θ ( s ) (cid:1)(cid:19) . Note that the mapping s ∈ R U θ ( s ) ∈ C × is of class C ( R ). In order to obtain a boundarycondition independent of the normal vector ν we introduce the unitary map U : L ( Str , C ) −→ L ( Str , C ) , U u := U θ u. (27)The operator D Γ , ( ε, m ) := U D Γ , ( ε, m ) U − is unitarily equivalent to D Γ ( ε, m ). As U θ is a boundedand C ( R ) function, its domain is given bydom ( D Γ , ( ε, m )) = U dom ( D Γ , ( ε, m ))= (cid:8) u = ( u , u ) ⊤ ∈ H ( Str , C ) :for all s ∈ R u ( s, ±
1) = ∓ u ( s, ± (cid:9) . Moreover, for u ∈ dom ( D Γ , ( ε, m )), there holds D Γ , ( ε, m ) u = 11 − εtκ U θ ( − iσ γ ′ ) ∂ s ( U ∗ θ u ) + 1 ε ( − iσ ) ∂ t u + εtκ ′ − εtκ ) ( − iσ ) u + κ − εtκ ) ( − iσ ) u + mσ u, where we have used the identities U θ σ γ ′ U ∗ θ = σ , U θ σ ν U ∗ θ = σ , U θ σ U ∗ θ = σ . One also obtains U θ ( − iσ γ ′ ) ∂ s ( U ∗ θ ) = κ iσ )which finally gives D Γ , ( ε, m ) u = 11 − εtκ ( − iσ ) ∂ s u + 1 ε ( − iσ ) ∂ t u + εtκ ′ − εtκ ) ( − iσ ) u + mσ u. The proof is completed by setting E Γ ( ε, m ) := D Γ , ( ε, m ). (cid:3) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 12
Quadratic form of the square.
The aim of this section is to prove the following propositionabout the quadratic form of the square of the operator E Γ ( ε, m ) defined in Proposition 3. Throughoutthis section, we assume that Γ is of class C , in order to give a meaning to κ ′′ . Proposition 13.
Let us assume additionally that Γ is of class C . Then, for every u ∈ dom ( E Γ ( ε, m )) , there holds kE Γ ( ε, m ) u k L ( Str , C ) = Z Str − εtκ ) | ∂ s u − i κ σ u | dsdt + 1 ε Z Str | ∂ t u | dsdt + mε Z R (cid:0) | u ( s, | + | u ( s, − | (cid:1) ds + m k u k L ( Str , C ) − Z Str κ − εtκ ) | u | dsdt − Z Str ( εtκ ′ ) (1 − εtκ ) | u | dsdt − Z Str εtκ ′′ (1 − εtκ ) | u | dsdt . Before proving Proposition 13, we need the next lemma.
Lemma 14.
The set C ∞ ( Str , C ) ∩ dom ( E ( ε, is dense in dom ( E ( ε, for the operator norm.Proof of Lemma 14. Let u ∈ dom ( E ( ε, ϕ ∈ C ∞ ( Str , C ) ∩ dom ( E ( ε, h u, ϕ i L ( Str , C ) + hE ( ε, u, E ( ε, ϕ i L ( Str , C ) . In particular, picking ϕ ∈ C ∞ ( Str , C ) yields − ∂ s u − ε ∂ t u = − u in D ′ ( Str , C ) and then in L ( Str , C ). Define w := − iσ ∂ s u − iε σ ∂ t u and consider w and u , the extension by 0 of w and u to the whole plane R , respectively. There holds for all χ ∈ C ∞ ( R , C ) h ( − iσ ∂ s − iε σ ∂ t ) w , χ i D ′ ( R , C ) , D ( R , C ) = h w , ( iσ ∂ s + iε σ ∂ t ) χ i D ′ ( R , C ) , D ( R , C ) = h w, ( − iσ ∂ s − iε σ ∂ t ) χ i L ( Str , C ) = hE ( ε, u, E ( ε, χ i L ( Str , C ) = −h u, χ i L ( Str , C ) = −h u , χ i D ′ ( R , C ) , D ( R , C ) . It yields ( − iσ ∂ s − iε σ ∂ t ) w ∈ L ( R , C ) and looking at its Fourier transform we get w ∈ H ( R , C ) which gives w ∈ H ( Str , C ) (see [8, Prop. 9.18.]). Moreover, in D ′ ( Str , C ) thereholds ( − iσ ∂ s − iε σ ∂ t ) w = ( − iσ ∂ s − iε σ ∂ t ) u = − ( − iσ ∂ s − iε σ ∂ t ) u = − w. Now pick a sequence ( w n ) n ∈ N ∈ C ∞ ( Str , C ) converging to w in the H ( Str , C )-norm. Remarkthat in particular, we get kE ( ε, w n − w ) k L ( Str , C ) ≤ max(1 , ε − ) k∇ ( w n − w ) k L ( Str , C ) → , when n → + ∞ . Hence, there holds h w, w n i L ( Str , C ) = −h ( − iσ ∂ s − iε σ ∂ t ) w, w n i L ( Str , C ) = −h ( − iσ ∂ s − iε σ ∂ t ) w, w n i D ′ ( Str , C ) , D ( Str , C ) = − (cid:28) ( − iσ ∂ s − iε σ ∂ t ) w, ( − iσ ∂ s − iε σ ∂ t ) w n (cid:29) D ′ ( Str , C ) , D ( Str , C ) = −hE ( ε, w, E ( ε, w n i L ( Str , C ) . Letting n → + ∞ , we obtain w = 0 and thus u = ( iσ ∂ s + iε σ ∂ t ) w = 0, which concludes theproof. (cid:3) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 13
Next, we prove Proposition 13.
Proof of Proposition 13.
Let u ∈ dom ( E Γ ( ε, m )), remark that there holds kE Γ ( ε, m ) u k L ( Str , C ) = Z Str − εtκ ) | ∂ s u | dsdt + 1 ε Z Str | ∂ t u | dsdt + Z Str ε t κ ′ − εtκ ) | u | dsdt + m Z Str | u | dsdt + 2 ℜ (cid:18) h − εtκ ( − iσ ) ∂ s u, ε ( − iσ ) ∂ t u i L ( Str , C ) (cid:19)| {z } := I + 2 ℜ (cid:18) h − εtκ ( − iσ ) ∂ s u, εtκ ′ − εtκ ) ( − iσ ) u i L ( Str , C ) (cid:19)| {z } := I + m ℜ (cid:18) h − εtκ ( − iσ ) ∂ s u, σ u i L ( Str , C ) (cid:19)| {z } := I + 2 ℜ (cid:18) h ( − iσ ) ∂ t u, tκ ′ − εtκ ) ( − iσ ) u i L ( Str , C ) (cid:19)| {z } := I + mε ℜ (cid:0) h ( − iσ ) ∂ t u, σ u i L ( Str , C ) (cid:1)| {z } := I + m ℜ (cid:18) h εtκ ′ − εtκ ) ( − iσ ) u, σ u i L ( Str , C ) (cid:19)| {z } := I . If u ∈ C ∞ ( Str , C ) ∩ dom ( E Γ ( ε, m )) after integration by parts in the s and t -variables we get: h − εtκ ( − iσ ) ∂ s u, ε ( − iσ ) ∂ t u i L ( Str , C ) = −h εtκ ′ (1 − εtκ ) ( − iσ ) u, ε ( − iσ ) ∂ t u i L ( Str , C ) − h ( − iσ u ) , ε (1 − εtκ ) ( − iσ ) ∂ t ∂ s u i L ( Str , C ) = −h ε ( − iσ ) ∂ t u, − εtκ ( − iσ ) ∂ s u i L ( Str , C ) − h εtκ ′ (1 − εtκ ) ( − iσ ) u, ε ( − iσ ) ∂ t u i L ( Str , C ) − h κ (1 − εtκ ) ( iσ ) u, ∂ s u i L ( Str , C ) − Z R (cid:20) ε (1 − εtκ ) h ( − iσ ) u, ( − iσ ) ∂ s u i (cid:21) t = − ds In particular, it yields I = − I + 2 ℜ (cid:18) h κ − εtκ ) ( − iσ ) u, ∂ s u i L ( Str , C ) (cid:19) − Z R (cid:20) ε (1 − εtκ ) h ( − iσ ) u, ( − iσ ) ∂ s u i (cid:21) t = − ds. PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 14
Now, remark that the boundary conditions yield u ( s, ±
1) = ∓ u ( s, ± , ∂ s u ( s, ±
1) = ∓ ∂ s u ( s, ± . Thus, we obtain h ( − iσ ) u ( s, ± , ( − iσ ) ∂ s u ( s, ± i C = − i (cid:0) u ( s, ± ∂ s u ( s, ± − u ( s, ± ∂ s u ( s, ± (cid:1) = 0 . In particular, there holds k − εtκ ∂ s u k L ( Str , C ) + I + I = k − εtκ ( ∂ s − i κ σ ) u k L ( Str , C ) − Z Str κ − εtκ ) k u k C dsdt. Hence, we obtain kE Γ ( ε, m ) u k = Z Str − εtκ ) | ( ∂ s − i κ σ ) u | dsdt + 1 ε Z Str | ∂ t u | dsdt + Z Str ε t κ ′ − εtκ ) | u | dsdt + m Z Str | u | dsdt + I + I + I + I . Taking into account the boundary conditions, an integration by parts in the t -variable yields I = 2 ℜ (cid:0) h ( − iσ ) ∂ t u, σ u i L ( Str , C ) (cid:1) = Z R | u ( s, | + | u ( s, − | ds. Similarly, we have I = 2 ℜ (cid:18) h − εtκ ( − iσ ) ∂ s u, εtκ ′ − εtκ ) ( − iσ ) u i L ( Str , C ) (cid:19) = 2 ℜ (cid:18) h − εtκ ∂ s u, εtκ ′ − εtκ ) u i L ( Str , C ) (cid:19) = Z Str εtκ ′ − εtκ ) ∂ s ( | u | ) dsdt. However, an integration by part in the s -variable yields I = − Z Str (cid:16) εtκ ′′ − εtκ ) + 32 ( εtκ ′ ) (1 − εtκ ) (cid:17) | u | dsdt. Finally, a last integration by parts in the s -variable yields I = − I and combining the previousequations, we obtain the proposition. (cid:3) Self-adjointness.
In this paragraph we prove that D Γ ( ε, m ) is self-adjoint using the Kato-Rellich theorem (see, e.g., [18, Thm. 4.3.]). Proposition 15.
The operator D Γ ( ε, m ) is self-adjoint. Before going through the proof of Proposition 15, we need a few lemmata regarding the operator E ( ε, m ) introduced in (21). The first Lemma is a consequence of Proposition 13, taking into accountthat in this special case κ = 0 and m = 0. Lemma 16.
For all u ∈ dom ( E ( ε, , there holds kE ( ε, u k L ( Str , C ) = k ∂ s u k L ( Str , C ) + 1 ε k ∂ t u k L ( Str , C ) . Lemma 17.
The operator D Γ ( ε, m ) is symmetric.Proof of Lemma 17. Let u, v ∈ dom ( D Γ ( ε, m )), there holds: hD Γ ( ε, m ) u, v i L (Ω ε , C ) = h u, D Γ ( ε, m ) v i L (Ω ε , C ) + Z ∂ Ω ε h u, iσ · ν ε v i C ds ( x ) , (28) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 15 where ds ( x ) is the one-dimensional Hausdorff measure on ∂ Ω ε . Now, for all x ∈ ∂ Ω ε , there holds h u ( x ) , iσ · ν ε ( x ) v ( x ) i C = h u ( x ) , iσ · ν ε ( x )( − iσ σ · ν ε ( x )) v ( x ) i C = −h u ( x ) , σ v ( x ) i C = −h− iσ σ · ν ε ( x ) u ( x ) , σ v ( x ) i C = i h σ · ν ε ( x ) u ( x ) , v ( x ) i C = −h u ( x ) , iσ · ν ε ( x ) v ( x ) i C . Thus, the boundary term in (28) vanishes and D Γ ( ε, m ) is a symmetric operator. (cid:3) Proof of Proposition 15.
Instead of working with the operator D Γ ( ε, m ), we work with the unitarilyequivalent operator E Γ ( ε, m ) introduced in Proposition 3. Moreover, as the multiplication operatorby σ is bounded and self-adjoint in L ( Str , C ) we set m = 0 without loss of generality.Remark that dom ( E Γ ( ε, m )) = dom ( E ( ε, m )) where E ( ε, m ) is defined in (21) and that for u ∈ dom ( E Γ ( ε, E Γ ( ε, u = E ( ε, u + V ( ε ) , where the perturbation operator V ( ε ) is defined as V ( ε ) := εtκ − εtκ ( − iσ ) ∂ s + εtκ ′ − εtκ ) ( − iσ ) , dom ( V ( ε )) := dom ( E ( ε, . (29)Remark that V ( ε ) is a symmetric operator because V ( ε ) is the difference of two symmetric operators: E ( ε,
0) is self-adjoint thus symmetric (see Proposition 12) and E Γ ( ε,
0) is symmetric because it isunitarily equivalent to a symmetric operator (see Lemma 17 and Proposition 3).Now, remark that for u ∈ C ∞ ( Str , C ) ∩ dom ( E ( ε, k V ( ε ) u k L ( Str , C ) ≤ ε k κ k L ∞ ( R ) − ε k κ k L ∞ ( R ) k ∂ s u k L ( Str , C ) + ε k κ ′ k L ∞ ( R ) (cid:0) − ε k κ k L ∞ ( R ) (cid:1) k u k L ( Str , C ) . Using Lemma 16 we obtain k V ( ε ) u k L ( Str , C ) ≤ ε k κ k L ∞ ( R ) − ε k κ k L ∞ ( R ) kE ( ε, u k L ( Str , C ) + ε k κ ′ k L ∞ ( R ) (cid:0) − ε k κ k L ∞ ( R ) (cid:1) k u k L ( Str , C ) (30)and by density of C ∞ ( Str , C ) ∩ dom ( E ( ε, E ( ε, u ∈ dom ( E ( ε, ε k κ k L ∞ ( R ) − ε k κ k L ∞ ( R ) < . As V ( ε ) is symmetric and E ( ε, E ( ε, E Γ ( ε,
0) is self-adjoint. (cid:3)
Invariance of the essential spectrum.
In this paragraph we prove that the essential spec-trum of E Γ ( ε, m ) is the same as the one of E ( ε, m ). This is the purpose of the following proposition. Proposition 18.
There holds Sp ess ( D Γ ( ε, m )) = (cid:0) − ∞ , − p m + ε − E ( mε ) (cid:3) ∪ (cid:2)p m + ε − E ( mε ) , + ∞ (cid:1) . Proof of Proposition 18.
Instead of working with the operator D Γ ( ε, m ) we work with the unitarilyequivalent operator E Γ ( ε, m ). Our aim is to apply Weyl’s criterion [28, Thm. XIII.14] and for thispurpose we define W := ( E Γ ( ε, m ) + i ) − − ( E ( ε, m ) + i ) − and prove that W is a compact operator in L ( Str , C ). To this aim, given f, g ∈ L ( Str , C ) define u := ( E Γ ( ε, m ) + i ) − f, v := ( E ( ε, m ) − i ) − g . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 16
There holds hW f, g i L ( Str , C ) = h ( E Γ + i ) − f, g i L ( Str , C ) − h ( E + i ) − f, g i L ( Str , C ) = h u, g i L ( Str , C ) − h f, v i L ( Str , C ) = h u, ( E − i ) v i L ( Str , C ) − h ( E Γ + i ) u, v i L ( Str , C ) = − h V u, v i L ( Str , C ) = − h ( E − i ) − V ( E Γ + i ) − f, g i L ( Str , C ) , (31)where the perturbation V is defined in (29).Observe that V = a∂ s + ∂ s a , where a := 12 (cid:18) − εκt − (cid:19) ( − iσ ) . Then we get −W = ( E − i ) − V ( E Γ + i ) − = ( E − i ) − a ∂ s ( E Γ + i ) − + ( E − i ) − ∂ s a ( E Γ + i ) − . Here a ( E Γ + i ) − and ( E − i ) − a are compact operators in L ( Str , C ) due to hypothesis (A) (thelatter operator is compact because its adjoint a ( E + i ) − is compact). At the same time, ∂ s ( E Γ + i ) − and ( E − i ) − ∂ s are bounded operators in L ( Str , C ) (the latter operator is bounded because itsadjoint − ∂ s ( E + i ) − is bounded). Then the compactness of W follows by the well-known fact thatcompact operators are *-both-sided ideal in the space of bounded operators. (cid:3) Proof of Theorem 2.
We are now in a good position to prove Theorem 2.
Proof of Theorem 2.
Thanks to Proposition 15 and Proposition 18, the only thing left to prove isthe symmetry of the spectrum of D Γ ( ε, m ). It is a consequence of the invariance of the system undercharge conjugation, corresponding to the operator C := σ C where C is the complex conjugation operator. A straightforward computation shows that for all u ∈ dom ( D Γ ( ε, m )) we have C u ∈ dom ( D Γ ( ε, m )) and D Γ ( ε, m )( C u ) = − C D Γ ( ε, m ) u. In particular, any Weyl sequence ( u n ) n ∈ N associated with λ ∈ Sp ( D Γ ( ε, m )) generates a Weylsequence ( C u n ) n ∈ N associated with − λ which proves that the spectrum of D Γ ( ε, m ) is symmetricand concludes the proof of Theorem 2. (cid:3) Thin waveguide limit
In this section we prove Theorem 4, which deals with the thin waveguide limit ε →
0. We firstshow that, up to a renormalization, the operator E Γ ( ε, m ) defined in Proposition 3 converges to theone-dimensional Dirac operator (7) in the norm resolvent sense.The proof is achieved in two different steps. First, in Section 4.1, we deal with the case of astraight strip and then, in Section 4.2, we consider the curved waveguide.Roughly speaking, the main idea of the proof is to project onto the eigenfunctions of the transversepart of the operator. It turns out that after renormalization, all tranverse modes converge to zeroexcept the first positive and negative one. The operator E Γ ( ε, m ) restricted to these two modes isunitarily equivalent to a one-dimensional Dirac operator as defined in (7).4.1. Convergence for the straight strip.
For k ≥
1, let π k denote the projector in L (( − , , C )on the vector space span( u + k , u − k ), where u ± k are given in Corollary 11. Similarly, we consider theprojectors in L (( − , , C ) defined by p ± := {± x> } ( T ) where T is defined in Section 2.1. Theseprojectors can be extended to L ( Str , C ) setting for u ∈ L ( Str , C )Π k u := π k u, P ± u := p ± u. (32) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 17
For further use, we renormalize the operator E ( ε, m ) as follows C ( ε, m ) := E ( ε, m ) − π ε (cid:0) P + − P − (cid:1) . (33)To investigate the behavior of the resolvent operator ( C ( ε, m ) − i ) − in the thin waveguide regime ε →
0, we consider the unitary map U : L ( Str , C ) → Π L ( Str , C ) × Π ⊥ L ( Str , C ) , ( U v ) := (Π v, Π ⊥ v ) ⊤ , (34)and remark that there holds U ( C ( ε, m ) − i ) U − = (cid:18) C ( ε, m ) − i Π C ( ε, m )Π ⊥ Π ⊥ C ( ε, m )Π C ⊥ ( ε, m ) − i (cid:19) , (35)where we have set for all k ≥ C k ( ε, m ) = Π k C ( ε, m )Π k , C ⊥ ( ε, m ) := Π ⊥ C ( ε, m )Π ⊥ . (36)In the remaining part of this paragraph, we will make an extensive use of block operator matrixtheory to investigate (35) (see [31] for an extensive discussion).4.1.1. A few lemmata.
The first lemma is about the operators C k ( ε, m ) defined in (36). It statesthat they are unitarily equivalent to one-dimensional Dirac operators (see (7)). Lemma 19.
Let k ≥ and consider the unitary map U k : Π k L ( Str , C ) → L ( R , C ) defined by U k v := (cid:18) h v, u + k i L (( − , , C ) h v, u − k i L (( − , , C ) (cid:19) . There holds U k C k ( ε, m ) U − k = D D (( k − π ε + m e,k ) where m e,k := (cid:26) if k is even, kπ m if k is odd. In particular, there holds Sp ( C k ( ε, m )) = (cid:0) − ∞ , − ( k − π ε − m e,k (cid:3) ∪ (cid:2) ( k − π ε + m e,k , + ∞ (cid:1) . Proof of Lemma 19.
Let us pick f = (cid:18) f + f − (cid:19) ∈ H ( R , C ) and consider C k ( ε, m ) U − k f = C k ( ε, m )( f + u + k + f − u − k )= Π k (( − iσ ) ∂ s + 1 ε ( − iσ ) ∂ t + mσ )( f + u + k + f − u − k )= (cid:0) − i ( f + ) ′ (cid:1) u − k + (cid:0) − i ( f − ) ′ (cid:1) u + k + ( k − π ε (cid:16) f + u + k − f − u − k (cid:17) + m Π k ( f + σ u + k + f − σ u − k )= (cid:0) − i ( f + ) ′ (cid:1) u − k + (cid:0) − i ( f − ) ′ (cid:1) u + k + ( k − π ε (cid:16) f + u + k − f − u − k (cid:17) + mf + ( h σ u + k , u + k i L (( − , , C ) u + k + h σ u + k , u − k i L (( − , , C ) u − k )+ mf − ( h σ u − k , u + k i L (( − , , C ) u + k + h σ u − k , u − k i L (( − , , C ) u − k ) . However, using that σ u ± k = u ∓ k as well as the anti-commutation rules of the Pauli matrices we get h σ u + k , u + k i L (( − , , C ) = −h σ u − k , u − k i L (( − , , C ) , h σ u + k , u − k i L (( − , , C ) = −h σ u − k , u + k i L (( − , , C ) . Now, a simple computation gives h σ u + k ( t ) , u − k ( t ) i C = 0 , h σ u + k , u + k i L (( − , , C ) = (cid:26) k is even, kπ if k is odd, (37) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 18 and we set m e,k := m h σ u + k , u + k i L (( − , , C ) . In particular, there holds C k ( ε, m ) U − k f = (cid:0) − i ( f + ) ′ (cid:1) u − k + (cid:0) − i ( f − ) ′ (cid:1) u + k + ( k − π ε (cid:16) f + u + k − f − u − k (cid:17) + m e,k f + u + k − m e,k f − u + k , so that U k C k ( ε, m ) U − k f = (cid:16) − iσ dds + ( m e,k + ( k − π ε ) σ (cid:17) f = D D ( m e,k + ( k − π ε ) f , (38)and the claim follows. (cid:3) Remark 20.
Notice that for k = 1, the one-dimensional Dirac operator in (38) does not dependon ε , that is U C ( ε ) U − = (cid:16) − iσ dds + 2 π mσ (cid:17) = D D (2 π − m ) . The next lemma concerns the off-diagonal operators Π j E ( ε, m )Π k for j, k ∈ N and j = k . Lemma 21.
Let k, j ≥ such that j = k . The operator Π j E ( ε, m )Π k satisfies for all u ∈ dom ( E ( ε, m )) : Π j E ( ε, m )Π k u = m Π j σ Π k u. Hence Π j E ( ε, m )Π k can be extended uniquely into a bounded operator in L ( Str , C ) with sameoperator norm.Proof of Lemma 21. Let v ∈ dom ( E ( ε, m )) and k, j ≥ k = j . Set Π k v = f + u + k + f − u − k ∈ dom ( E ( ε, m )), there holdsΠ j E ( ε, m )Π k v = Π j (cid:16) ( − i ( f + ) ′ ) u − k + ( − i ( f − ) ′ ) u + k + k π ε ( f + u + k − f − u − k ) (cid:17) + m Π j σ Π k v = m Π j σ Π k v. As m Π j σ Π k is a bounded operator in L ( Str , C ) and dom ( E ( ε, m )) is dense in L ( Str , C ) wededuce that Π j E ( ε, m )Π k can be extended uniquely to a bounded operator in L ( Str , C ) and thisoperator acts as m Π j σ Π k . (cid:3) Proposition 22.
Let C ⊥ ( ε, m ) be the operator defined in (36) . The operator C ⊥ ( ε, m ) − i acting in Π ⊥ L ( Str , C ) is boundedly invertible and there exists C > and ε > such that for all ε ∈ (0 , ε ) there holds k ( C ⊥ ( ε, m ) − i ) − k B (Π ⊥ L ( Str , C )) ≤ Cε.
Remark 23.
In Proposition 22, we used the notation B ( H ) which for a complex Hilbert-space H stands for the space of bounded operators on H . Similarly, if H and H are two complex Hilbertspaces B ( H , H ) denotes the set of bounded operators from H to H . Proof of Proposition 22.
First, remark that C ⊥ ( ε, m ) is a self-adjoint operator when acting inΠ ⊥ L ( Str , C ) with domain Π ⊥ dom ( E ( ε, m )). Hence, the operator C ⊥ ( ε, m ) − i is boundedlyinvertible in Π ⊥ L ( Str , C ). Second, observe that on Π ⊥ L ( Str , C ) there holds C ⊥ ( ε, m ) = (cid:0) X j ≥ Π j (cid:1) C ( ε, m ) (cid:0) X k ≥ Π k (cid:1) = X j ≥ (Π j C ( ε, m )Π j ) + X j,k ≥ j = k (Π j C ( ε, m )Π k )= X j ≥ (Π j C ( ε, m )Π j ) | {z } := G ( ε,m ) + m X j,k ≥ j = k Π j σ Π k | {z } := B , PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 19 where we have used Lemma 21 in the last equation, observing thatΠ j C ( ε, m )Π k = Π j E ( ε, m )Π k , if j = k .Remark that, as defined, the operator G ( ε, m ) is self-adjoint and B ∈ B (Π ⊥ L ( Str , C )). Indeed, wehave X j,k ≥ j = k Π j σ Π k = X j ≥ Π j σ X k ≥ j = k Π k = X j ≥ Π j σ (Π ⊥ − Π j ) = Π ⊥ σ Π ⊥ − X j ≥ Π j σ Π j . Now, the first term on the right-hand side is a bounded operator in Π ⊥ L ( Str , C ) while for thesecond we can argue as follows. Let u ∈ Π ⊥ L ( Str , C ), there holds (cid:13)(cid:13)(cid:13) X j ≥ Π j σ Π j u (cid:13)(cid:13)(cid:13) ⊥ L ( Str , C ) = X j ≥ k Π j σ Π j u k ⊥ L ( Str , C ) ≤ X j ≥ k Π j u k ⊥ L ( Str , C ) = k u k ⊥ L ( Str , C ) . (39)Moreover, we have( C ⊥ ( ε, m ) − i ) − = ( G ( ε, m ) − i ) − (cid:0) mB ( G ( ε, m ) − i ) − (cid:1) − . (40)Now, we need to estimate k ( G ( ε, m ) − i ) − k B (Π ⊥ L ( Str , C )) = dist( i, Sp ( G ( ε, m ))) − . Recall that byconstruction we have G ( ε, m ) = M k ≥ C k ( ε, m ) , see [28, p. 268] for the definition of the direct sum of self-adjoint operators. In particular, by [28,Thm. XIII.85], there holds Sp ( G ( ε, m )) = [ k ≥ Sp ( C k ( ε, m )) = (cid:0) − ∞ , − π ε (cid:3) ∪ (cid:2) π ε , + ∞ (cid:1) . Indeed, thanks to Lemma 19 for all k ≥ Sp ( C k ( ε, m )) = (cid:0) − ∞ , − ( k − π ε − m e,k (cid:3) ∪ (cid:2) ( k − π ε + m e,k , + ∞ (cid:1) and for all k ≥ k ≥ n m e,k + ( k − π ε o = π ε . Hence, we get dist( i, Sp ( G ( ε, m ))) = q π ε and we obtain k ( G ( ε, m ) − i ) − k B (Π ⊥ L ( Str , C )) = 1 q π ε . In particular, we get k ( G ( ε, m ) − i ) − k B (Π ⊥ L ( Str , C )) = 4 π ε + O ( ε ) , when ε → . (41)Next, remark that by (39) there holds k B k B (Π ⊥ L ( Str , C )) ≤ k (cid:0) mB ( G ( ε, m ) − i ) − (cid:1) − k B (Π ⊥ L ( Str , C )) = 1 + O ( ε ) , when ε → . (42)Finally, combining (41) and (42), (40) yields k ( C ⊥ − i ) − k B (Π ⊥ L ( Str , C )) ≤ π ε + O ( ε ) , when ε → . It concludes the proof of Proposition 22. (cid:3)
PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 20
Proof of Theorem 4 in the case of the straight strip.
In this paragraph we prove Theorem 4in the special case of a straight strip but first, we need the next proposition whose proof is a directapplication of block operator matrices theory.
Proposition 24.
Recall that U is the unitary map defined in (34) . There holds U ( C ( ε, m ) − i ) − U − = (cid:18) C , ( ε, m ) C , ( ε, m ) C , ( ε, m ) C , ( ε, m ) (cid:19) where C , ( ε, m ) := ( C ( ε, m ) − i ) − + m ( C ( ε, m ) − i ) − Π σ Π ⊥ S ( i ) − Π ⊥ σ Π ( C ( ε, m ) − i ) − ,C , ( ε, m ) := − m ( C ( m, ε ) − i ) − Π σ Π ⊥ S ( i ) − ,C , ( ε, m ) := − m S ( i ) − Π ⊥ σ Π ( C ( ε, m ) − i ) − ,C , ( ε, m ) := S ( i ) − . Here, S ( i ) denotes the Schur complement: S ( i ) := C ⊥ ( ε, m ) − i − m Π ⊥ σ Π ( C ( ε, m ) − i ) − Π σ Π ⊥ . (43) Proof of Proposition 24.
According to the notation of [31, Thm. 2.3.3], we set A := C ( ε, m ) , B := m Π σ Π ⊥ , C := m Π ⊥ σ Π , D := C ⊥ ( ε, m ) , where we have used Lemma 21 to rewrite the operators B and C .Now, we check all the hypothesis of [31, Thm. 2.3.3]: • dom ( A ) = Π dom ( E ( ε, m )) ⊂ dom ( C ) = Π L ( Str , C ), • A is self-adjoint as an operator acting in Π L ( Str , C ) thus i / ∈ Sp ( A ), • as A is self-adjoint and B is bounded, the operator ( A − i ) − B is bounded in Π ⊥ L ( Str , C ), • the operator S ( i ) is closed because D is self-adjoint and the operator Π ⊥ σ Π ( A − i ) − Π σ Π ⊥ ∈ B (Π ⊥ L ( Str , C )) (hence both are closed).Thus, [31, Thm. 2.3.3] yields U ( C ( ε, m ) − i ) − U − = (cid:18) C , ( ε, m ) C , ( ε, m ) C , ( ε, m ) C , ( ε, m ) (cid:19) with C , ( ε, m ) := ( A − i ) − (cid:16) + B S ( i ) − C ( A − i ) − (cid:17) C , ( ε, m ) := − ( A − i ) − B S ( i ) − C , ( ε, m ) := −S ( i ) − C ( A − i ) − C , ( ε, m ) := S ( i ) − . This finishes the proof. (cid:3)
We are now in a good position to prove (8) in Theorem 4 for the straight waveguide.
Proposition 25.
There exists a unitary map V such that V : L ( Str , C ) → L ( R , C ) ⊕ Π ⊥ L ( Str , C ) and there holds V (cid:0) E ( ε, m ) − π ε ( P + − P − ) − i (cid:1) − V − = (cid:0) D D (2 π − m ) − i (cid:1) − ⊕ O ( ε ) , in the operator norm, where P ± are the projectors defined in (32) .Proof of Proposition 25. The proof is performed in three steps. In the first two steps we estimatethe norm of the bounded operators ( C ( ε, m ) − i ) − and the Schur complement S ( i ) − (defined in(43)). In the last step, we use Proposition 24 to obtain an asymptotic expansion of the operator U ( C ( ε, m ) − i ) − U − . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 21
Step 1.
Thanks to Lemma 19, we know that Sp ( C ( ε, m )) = ( −∞ , − π m ] ∪ [ π m, + ∞ ). In particular,there holds k ( C ( ε, m ) − i ) − k B (Π L ( Str , C )) = 1dist( i, Sp ( C ( ε, m ))) = 1 q π m . (44) Step 2.
Remark that there holds S ( i ) − = (cid:0) − m ( C ⊥ ( ε, m ) − i ) − Π ⊥ σ Π ( C ( ε, m ) − i ) − Π σ Π ⊥ (cid:1) − ( C ⊥ ( ε, m ) − i ) − and in particular, we have k ( C ⊥ ( ε, m ) − i ) − Π ⊥ σ Π ( C ( ε, m ) − i ) − Π σ Π ⊥ k B (Π ⊥ L ( Str , C )) ≤ k ( C ⊥ ( ε, m ) − i ) − k B (Π ⊥ L ( Str , C )) k ( C ( ε, m ) − i ) − k B (Π L ( Str , C )) ≤ C q π m ε := ˜ Cε, when ε → . Here, the first inequality is obtained using that σ is a unitary operator from L ( Str , C ) onto itselfand that Π and Π ⊥ , being orthogonal projectors, are bounded operators with norm smaller than 1.The second inequality is a consequence of (44) and Proposition 22. In particular, using a Neumannseries and Proposition 22, it yields the existence of C ′ > ε > ε ∈ (0 , ε )there holds kS ( i ) − k B (Π ⊥ L ( Str , C )) ≤ C ′ ε. (45) Step 3.
Thanks to Proposition 24 there holds U ( C ( ε, m ) − i ) − U − = (cid:18) ( C ( ε, m ) − i ) −
00 0 (cid:19) + (cid:18) R , ( ε, m ) C , ( ε, m ) C , ( ε, m ) C , ( ε, m ) (cid:19) , where we have set R , ( ε, m ) = m ( C ( ε, m ) − i ) − Π σ Π ⊥ S ( i ) − Π ⊥ σ Π ( C ( ε, m ) − i ) − . Now, we examine the norm of each bounded operator appearing in the second block matrix on theright-hand side. Remark that by (44) and (45), for all ε ∈ (0 , ε ) there holds k R , ( ε, m ) k B (Π L ( Str , C )) ≤ m k ( C ( ε, m ) − i ) − k B (Π L ( Str , C )) kS ( i ) − k B (Π ⊥ L ( Str , C )) ≤ m C ′ π m ε. (46)Similarly, for ε ∈ (0 , ε ) there holds k C , ( ε, m ) k B (Π ⊥ L ( Str , C ) , Π L ( Str , C )) ≤ m ˜ C q π m ε (47)and k C , ( ε, m ) k B (Π L ( Str , C ) , Π ⊥ L ( Str , C )) ≤ m ˜ C q π m ε. (48)Gathering (46), (47), (48) and (45) we get U ( C ( ε, m ) − i ) − U − = (cid:18) ( C ( ε, m ) − i ) −
00 0 (cid:19) + O ( ε ) , when ε → . To conclude, we introduce the unitary map V : L ( Str , C ) → L ( R , C ) ⊕ Π ⊥ ( L ( Str , C )) , ( V u ) := (cid:0) U Π u, Π ⊥ u (cid:1) , PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 22 where the unitary map U is defined in Lemma 19. When ε →
0, there holds V ( C ( ε, m ) − i ) − V − = (cid:18) U ( C ( ε, m ) − i ) − U −
00 0 (cid:19) + O ( ε )= (cid:18) ( D D (2 π − m ) − i ) −
00 0 (cid:19) + O ( ε )= ( D D (2 π − m ) − i ) − ⊕ O ( ε ) . (cid:3) Convergence for the curved waveguide.
This paragraph is devoted to the proof of Theo-rem 4. Once again, we use a perturbation argument. We start with a few auxiliary results.The first lemma deals with the quadratic form q m ( u ) := k ( − iσ ) u ′ k L (( − , , C ) + m (cid:0) k u (1) k L (( − , , C ) + k u ( − k L (( − , , C ) (cid:1) , dom ( q m ) := { u = ( u , u ) ⊤ ∈ H (( − , , C ) : u ( ±
1) = ∓ u ( ± } . (49)Remark that q m is the quadratic form associated with the operator T (0 , m ) − m , where T (0 , m )is defined in (14), as can be seen in (15) and below. Lemma 26.
Let u ∈ dom ( q m ) , there holds q m ( u ) ≥ E ( m ) k π u k L (( − , , C ) + q ( π ⊥ u ) , where the projector π is defined in § π ⊥ = − π .Proof of Lemma 26. Let u ∈ dom ( q m ), there holds q m ( u ) = q m ( π u + π ⊥ u ) = q m ( π u ) + q m ( π ⊥ u ) + 2 ℜ ( q m ( π u, π ⊥ u )) ≥ E ( m ) k π u k L (( − , , C ) + q ( π ⊥ u )+ 2 ℜ ( q m ( π u, π ⊥ u )) , (50)where we have used the min-max principle (Proposition 6) and bounded from below the quadraticform q m by q . Now, remark that for all v ∈ dom ( q m ) there holds q m ( v ) = k ( − iσ ) v ′ + mσ v k L ( Str , C ) − m k v k L ( Str , C ) . In particular, for the associated sesquilinear form it gives q m ( π v, π ⊥ v ) = h (cid:0) ( − iσ ) ddt + mσ (cid:1) π v, (cid:0) ( − iσ ) ddt + mσ (cid:1) π ⊥ v i L (( − , , C ) − m h π v, π ⊥ v i L (( − , , C ) = m (cid:16) hT π v, σ π ⊥ v i L (( − , , C ) + h σ π v, T π ⊥ v i L (( − , , C ) (cid:17) . Now, remark that hT π v, σ π ⊥ v i L (( − , , C ) = hT v, π σ π ⊥ v i L (( − , , C ) , h σ π v, T π ⊥ v i L (( − , , C ) = h π ⊥ σ π v, T u i L (( − , , C ) . If v ∈ dom ( q m ) = dom ( T ), then π ⊥ σ π u ∈ dom ( T ) and as T is self-adjoint there holds hT v, π σ π ⊥ v i L (( − , , C ) = h v, T π σ π ⊥ v i L (( − , , C ) = −h π ⊥ σ π v, T v i L (( − , , C ) , where we have used that T commutes with π and π ⊥ and that σ anti-commutes with σ . Inparticular, we obtain that q m ( π u, π ⊥ u ) = 0 which combined with equation (50) yields q m ( u ) ≥ E ( m ) k π u k L (( − , , C ) + q ( π ⊥ u ) , which is precisely Lemma 26. (cid:3) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 23
Lemma 27.
Let u ∈ dom ( E ( ε, m )) , there exists ε > and K > such that for all ε ∈ (0 , ε ) there holds k ( − iσ ) ∂ s u + mσ u k L ( Str , C ) ≤ kC ( ε, m ) u k L ( Str , C ) + K k u k L ( Str , C ) , where the operator C ( ε, m ) is defined in (33) .Proof of Lemma 27. Let u ∈ dom ( E ( ε, m )) and remark that there holds kC ( ε, m ) u k L ( Str , C ) = k ( − iσ ) ∂ s u + mσ u k L ( Str , C ) + 1 ε k ( − iσ ) ∂ t u − π P + − P − ) u k L ( Str , C ) | {z } := A + 1 ε ℜ ( h ( − iσ ) ∂ s u, ( − iσ ) ∂ t u − π P + − P − ) u i L ( Str , C ) ) | {z } := B + mε ℜ ( h σ u, ( − iσ ) ∂ t u i L ( Str , C ) ) | {z } := C − mπ ε ℜ ( h σ u, ( P + − P − ) u i L ( Str , C ) ) | {z } := D . (51)Now, we deal with each term appearing on the right-hand side of (51). For further use, for all k ≥
1, we set f ± k := h u, u ± k i L (( − , , C ) and recall that Π k denotes the projector defined in (32). Inparticular, for all k ≥
1, there holds k Π k u k L ( Str , C ) = Z R (cid:16) | f + k ( s ) | + | f − k ( s ) | (cid:17) ds Step 1.
In this step, we analyze the term A appearing in (51). We remark that( − iσ ∂ t − π P + − P − )) u = X k ≥ ( k − π f + k u + k − f − k u − k ) . (52)In particular, it gives A = π X k ≥ ( k − k Π k u k L ( Str , C ) . (53) Step 2.
A straightforward computation gives − iσ ∂ s u = X k ≥ − i ( f − k ) ′ u + k − i ( f + k ) ′ u − k . In particular, using (52), there holds h− iσ ∂ s u, (cid:0) − iσ ∂ t − π P + − P − ) (cid:1) u i L ( Str , C ) = π X k ≥ ( k − (cid:0) − i Z R ( f − k ) ′ ( s ) f + k ( s ) ds + i Z R ( f + k ) ′ ( s ) f − k ( s ) ds (cid:1) . (54)Integrating by parts, we find − i Z R ( f − k ) ′ ( s ) f + k ( s ) ds + i Z R ( f + k ) ′ f − k ( s ) ds = i Z R ( f − k ) ′ ( s ) f + k ( s ) ds − i Z R ( f + k ) ′ ( s ) f − k ( s ) ds = − (cid:18) − i Z R ( f − k ) ′ ( s ) f + k ( s ) ds + i Z R ( f + k ) ′ ( s ) f − k ( s ) ds (cid:19) , PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 24 and then using (54) we get h− iσ ∂ s u, (cid:0) − iσ ∂ t − π P + − P − ) (cid:1) u i L ( Str , C ) = −h (cid:0) − iσ ∂ t − π P + − P − ) (cid:1) u, − iσ ∂ s u i L ( Str , C ) . In particular, we obtain B = 2 ℜ ( h− iσ ∂ s u, (cid:0) − iσ ∂ t − π P + − P − ) (cid:1) u i L ( Str , C ) ) = 0 . (55) Step 3.
In this step we deal with the term C . Integrating by parts as in (16), we obtain: C = Z R | u ( s, | + | u ( s, − | ds. (56) Step 4.
It remains to deal with the term D . To do so we remark that: h σ u, ( P + − P − ) u i L ( Str , C ) = h Π σ Π u, ( P + − P − ) u i L ( Str , C ) | {z } := α + h Π ⊥ σ Π ⊥ u, ( P + − P − ) u i L ( Str , C ) | {z } := β + h Π σ Π ⊥ u, ( P + − P − ) u i L ( Str , C ) | {z } := γ + h Π ⊥ σ Π u, ( P + − P − ) u i L ( Str , C ) | {z } := δ . (57)Now, in each of the next substep, we deal with the terms appearing on the right-hand side of (57). Substep 4.1 . Remark that there holds α = h f +1 σ u +1 + f − σ u − , f +1 u +1 − f − u − i L ( Str , C ) = h σ u +1 , u +1 i L (( − , , C ) k f +1 k L ( R ) − h σ u − , u − i L (( − , , C ) k f − k L ( R ) − h σ u +1 , u − i L (( − , , C ) h f +1 , f − i L ( R ) + h σ u − , u +1 i L (( − , , C ) h f − , f +1 i L ( R ) . Thanks to (37) we get α = 2 π k Π u k L ( Str , C ) . (58) Substep 4.2 . We handle the term β by obtaining the following upper-bound thanks to the Cauchy-Schwarz inequality: | β | = |h Π ⊥ σ Π ⊥ u, ( P + − P − ) u i L ( Str , C ) | ≤ k Π ⊥ u k L ( Str , C ) . (59) Substep 4.3 . Now, let us focus on the two off-diagonal terms γ and δ . We only deal with γ , thecomputations for δ being similar.A direct computation shows that h σ u − k , u +1 i C = −h σ u + k , u − i C , h σ u − k , u − i C = −h σ u + k , u +1 i C . Then we get Π σ Π ⊥ u = (cid:0) X k ≥ a k f + k − b k f − k (cid:1) u +1 + (cid:0) X k ≥ b k f + k − a k f − k (cid:1) u − , where we have set for k ≥ a k := h σ u + k , u +1 i L (( − , , C ) = 4 π sin ( π ( k + 1))( k + 1) , (60) b k := h σ u + k , u − i L (( − , , C ) = 4 π sin ( π ( k − k − . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 25
Thus, we find γ = X k ≥ Z R h ( a k − σ b k ) (cid:18) f + k f − k (cid:19) , (cid:18) f +1 f − (cid:19) i C ds. (61)A similar computation gives δ = X k ≥ Z R h (cid:18) f +1 f − (cid:19) , ( a k + σ b k ) (cid:18) f + k f − k (cid:19) i C ds. (62)In particular, using (61) and (62) we get γ + δ = 2 ℜ (cid:16) X k ≥ a k Z R h (cid:18) f +1 f − (cid:19) , (cid:18) f + k f − k (cid:19) i C ds (cid:17) + 2 i ℑ (cid:16) X k ≥ b k Z R h (cid:18) f +1 f − (cid:19) , σ (cid:18) f + k f − k (cid:19) i C ds (cid:17) . (63)Using (57), (58) and (63) we obtain D = 4 π k Π u k L ( Str , C ) + 2 ℜ ( β ) + 4 ℜ (cid:16) X k ≥ a k Z R h (cid:18) f +1 f − (cid:19) , (cid:18) f + k f − k (cid:19) i C ds (cid:17) . In particular, using the Cauchy-Schwartz inequality we get D ≤ π k Π u k L ( Str , C ) + 2 | β | + 4 X k ≥ (cid:16) | a k |k Π u k L ( Str , C ) k Π k u k L ( Str , C ) (cid:17) . (64)Now, let us fix c > a, b ∈ R and ε >
0, we recall the elementaryinequality ab ≤ cε a + cε b that we use to get for all k ≥ | a k |k Π u k L ( Str , C ) k Π k u k L ( Str , C ) ≤ cε a k k Π u k L ( Str , C ) + 12 cε k Π k u k L ( Str , C ) . Then, summing up for k ≥
2, we get X k ≥ (cid:16) | a k |k Π u k L ( Str , C ) k Π k u k L ( Str , C ) (cid:17) ≤ cεS k Π u k L ( Str , C ) + 12 cε k Π ⊥ u k L ( Str , C ) , (65)where we have set S = P k ≥ a k < + ∞ because a k = O ( k − ) when k → + ∞ by (60). Taking intoaccount (59) and (65), (64) gives D ≤ ( 4 π + 2 cSε ) k Π u k L ( Str , C ) + 2(1 + 1 cε ) k Π ⊥ u k L ( Str , C ) . (66) Step 5.
In this step we conclude the proof. Using (53), (55) and (56), (51) becomes kC ( ε, m ) u k L ( Str , C ) = k ( − iσ ∂ s + mσ ) u k L ( Str , C ) + π ε X k ≥ ( k − k Π k u k L ( Str , C ) + mε Z R (cid:16) | u ( s, | + | u ( s, − | (cid:17) ds − mπ ε D = k ( − iσ ∂ s + mσ ) u k L ( Str , C ) + π ε X k ≥ ( k − k Π k u k L ( Str , C ) + 1 ε ( q mε ( u ) − q ( u )) − mπ ε D = k ( − iσ ∂ s + mσ ) u k L ( Str , C ) + π ε X k ≥ ( k − k Π k u k L ( Str , C ) − π ε k Π u k L ( Str , C ) + 1 ε ( q mε ( u ) − q (Π ⊥ u )) − mπ ε D, PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 26 where the quadratic forms q εm and q are defined in (49). Notice that in the above formula we usedthe fact that q ( u ) = q (Π u ) + q (Π ⊥ u ) = π k Π u k L ( Str , C ) + q (Π ⊥ u ) . Using Lemma (26), this last inequality becomes kC ( ε, m ) u k L ( Str , C ) ≥ k ( − iσ ∂ s + mσ ) u k L ( Str , C ) + π ε k Π ⊥ u k L ( Str , C ) + 1 ε (cid:16) E ( mε ) − π (cid:17) k Π u k L ( Str , C ) − mπ ε D and (66) yields kC ( ε, m ) u k L ( Str , C ) ≥ k ( − iσ ∂ s + mσ ) u k L ( Str , C ) + 1 ε (cid:16) π − mπ c − mπ ε (cid:17) k Π ⊥ u k L ( Str , C ) + 1 ε (cid:16) E ( mε ) − π − mε − mπcS ε (cid:17) k Π u k L ( Str , C ) . (67)Now, we choose c > mπ and remark that there exists ε > ε ∈ (0 , ε ) there holds π − mπ c − mπ ε > . (68)Moreover, thanks to (iv) of Proposition 10, there exists ε and K > ε ∈ (0 , ε ) E ( mε ) − π − mε − mπcS ε > − Kε . (69)Setting ε := min( ε , ε ) and taking into account (68) and (69) in (67) we obtain that for all ε ∈ (0 , ε ) there holds K k Π u k L ( Str , C ) + kC ( ε, m ) u k L ( Str , C ) ≥ k ( − iσ ∂ s + mσ ) u k L ( Str , C ) . The proof of Lemma 27 is completed remarking that k Π u k L ( Str , C ) ≤ k u k L ( Str , C ) . (cid:3) We are now in a good position to prove Theorem 4.
Proof of Theorem 4.
Let us set C Γ ( ε, m ) := E Γ ( ε, m ) − π ε ( P + − P − ) . and remark that C Γ ( ε, m ) = C ( ε, m ) + V ( ε ) , where C ( ε, m ) is defined in (33) and the symmetric operator V ( ε ) is defined in (29).Consider the operator( C Γ ( ε, m ) − i ) − = ( C ( ε, m ) − i + V ( ε )) − = ( C ( ε, m ) − i ) − (cid:0) + V ( ε )( C ( ε, m ) − i ) − (cid:1) − . We claim that there exists ε > K ′ > ε ∈ (0 , ε ) there holds k V ( ε )( C ( ε, m ) − i ) − k B ( L ( Str , C )) ≤ K ′ ε. Indeed, for u ∈ L ( Str , C ), there holds k V ( ε )( C ( ε, m ) − i ) − u k L ( Str , C ) ≤ ε k κ k L ∞ ( R ) − ε k κ k L ∞ ( R ) k ( − iσ ) ∂ s ( C ( ε, m ) − i ) − u k L ( Str , C ) + ε k κ ′ k L ∞ ( R ) − ε k κ k L ∞ ( R ) ) k ( C ( ε, m ) − i ) − k B ( L ( Str , C )) k u k L ( Str , C ) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 27
One remarks that k ( − iσ ) ∂ s ( C ( ε, m ) − i ) − u k L ( Str , C ) = k (cid:0) ( − iσ ) ∂ s + mσ − mσ (cid:1) ( C ( ε, m ) − i ) − u k L ( Str , C ) ≤ k (cid:0) ( − iσ ) ∂ s + mσ (cid:1) ( C ( ε, m ) − i ) − u k L ( Str , C ) + m k ( C ( ε, m ) − i ) − u k L ( Str , C ) . Hence, by Lemma 27, there exists
K > ε > ε ∈ (0 , ε ) there holds: k ( − iσ ) ∂ s ( C ( ε, m ) − i ) − u k L ( Str , C ) ≤ kC ( ε, m )( C ( ε, m ) − i ) − u k L ( Str , C ) + ( m + K ) k ( C ( ε, m ) − i ) − u k L ( Str , C ) ≤ k ( C ( ε, m ) − i )( C ( ε, m ) − i ) − u k L ( Str , C ) + (1 + m + K ) k ( C ( ε, m ) − i ) − u k L ( Str , C ) ≤ k u k L ( Str , C ) + ( m + 1 + K ) k ( C ( ε, m ) − i ) − k B ( L ( Str , C )) k u k L ( Str , C ) . Remarking that k ( C ( ε, m ) − i ) − k B ( L ( Str , C )) = dist( i, Sp ( C ( ε, m ))) − ≤ ε ∈ (0 , ε ) such that k V ( ε )( C ( ε, m ) − i ) − u k L ( Str , C ) ≤ K ′ ε k u k L ( Str , C ) , for some constant K ′ >
0. Thus, developing in Neumann series and using (70) we get( C Γ ( ε, m ) − i ) − = ( C ( ε, m ) − i ) − + O ( ε )and the theorem is proved applying Proposition 25. (cid:3) Non-relativistic limit
This section is devoted to the proof of Proposition 8 and Theorem 9. In the sequel we will assume ε > m → + ∞ . We start by proving Proposition8 before turning to the proof of Theorem 9.5.1. Proof of Proposition 8.
We start by computing the quadratic form associated with theoperator D Γ ( ε, m ) . Lemma 28.
Given u ∈ dom ( D Γ ( ε, m )) , there holds kD Γ ( ε, m ) u k L (Ω ε , C ) = k∇ u k L (Ω ε , C ) + m k u k L (Ω ε , C ) + Z ∂ Ω ε ( m − κ ε | u | ds , where κ ε is the signed curvature of the boundary ∂ Ω ε with respect to the outer normal ν ε .Proof of Lemma 28. Recalling Definition (4), we get kD Γ ( ε, m ) u k L ( Str , C ) = k ( − iσ · ∇ ) u k L ( Str , C ) + m k u k L ( Str , C ) + 2 m ℜ (cid:0) h σ u, − iσ · ∇ u i L (Ω ε , C ) (cid:1) . (71)Thanks to the boundary conditions in (4), arguing as in the proof of [26, Prop. 3.3, 3.5] one finds k ( − iσ · ∇ ) u k L ( Str , C ) = k∇ u k L ( Str , C ) − Z ∂ Ω ε κ ε | u | ds . (72)Let us turn to the last term in (71). Observe that σ ( σ · ∇ ) = − ( σ · ∇ ) σ , so that integrating byparts we get h σ u, − iσ · ∇ u i L (Ω ε , C ) = −h− iσ · ∇ u, σ u i L (Ω ε , C ) + h σ u, − iσ · ν ε u i L ( ∂ Ω ε , C ) . Then, taking into account the boundary conditions given in (4), we find2 m ℜ (cid:16) h σ u, − iσ · ∇ u i L (Ω ε , C ) (cid:17) = m h σ u, − iσ · ν ε u i L ( ∂ Ω ε , C ) = m k u k L ( ∂ Ω ε , C ) , (73) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 28 and Lemma 28 follows combining (72) and (73) in (71). (cid:3)
Let us introduce the quadratic forms q m ( u ) := kD Γ ( ε, m ) u k L (Ω ε , C ) − m k u k L (Ω ε , C ) , dom ( q m ) := dom ( D Γ ) , and q ∞ ( u ) := k∇ u k L (Ω ε , C ) , dom ( q ∞ ) := H (Ω ε , C ) . Notice that q ∞ is the quadratic form of the Dirichlet Laplacian L Γ ( ε ) defined in (10). In the followingwe shall consider the min-max values of the forms above, as introduced in Definition 5. We are nowin a good position to prove Proposition 8. Proof of Proposition 8.
Observe that dom ( q ∞ ) ⊂ dom ( q m ) and that by Lemma 28 if u ∈ dom ( q ∞ )we have q ∞ ( u ) = q m ( u ). Then, by Proposition 6, we immediately get for all j ∈ N : µ j ( q m ) ≤ µ j ( q ∞ ) . (74)Recall that by Theorem 2, ε − E ( mε ) is the bottom of the essential spectrum of D Γ ( ε, m ) − m .Now, fix j ∈ N with j < N Γ + 1 (with the convention that N Γ + 1 = + ∞ if N Γ = + ∞ ). Then, byProposition 6, (v) of Proposition 10 and Proposition 7, we get for all j ∈ { , . . . , j } : µ j ( q m ) − E ( mε ) ε ≤ µ j ( q ∞ ) − E ( mε ) ε ≤ µ j ( q ∞ ) − π ε | {z } < + Cm , for some constant
C >
0. Then the claim follows taking m large enough. (cid:3) Finite waveguides.
In our way to prove Theorem 9, we need to investigate the min-maxvalues of quadratic forms in finite waveguides. To this aim, for
R > ε into the following three domainsΩ Rε := { γ ( s ) + εtν ( s ) : | s | < R , t ∈ ( − , } , Ω R, ± ε := { γ ( s ) + εtν ( s ) : ± s > R , t ∈ ( − , } , and consider the following four forms: q R ∞ ( u ) := k∇ u k L (Ω Rε , C ) , dom (cid:0) q R ∞ (cid:1) := H (Ω Rε , C ) ,q Rm ( u ) := kD Γ ( ε, m ) u k L (Ω Rε , C ) − m k u k L (Ω Rε , C ) , dom (cid:0) q Rm (cid:1) := (cid:8) u ∈ H (Ω Rε , C ) : − iσ σ · ν ε u = u on ∂ Ω Rε ∩ ∂ Ω ε ,u = 0 on ∂ Ω Rε \ ∂ Ω ε (cid:9) ,q R, ± m ( u ) := kD Γ ( ε, m ) u k L (Ω R, ± ε , C ) − m k u k L (Ω R, ± ε , C ) , dom (cid:0) q R, ± m (cid:1) := (cid:8) u ∈ H (Ω R, ± ε , C ) : − iσ σ · ν ε u = u on ∂ Ω R, ± ε ∩ ∂ Ω ε ,u = 0 on ∂ Ω R, ± ε \ ∂ Ω ε (cid:9) . In the following we shall consider the min-max values of the above forms as introduced in Defi-nition 5.Arguing as in [1, Prop. 2.1] one can prove the following local convergence result whose proof isomitted.
Lemma 29.
For all
R > and j ∈ N , there holds lim m → + ∞ µ j ( q Rm ) = µ j ( q R ∞ ) . For further use, we need the following lemma.
Lemma 30.
For all j ∈ N there holds lim R → + ∞ µ j ( q R ∞ ) = µ j ( q ∞ ) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 29
Proof of Lemma 30.
Fix j ∈ N and observe that thanks to a Dirichlet bracketing argument one gets µ j ( q R ∞ ) ≥ µ j ( q ∞ ), for all R >
0. Thenlim inf R →∞ µ j ( q R ∞ ) ≥ µ j ( q ∞ ) . (75)Now, we need to prove the opposite inequality.Take a cut-off function θ ∈ C ∞ ( R ) such that 0 ≤ θ ≤ θ ( s ) = 1 for | s | ≤ and θ ( s ) = 0 for | s | ≥
1. Given
R >
0, define θ R ( s ) := θ ( R − s ) , s ∈ R . We introduce χ R := ( U ) − θ R , where U is the unitary map (24). For further use, we compute ∇ χ R .Since χ R ( γ ( s ) + εtν ( s )) = θ ( R − s ), we get ( ∂ s χ R = γ ′ (1 − εtκ ) ∂ χ R + γ ′ (1 − εtκ ) ∂ χ R = R − θ ′ ( R − s ) ,∂ t χ R = εν ∂ χ R + εν ∂ χ R = 0 , (76)where γ ′ = ( γ , γ ) ⊤ and ν = ( ν , ν ) ⊤ = ( − γ ′ , γ ′ ) ⊤ . Then (76) can be rewritten as (cid:18) γ ′ (1 − εtκ ) γ ′ (1 − εtκ ) − εγ ′ εγ ′ (cid:19) (cid:18) ∂ χ R ∂ χ R (cid:19) = (cid:18) R − θ ′ ( R − s )0 (cid:19) , (77)so that, inverting the matrix in (77) and after straightforward computations one finds for x = γ ( s ) + εtν ( s ) : ∇ χ R ( x ) = ∇ χ R ( γ ( s ) + εtν ( s )) = θ ′ ( R − s ) R (1 − εtκ ( s )) γ ′ ( s ) . (78)Take u = ( u , u ) ⊤ ∈ dom ( q ∞ ). As chosen, we have χ R u ∈ dom (cid:0) q R ∞ (cid:1) . Thus, we find q ∞ ( χ R u ) = q R ∞ ( χ R u ) . (79)On the other hand, we have q ∞ ( χ R u ) = X k =1 (cid:16) k χ R ∇ u k k L (Ω ε , C ) | {z } := a k + k u k ∇ χ R k L (Ω ε , C ) | {z } := b k + 2 ℜ (cid:0) h χ R ∇ u k , u k ∇ χ R i L (Ω ε , C (cid:1)| {z } := c k (cid:17) . (80)Let k ∈ { , } , we get a k ≤ k∇ u k k L (Ω ε , C ) . By (78) the second term b k can be estimated as b k ≤ k θ ′ k L ∞ ( R ) R (1 − ε k κ k L ∞ ( R ) ) k u k k L (Ω ε ) . Similarly, we obtain c k ≤ k ( ∇ χ R ) u k k L (Ω ε , C ) k χ R ∇ u k k L (Ω ε , C ) ≤ k θ ′ k L ∞ ( R ) R (1 − ε k κ k L ∞ ( R ) ) k u k k L (Ω ε ) k∇ u k k L (Ω ε , C ) ≤ k θ ′ k L ∞ ( R ) R (1 − ε k κ k L ∞ ( R ) ) ( k∇ u k k L (Ω ε , C ) + k u k k L (Ω ε ) ) . Combining the above estimates with (79) and (80), we obtain the existence of R > C >
R > R there holds q R ∞ ( χ R u ) ≤ (cid:16) CR (cid:17) q ∞ ( u ) + CR k u k L (Ω ε , C ) . (81) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 30
Now, by Definition 5, for η >
0, there exists W η ⊂ dom ( q ∞ ) a j -th dimensional vector space suchthat µ j ( q ∞ ) ≤ sup u ∈ W η \{ } q ∞ ( u ) k u k L (Ω ε , C ) ≤ µ j ( q ∞ ) + η. (82)Remark that if ( u η , . . . , u ηj ) is an orthonormal basis of W η , then there exists R := R ( η ) > R > R the family ( χ R u η , . . . , χ R u ηj ) is a basis in L (Ω Rε , C ) of the vector space W Rη := { χ R u : u ∈ span( u η , . . . , u ηj ) } . Indeed, for all k, p ∈ { , . . . , j } there holds h χ R u ηk , χ R u ηp i L (Ω Rε , C ) = δ k,p − Z Ω Rε (1 − χ R ) h u ηk , u ηp i C dx. Hence, by the dominated convergence theorem, the second term on the right-hand side of the aboveequation converges to 0 as R → + ∞ and there exists R > R > R anddim( W Rη ) = j .Consequently, as W Rη ⊂ dom (cid:0) q R ∞ (cid:1) , for all u ∈ W η \ { } , the min-max principle (Proposition 6),(81) and (82) give µ j ( q R ∞ ) k χ R u k L (Ω Rε , C ) k u k L (Ω ε , C ) ≤ (1 + CR ) q ∞ ( u ) k u k L (Ω ε , C ) + CR ≤ (1 + CR )( µ j ( q ∞ ) + ε ) + CR . (83)Observe that by dominated convergence one also gets k χ R u k L (Ω Rε , C ) → k u k L (Ω ε , C ) , as R → ∞ .Thus, letting R → ∞ in (83), we obtain the inequalitylim sup R →∞ µ j ( q R ∞ ) ≤ µ j ( q ∞ ) + η . As this is true for all η >
0, combining it with (75) we get Lemma 30. (cid:3)
We conclude this paragraph with the following lemma.
Lemma 31.
Let us assume additionally that Γ is of class C , that κ ′ ( s ) → and κ ′′ ( s ) → when | s | → + ∞ and let R > . For all u ∈ dom (cid:0) q R, ± m (cid:1) there holds µ ( q R, ± m ) ≥ E ( mε ) ε − η ± ( R ) , where η ± ≥ does not depend on m and verifies η ± ( R ) → when R → + ∞ .Proof of Lemma 31. Let u ∈ dom (cid:0) q R, ± m (cid:1) and consider u its extension by 0 to the whole waveguideΩ ε . Remark that u ∈ dom ( q m ) and set v = ( U U U ) u where the unitary maps U , U and U are defined in (24), (25) and (27) respectively. By Proposition 13, and using the min-max principleon the operator acting in the t -variable we get q R, ± m ( u ) = q m ( u ) ≥ E ( mε ) ε k v k L ( Str , C ) − Z Str κ − εtκ ) | v | dsdt − Z Str ( εtκ ′ ) (1 − εtκ ) | u | dsdt − Z Str εtκ ′′ (1 − εtκ ) | u | dsdt = E ( mε ) ε k v k L ( Str , C ) − Z Str R, ± κ − εtκ ) | v | dsdt − Z Str R, ± ( εtκ ′ ) (1 − εtκ ) | v | dsdt − Z Str R, ± εtκ ′′ (1 − εtκ ) | v | dsdt, where we have taken into account that v is supported in Str R, ± := { ( s, t ) ∈ R : ± s > R, t ∈ ( − , } . This last equality gives q R, ± m ( u ) ≥ E ( mε ) ε k u k L ( Str , C ) − η ± ( R ) k u k L ( Str , C )PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 31 with η ± ( R ) := sup {± s>R } n κ ( s )4(1 − ε k κ k L ∞ ( R ) ) + 54 ε κ ′ ( s ) (1 − ε k κ k L ∞ ( R ) ) + 12 ε | κ ′′ ( s ) | (1 − ε k κ k L ∞ ( R ) ) (cid:17)o . By (A) and by the additional assumptions on κ ′ and κ ′′ we get η ± ( R ) → R → + ∞ and theLemma is proved applying the min-max principle (Proposition 6). (cid:3) Convergence of min-max values for m → + ∞ . Combining the results of the previousparagraph we can prove the convergence of the min-max values in the large mass limit.
Proof of Theorem 9.
In this proof we assume that Γ is of class C , κ ′ ( s ) → κ ′′ ( s ) → | s | → + ∞ .Consider a partition of unity given by cut-off functions θ , θ , θ ∈ C ∞ ( R ), with 0 ≤ θ k ≤ k = 1 , ,
3, and such that θ + θ + θ = 1. We also assume that θ ( s ) = 0 if s ≥ − ,θ ( s ) = 0 if s ≤ ,θ ( s ) = 0 if | s | ≥ . Recall that U is the unitary map defined in (24) and for k ∈ { , , } , define χ k,R := ( U − θ k,R ) , where for s ∈ R we have set θ k,R ( s ) := θ k ( R − s ). In particular, arguing as in (78), we get for all x = γ ( s ) + tεν ( s ) ∈ Ω ε : ∇ χ k,R ( x ) = θ ′ k,R ( R − s ) R (1 − εtκ ) γ ′ ( s ) . (84)Let u = ( u , u ) ⊤ ∈ dom ( q m ), then by Lemma 28 and the fact that χ ,R + χ ,R + χ ,R = 1 we have q m ( u ) = X k =1 (cid:18)Z Ω ε | χ k,R ∇ u | dx + Z ∂ Ω ε ( m − κ ε | χ k,R u | ds (cid:19) . (85)Let us rewrite the first integral in (85). We have Z Ω ε | χ k,R ∇ u | dx = X j =1 Z Ω ε |∇ ( χ k,R u j ) − u j ∇ χ k,R | dx = X j =1 ( Z Ω ε |∇ ( χ k,R u j ) | dx + Z Ω ε | u j | |∇ χ k,R | dx − ℜ (cid:18)Z Ω ε h∇ ( χ k,R u j ) , u j ∇ χ k,R i dx (cid:19) ) . Moreover for j ∈ { , } , there holds2 ℜ (cid:18)Z Ω ε h∇ ( χ k,R u j ) , u j ∇ χ k,R i dx (cid:19) =2 Z Ω ε | u j | |∇ χ k,R | dx + 12 Z Ω ε h∇ ( χ k,R ) , ∇ ( | u j | ) i dx . Recall that P k =1 χ k,R = 1, so that, summing up with respect to k ∈ { , , } , the last term in theabove formula vanishes. Thus, we find q m ( u ) = q R , − m ( χ ,R u ) + q R , + m ( χ ,R u ) + q Rm ( χ ,R u ) − Z Ω ε W R | u | dx , (86)where W R := P k =1 |∇ χ k,R | and k W R k L ∞ (Ω ε ) ≤ CR , for some constant C >
0, by (84).Now, fix j ∈ N and consider the isometry I : L (Ω ε , C ) → L (Ω R , − ε , C ) × L (Ω R , − ε , C ) × L (Ω Rε , C ) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 32 defined by I u = ( χ ,R u, χ ,R u, χ ,R u ). Let W ⊂ dom ( q m ) be a vector space of dimension j , by(86), there holds (cid:18) sup u ∈ W \{ } q m ( u ) k u k L (Ω ε , C ) (cid:19) + CR ≥ sup v =( v ,v ,v ) ∈ ( I W ) \{ } q R , − m ( v ) + q R , + m ( v ) + q Rm ( v ) k v k L (Ω R , − ε , C ) + k v k L (Ω R , + ε , C ) + k v k L (Ω Rε , C ) . As I is an isometry we get dim( I W ) = j and by definition of the cut-off functions χ k,R ( k ∈ { , , } ),we also have ( I W ) ⊂ D := dom (cid:16) q R , − m (cid:17) × dom (cid:16) q R , + m (cid:17) × dom (cid:0) q Rm (cid:1) . In particular, there holds (cid:18) sup u ∈ W \{ } q m ( u ) k u k L (Ω ε , C ) (cid:19) + CR ≥ inf V ⊂ D dim( V )= j sup v =( v ,v ,v ) ∈ V \{ } q R , − m ( v ) + q R , + m ( v ) + q Rm ( v ) k v k L (Ω R , − ε , C ) + k v k L (Ω R , + ε , C ) + k v k L (Ω Rε , C ) . Now, taking the infimum over all vector spaces W ⊂ dom ( q m ) of dimension j and noting thatthe right-hand side is the j -th min-max value of the quadratic form of the tensor product of thethree self-adjoint operators associated with the quadratic forms q R , − m , q R , + m and q Rm respectively, themin-max principle (Proposition 6) yields: µ j ( q m ) + CR ≥ j -th smallest element of the set { µ j ( q Rm ) } j ∈ N [ { µ j ( q R , + m ) } j ∈ N [ { µ j ( q R , − m ) } j ∈ N . First, remark that by the min-max principle for all j ∈ N , m µ j ( q m ) is a non-decreasing functionon [0 , + ∞ ) and such that µ j ( q m ) ≤ µ j ( q ∞ ). In particular µ j ( q m ) has a limit when m → + ∞ .Now, pick j ∈ N such that j < N Γ + 1 (with the convention that N Γ + 1 = + ∞ if N Γ = + ∞ ).Recall that by Proposition 7 µ j ( q ∞ ) < π ε for all j ∈ { , . . . , j } . For all k ∈ N , by Lemma 31,there holds µ k ( q R , ± m ) ≥ µ ( q R , ± m ) ≥ E ( mε ) ε − η ± ( R ) , and η ± does not depend on m and η ± ( R ) → R → + ∞ . In particular, if one fixes α > R > R > R there holds η ± ( R ) < α . Now, using (v) of Proposition10, there exists m > m > m there holds E ( mε ) ε ≥ π − α . Choosing α = (cid:0) π ε − µ j ( q ∞ ) (cid:1) it gives µ ( q R , ± m ) ≥ π ε − (cid:0) π ε − µ j ( q ∞ ) (cid:1) (87)and by Lemma 30 there exists m > m ≥ m there holds µ j ( q Rm ) ≤ µ j ( q R ∞ ) ≤ µ j ( q ∞ ) + 14 (cid:0) π ε − µ j ( q ∞ ) (cid:1) ≤ µ j ( q ∞ ) + 14 (cid:0) π ε − µ j ( q ∞ ) (cid:1) . (88)As there holds µ j ( q ∞ ) + 14 (cid:0) π ε − µ j ( q ∞ ) (cid:1) < π ε − (cid:0) π ε − µ j ( q ∞ ) (cid:1) , (87) and (88) give that for all m > max( m , m ) and all R > R there holds µ j ( q m ) + CR ≥ µ j ( q Rm ) . Hence, taking the limit m → + ∞ then R → + ∞ in the last equation, by Lemma 29 and Lemma 30we obtain lim m → + ∞ µ j ( q m ) ≥ µ j ( q ∞ ) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 33
In particular, if N Γ = + ∞ , the proof is completed. Now assume that N Γ < + ∞ and let j ≥ N Γ +1.Let us prove that µ j ( q m ) converges to π ε . By Proposition 6 and Proposition 7, there holds µ j ( q m ) ≤ µ j ( q ∞ ) = π ε . In particular, let us consider the j -th smallest element of the set { µ j ( q Rm ) } j ∈ N [ { µ j ( q R , + m ) } j ∈ N [ { µ j ( q R , − m ) } j ∈ N . Either there exists k ≥ N Γ + 1 such that this element is µ k ( q Rm ) or p ∈ N such that this elementis µ p ( q R , ± m ). In the first case, there holds: − CR ≤ π ε − ( µ j ( q m ) + CR ) ≤ π ε − µ k ( q Rm ) ≤ π ε − µ N Γ +1 ( q Rm ) . Now, in the second case, there holds − CR ≤ π ε − ( µ j ( q m ) + CR ) ≤ π ε − µ p ( q R , ± m ) ≤ π ε − µ ( q R , ± m ) ≤ π ε − E ( mε ) ε + η ± ( R ) , where we have used Lemma 31. These two inequalities yield − CR ≤ π ε − ( µ j ( q m ) + CR ) ≤ min (cid:16) π ε − µ N Γ +1 ( q Rm ) , π ε − E ( mε ) ε + η ± ( R ) (cid:17) Now, taking the limit m → + ∞ and then R → + ∞ by Lemma 29, Lemma 30, (v) of Proposition10 and Lemma 31 we get lim m → + ∞ µ j ( q m ) = π (cid:3) A quantitative condition for the existence of bound states
The goal of this section is to obtain an explicit geometric condition on the curvature of the basecurve Γ which ensures that the operator D Γ ( ε, m ) have at least two bound states.To state it, whenever Γ is of class C , we introduce the well-known geometric potential (cf. [15,Eq. (3.9)]) V ε ( s, t ) := − κ ( s ) (1 − εtκ ( s )) − κ ′′ ( s ) εt (1 − εtκ ( s )) − κ ′ ( s ) ε t (1 − εtκ ( s )) . It depends on the geometry of the waveguide Ω ε through the curvature κ of the base curve Γ, itstwo derivatives and the radius ε of the tubular neighbourhood.The sufficient condition we obtain reads as follows. Proposition 32 ( Quantitative existence of bound states ) . Let us assume additionally that Γ is ofclass C and that supp κ ⊂ ( − L, L ) with L > . If I ε := − Z R Z − V ε ( s, t ) cos (cid:16) π t (cid:17) dt ds > , (89) then there exists m ∈ R such that for every m > m , Sp dis ( D Γ ( ε, m )) ≥ . (90) Moreover, there holds m ≤ ε h I ε (cid:16) π L ε + 2 L (cid:17) − i (91) PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 34
Note that the integral I ε is independent of m . Since V ε ( s, t ) → − κ ( s ) as ε →
0, uniformly in( s, t ) ∈ R × ( − , κ is notidentically equal to zero and ε is small enough.Compared to Proposition 8, Proposition 32 gives a quantitative geometric bound control on m to obtain the existence of bound states.We work with the square of the operator D Γ ( ε, m ) studying the min-max value µ ( D Γ ( ε, m ) )following the notation introduced in Definition 5. The main idea is that, thanks to Proposition 3and Proposition 13, we know that µ ( D Γ ( ε, m ) ) = inf u ∈ dom( E Γ ( ε,m )) \{ } kE Γ ( ε, m ) u k L ( Str , C ) k u k L ( Str , C ) . (92) Proof of Proposition 32.
In view of (92) and the symmetry of the spectrum of D Γ ( ε, m ) (see Theo-rem 2), it is enough to find a test function u ∈ dom ( E Γ ( ε, m )) such that q ( u ) := kE Γ ( ε, m ) u k L ( Str , C ) − (cid:0) m + ε − E ( mε ) (cid:1) k u k L ( Str , C ) < , (93)with dom ( q ) := dom ( E Γ ( ε, m )). Indeed, then necessarily we have µ ( q ) < η >
0, and define u η ( s, t ) := 1 √ ϕ η ( s ) cos (cid:0) π t (cid:1) e i θ ( s )2 e − i θ ( s )2 ! , where θ is defined in (26) and, for every η ∈ R , ϕ η ( s ) := | s | ≤ η , η − | s | η if η < | s | < η , | s | ≥ η . Remark that u η ∈ H ( Str , C ) ⊂ dom ( q ) and k u η k L ( Str , C ) = k ϕ η k L ( R ) = η . Using the boundarycondition, one easily checks the identity1 ε Z Str | ∂ t u η ( s, t ) | ds dt + εm Z R | u η ( s, − | ds + εm Z R | u η ( s, | ds = π ε k u η k L ( Str , C ) . Consequently, there holds q ( u η ) = ε − (cid:16) π − E ( mε ) (cid:17) k u η k L ( Str , C ) + Z Str | ( ∂ s − i κ σ ) u η ( s, t ) | (1 − εtκ ( s )) ds dt + Z Str V ε ( s, t ) | u η ( s, t ) | ds dt . (94)To deal with the second term on the right-hand side of (94), we set v η := e − i θ σ u η and remarkthat for all ( s, t ) ∈ Str there holds v η ( s, t ) = 1 √ ϕ η ( s ) cos (cid:0) π t (cid:1) (cid:18) (cid:19) , k v η ( s, t ) k C = k u η ( s, t ) k C . In particular, we remark that e − i θ σ ( ∂ s − i κ σ ) u η ( s, t ) = ( ∂ s v η )( s, t ) = 1 √ ϕ ′ η ( s ) cos (cid:0) π t (cid:1) (cid:18) (cid:19) . PECTRAL PROPERTIES OF RELATIVISTIC QUANTUM WAVEGUIDES 35
Consequently, we obtain q ( u η ) = ε − (cid:16) π − E ( mε ) (cid:17) k ϕ η k L ( R ) + Z R | ϕ ′ η ( s ) | Z − − κ ( s ) εt ) cos (cid:16) π t (cid:17) dt ds (95)+ Z R | ϕ η ( s ) | V ε ( s, t ) cos (cid:16) π t (cid:17) dt ds. Now we employ the hypothesis that the curvature κ (and therefore also its derivatives κ ′ and κ ′′ )is compactly supported and choose η ≥ L . Then the last line equals − I ε and the second line equals k ϕ ′ η k L ( R ) = η . In summary, q ( u η ) = ε − (cid:16) π − E ( mε ) (cid:17) η + 2 η − I ε . Using in (19) the elementary bound tan( x ) ≤ x − π valid for every x ∈ ( π , π ], we get the estimate p E ( mε ) ≥ π mε mε , which remains true for m = 0, though it becomes rather too crude for small masses. Consequently,using the elementary inequality (1 + 4 mε ) ≤ mε ), we get q ( u η ) ≤ π ε mε (1 + 2 mε ) η + 2 η − I ε ≤ π ε mε ) 83 η + 2 η − I ε . Setting η := L √ mε ≥ L , we find q ( u η ) ≤ (cid:18) π L ε + 2 L (cid:19) √ mε − I ε . Therefore, if I ε >
0, we see that q ( u η ) is negative whenever m ≥ ˜ m , where ˜ m coincides with theright-hand-side of (91). It concludes the proof of Proposition 32. (cid:3) Remark 33.
The hypothesis that κ is compactly supported is apparently just a technical conditionin order to simplify the expression (95). A series of alternative sufficient conditions which guaranteethat q ( u η ) is negative could be obtained by playing with (95). Acknowledgment.
The research of D.K. was partially supported by the EXPRO grant No. 20-17749X of the Czech Science Foundation (GACR). W.B. is member of GNAMPA as part of INdAM.
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Centro De Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri 3, I-56100 , Pisa, Italy.
Email address : [email protected] (P. Briet) Aix-Marseille Universit´e, Universit´e de Toulon, CNRS, CPT, Marseille, France.
Email address : [email protected] URL : (D. Krejˇciˇr´ık) Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,Czech Technical University in Prague, Trojanova 13, 12000 Prage 2
Email address : [email protected] URL : http://nsa.fjfi.cvut.cz/david/ (T. Ourmi`eres-Bonafos) Aix-Marseille Universit´e, CNRS, Centrale Marseille, I2M, Marseille, France.
Email address : [email protected] URL ::