Quantum singular operator limits of thin Dirichlet tubes via Γ -convergence
aa r X i v : . [ m a t h - ph ] O c t QUANTUM SINGULAR OPERATOR LIMITS OF THINDIRICHLET TUBES VIA Γ -CONVERGENCE C´ESAR R. DE OLIVEIRA
Abstract.
The Γ-convergence of lower bounded quadratic forms is used tostudy the singular operator limit of thin tubes (i.e., the vanishing of the crosssection diameter) of the Laplace operator with Dirichlet boundary conditions; aprocedure to obtain the effective Schr¨odinger operator (in different subspaces)is proposed, generalizing recent results in case of compact tubes. Finally, afterscaling curvature and torsion the limit of a broken line is briefly investigated.Keywords: quantum thin tubes, singular operators, Γ-convergence, brokenline.
Contents
1. Introduction 12. Strong Resolvent and Γ Convergences 42.1. Γ-Convergence 42.2. Γ and Resolvent Convergences 62.3. Norm Resolvent Convergence 73. Singular Limit of Dirichlet Tubes 93.1. Tube and Hamiltonian 93.2. Confinement: Statements 143.3. Confinement: Convergence 173.4. Spectral Possibilities 213.5. Bounded Tubes 224. Broken-Line Limit 244.1. h V eff n i 6 = 0 254.2. h V eff n i = 0 26References 271. Introduction
Among the tools for studying limits of self-adjoint operators in Hilbert spacesare the resolvent convergence and the sesquilinear form convergence. In the con-text of quantum mechanics, it is well known that in case of monotone sequences ofoperators these approaches are strictly related, as discussed, for example, in Sec-tion 10.4 of [1] and Section VIII.7 (Supplementary Material) of [2]. These form
Mathematics Subject Classification. convergences have the advantage of dealing with some singular limits in quantummechanics, which is well exemplified by mathematical arguments supporting theAharonov-Bohm Hamiltonian [3, 4].Let denote the identity operator and 0 ≤ T j be a sequence of positive (oruniformly lower bounded in general) self-adjoint operators acting in the Hilbertspace H . From the technical point of view what happens is that the monotoneincreasing of the sequence of resolvent operators R − λ ( T j ) := ( T j + λ ) − ( λ >
0) implies the monotone decreasing of the corresponding sequence of sesquilinearforms and vice versa. Due to monotonicity, in such cases one clearly understandsthe existence of limits and have some insight in limit form domains. However, inprinciple it is not at all clear what are the relations between strong resolvent con-vergence of operators and sesquilinear form convergence in more general cases, say,if one requires that the self-adjoint operators or closed forms are only uniformlybounded from below. It happens that such relations are well known among peopleinterested in variational convergences and applied mathematics, and it is directlyrelated to the concept of Γ-convergence; but when addressing sesquilinear formsthese variational problems are usually formulated in real Hilbert spaces and thetheory has been developed under this condition. However, in quantum mechanicsthe Hilbert spaces are usually complex, and an adaptation of the main results tocomplex Hilbert spaces will appear elsewhere [5]. Due to the particular class ofapplications we have in mind, here it will be enough to reduce some key argumentsto real Hilbert spaces.In Section 2 the very basics of Γ-convergence are recalled in a suitable way; fordetails the reader is referred to the important monographs in the field [6, 7].As an application of Γ-convergence of forms we study the limit operator obtainedfrom the Laplacian (with Dirichlet boundary conditions) restricted to a tube in R that shrinks to a smooth curve. This problem has been considered in the interestingwork [8], where the strong Γ-convergence was employed to study the limit operatorand convergence of eigenvalues and eigenvectors, but the curve was supposed to havefinite length L and the convergence restricted to the subspace of vectors of the form w ( s ) u ( y ) (“the first sector”), with w ∈ H [0 , L ] (i.e., the usual Sobolev space) and u being the first eigenfunction of the restriction of the Laplacian to the tube crosssection ( s denotes the curve arc length and y = ( y , y ) the cross section variables).Here we indicate how the proofs in [8] can be worked out to get an effective operatorin case of curves of infinite length; in fact it will be necessary to go a step furtherand we prove both strong and weak Γ-convergences of forms which imply the strongresolvent convergence of operators to an effective Schr¨odinger operator on the curve.We also give an alternative proof of the spectral convergence discussed in [8] in caseof finite length curves: compactness arguments due to the boundedness of the tubein this case will be essential; our proof also clarifies the mechanism behind thespectral convergence. These issues are discussed in Section 3.The use of quadratic forms, and the possibility of considering Γ-convergence, inthe application mentioned above is important because due to the intricate geom-etry of the allowed tubes the expressions of the actions of the involved operatorsare rather complicated, with the presence of mixed derivatives and nonlinear co-efficients. In case the curve lives in R there are results about the limit operatorin [9, 10] and the effective potential is written in terms of the curvature; the mainnovelty in case of R considered in [8] is the additional presence of “twisting” and UANTUM RESOLVENT AND Γ CONVERGENCES 3 torsion in the effective potential, since the case of untwisted tubes has also beenpreviously studied in [11] (see also [12, 13]). Since in both R and untwisted tubesin R we have simpler expressions for the involved operators, it is possible to dealdirectly with the strong resolvent convergence and consider more general spacesthan that related only to the first eigenfunction of the laplace operator in the crosssection (i.e., the first sector). So, based on the Γ-convergence of sesquilinear forms,we propose here a rather natural procedure to handle the strong convergence incase of vectors of the form w ( s ) u n ( y ), i.e., the ( n + 1)th sector spanned by thegeneral eigenvector u n of the Laplacian restricted to the cross section, and we willimpose that each eigenvalue of this operator is simple in order to simplify the im-plementation of the Γ-convergence of b εn and the strong resolvent convergence ofthe associated self-adjoint operators H εn , n ≥
1, which are introduced in Defini-tion 2. It turned out that the resulting effective operator depends on the sectorconsidered, an effect not present in R [9, 10]. This “new” effect is understoodsince in R there is a separation of variables s and y (see the top of page 14 of[10], but be aware of the different notations), which in general does not occur in R and so the necessity of introducing a specific procedure. These developmentsare also presented in Section 3. Roughly speaking, from the dynamical point ofview the dependence of the limit operator on the sector of the Hilbert space wouldcorrespond to the dependence of the effective potential on the initial condition, aphenomenon already noticed in [14, 15].It must be mentioned that the author has no intention to claim that Γ-conver-gence should replace the traditional operator techniques in such kind of problemswith no separation of variables [16, 17, 18, 19], including some important spectralresults for twisted tubes [22]. Further, we need a combination of strong and weakΓ-convergences of quadratic forms in order to get strong resolvent convergence ofthe related operators, and other assumptions and tools must be added to get normresolvent convergence (see, for instance, Subsection 3.5). The Γ-convergence in thiscontext must be seen as just another available tool for mathematical physicists.With the effective limit operator at hand, we finally discuss the limit of a smoothcurve approaching a broken line, similarly to [9, 10], but here we confine ourselvesto take one limit at a time, that is, first we constrain the particle motion fromthe tube to the curve (as discussed above), then we take the limit when the curveapproaches the broken line; we shall closely follow a discussion in [10]. Besides thenegative term related to the curvature, in our case the effective potential V eff ( s )has the additional presence of a positive term related to “twist” and torsion so thatthe technical condition Z R V eff ( s ) ds = 0assumed in [10] might not hold in some cases. We have then checked that it isstill possible to follow the same proofs, but with suitable adaptations, even if theabove integral vanishes. The bottom line is that different self-adjoint realizationsfor the operator on the broken line are found. These broken-line limits are shortlydiscussed in Section 4.Of course in this introduction we have skipped many technical details, some ofthem are fundamental to a correct understanding of the contents of this work. Inthe following sections I shall try to fill out those gaps. I expect this work will C´ESAR R. DE OLIVEIRA motivate researchers to seriously consider the Γ-convergence of forms as a usefulway to study (singular) limits of observables in quantum mechanics.2.
Strong Resolvent and Γ Convergences -Convergence.
Our sequences (more properly they should be called fami-lies) of self-adjoint operators T ε , with domain dom T ε in a separable Hilbert space H , and the corresponding closed sesquilinear forms b ε will be indexed by the pa-rameter ε > ε → T (resp. b ) of T ε (resp. b ε ). The domain of T will not be supposed tobe dense in H and its closure will be denoted by H = dom T (with rng T ⊂ H );usually this is indicated by simply saying that “ T is self-adjoint in H .”We assume that a sesquilinear form b ( ζ, η ) is linear in the second entry andantilinear in the first one. As usual the real-valued function ζ b ( ζ, ζ ) will besimply denoted by b ( ζ ) and called the associated quadratic form; we will use theterms sesquilinear and quadratic forms almost interchangeably, since usually thecontext makes it clear which one is being referred to. It will also be assumed that b is positive (or lower bounded in general) and b ( ζ ) = ∞ if ζ does not belong toits domain dom b ; this is important in order to guarantee that in some cases b islower semicontinuous, which is equivalent to b be the sesquilinear form generatedby a positive self-adjoint operator T , that is, b ( ζ, η ) = h T / ζ, T / η i , ζ, η ∈ dom b = dom T / ;see Theorem 9.3.11 in [1]. By allowing b ( ζ ) = ∞ we have a handy way to work inthe larger space H instead of only in H = dom T . If λ ∈ R , then b + λ indicates thesesquilinear form ( b + λ )( ζ, η ) := b ( ζ, η ) + λ h ζ, η i , whose corresponding quadraticform is b ( ζ ) + λ k ζ k .It is known that (Lemma 10.4.4 in [1]), for any λ >
0, one has b ε ≤ b ε iff R − λ ( T ε ) ≤ R − λ ( T ε ), so that a sequence of quadratic forms is monotone iff thecorresponding sequence of resolvent operators is monotone. This has been ex-plored in the quantum mechanics literature in order to get strong resolvent limitsof self-adjoint operators through the study of quadratic forms (see, for instance,Section 10.4 of [1] and Section VIII.7 of [2]). For more general sequences of opera-tors, the form counterpart of the strong resolvent convergence is not so direct andit was found that the correct concept comes from the so-called Γ-convergence [6].In what follows the concept of Γ-convergence will be recalled in a suitable way, andthen applied to the study of singular limits of Dirichlet tubes in other sections.The general concept of Γ-convergence is not restricted to quadratic forms andcan be applied to quite general topological spaces, but in this section the ideas willmostly be suitably adapted to our framework; e.g., we try to restrict the discussionto Hilbert spaces and to lower semicontinuous functions, since the quadratic formswe are interested in have this property. The general theory is nicely presented in thebook [6], to which we will often refer. H always denote a separable Hilbert spaceand B ( ζ ; δ ) the open ball cantered at ζ ∈ H of radius δ >
0; finally R := R ∪ {∞} and the symbol l.sc. will be a shorthand to lower semicontinuous . Definition 1.
The lower Γ -limit of a sequence of l.sc. functions f ε : H → R is thefunction f − : H → R given by f − ( ζ ) = lim δ → lim inf ε → inf { f ε ( η ) : η ∈ B ( ζ ; δ ) } , ζ ∈ H . UANTUM RESOLVENT AND Γ CONVERGENCES 5
The upper Γ -limit f + ( ζ ) of f ε is defined by replacing lim inf by lim sup in the aboveexpression. If f − = f + =: f we say that such function is the Γ -limit of f ε and itwill be denoted by f = Γ - lim ε → f ε . Remark 1.
It was assumed in Definition 1 that the topology of H is the usualnorm topology, and in this case we speak of strong Γ-convergence . If the weaktopology is considered, the balls B ( ζ ; δ ) must be replaced by the set of all openweak neighborhoods of ζ [6] , and in this case we speak of weak Γ-convergence .Both concepts will be important here, and in general they are not equivalent sincethe norm is not continuous in the weak topology (see Example 6.6 in [6] ). Whenconvenient, the symbols f ε SΓ −→ f, f ε WΓ −→ f will be used to indicate that f ε Γ -converges to f in the strong and weak sense in H ,respectively. See also Proposition 1 and Remark 4. Example 1.
The sequence f ε : R → R , f ε ( x ) = sin( x/ε ) , Γ -converges to theconstant function f ( x ) = − as ε → . This simple example nicely illustrates theproperty of “convergence of minima” that motivated the introduction of the Γ -con-vergence. In quantum mechanics one is used to the convergence of averages, so thatthe natural guess (if any) for the limit in this example would be the null function. Remark 2.
The Γ -convergence is usually different from both the pointwise andweak limit of functions and roughly it can be illustrated as follows. Assume oneis studying a heterogeneous material subject to strong tensions whose intensity insome regions is measured by the parameter /ε and it will undergo a kind of phasetransition as ε → ; for each ε > one computes the equilibrium configurationof this material via a minimum of certain energy functional; this transition wouldbe computed as the limit of the equilibria (i.e., minima of such functionals), andthis limit (i.e., minimum of an “effective limit functional”) would be quite singular.In many instances Γ -convergence is the correct concept to describe such situations [7, 6] and its importance in the asymptotics of variational problems relies here. Remark 3.
From some points of view the notion of Γ -convergence is quite subtle,as exemplified by the following facts: a) Due to the prominent role played by minima, in general Γ - lim ε → f ε = − (cid:16) Γ - lim ε → ( − f ε ) (cid:17) . b) Assume that f = Γ - lim ε → f ε and g = Γ - lim ε → g ε ; it may happen that ( f ε + g ε ) is not Γ -convergent. c) Clearly it is not necessary to restrict the definition of Γ -convergence to l.sc.functions. If f ε = f , for all ε , and f is not l.sc., then Γ - lim ε → f is thegreatest lower semicontinuous function majorized by f (the so-called l.sc.envelope of f ), and so different from f ! Since H satisfies the first axiom of countability, Proposition 8.1 of [6] implies thefollowing handy characterization of strong and weak Γ-convergence: Proposition 1.
The sequence f ε : H → R strongly Γ -converges to f (that is, f ε SΓ −→ f ) iff the following two conditions are satisfied: C´ESAR R. DE OLIVEIRA i) For every ζ ∈ H and every ζ ε → ζ in H one has f ( ζ ) ≤ lim inf ε → f ε ( ζ ε ) . ii) For every ζ ∈ H there exists a sequence ζ ε → ζ in H such that f ( ζ ) = lim ε → f ε ( ζ ε ) . Remark 4.
If instead of strong convergence ζ ε → ζ one considers weak conver-gence ζ ε ⇀ ζ in Proposition 1, then we have a characterization of f ε WΓ −→ f . Herethis characterization will be used in practice, and so it justifies the lack of detailswith respect to weak Γ -convergence in Remark 1. Recall that a function f : H → R is coercive if for every x ∈ R the set f − ( −∞ , x ]is precompact in H . Since H is reflexive, it turns out that a function f is coercive inthe weak topology of H iff lim k ζ k→∞ f ( ζ ) = ∞ . A sequence of functions f ε : H → R is equicoercive if there exists a coercive ϕ : H → R such that f ε ≥ ϕ , for all ε > Theorem 1.
Assume that f ε : H → R Γ -converges to f and let ζ ε be a minimizerof f ε , for all ε . Then any cluster point of ( ζ ε ) is a minimizer of f . Further, if f ε is equicoercive and f has a unique minimizer x , then ζ ε converges to x . Theorem 2.
Let b ε , b ≥ β > −∞ be closed and (uniformly) lower bounded sesqui-linear forms in H and f : H → R a continuous function. Then b ε Γ -converges to b iff ( b ε + f ) Γ -converges to ( b + f ) . In particular, in case of both weak and strong Γ -convergences in H , it holds for the functional f ( · ) = h η, ·i + h· , η i , defined foreach fixed η ∈ H . and Resolvent Convergences. Now we recall the main results with re-spect to the relation between the strong resolvent convergence of self-adjoint oper-ators and Γ-convergence of the associated sesquilinear forms [6, 5].
Theorem 3.
Let b ε , b be positive (or uniformly lower bounded) closed sesquilin-ear forms in the Hilbert space H , and T ε , T the corresponding associated positiveself-adjoint operators. Then the following statements are equivalent: i) b ε SΓ −→ b and, for each ζ ∈ H , b ( ζ ) ≤ lim inf ε → b ε ( ζ ε ) , ∀ ζ ε ⇀ ζ in H . ii) b ε SΓ −→ b and b ε WΓ −→ b . iii) b ε + λ SΓ −→ b + λ and b ε + λ WΓ −→ b + λ , for some λ > (and so for all λ ≥ ). iv) For all η ∈ H and λ > , the sequence min ζ ∈H (cid:20) b ε ( ζ ) + λ k ζ k + 12 ( h η, ζ i + h ζ, η i ) (cid:21) converges to min ζ ∈H (cid:20) b ( ζ ) + λ k ζ k + 12 ( h η, ζ i + h ζ, η i ) (cid:21) . UANTUM RESOLVENT AND Γ CONVERGENCES 7 v) T ε converges to T in the strong resolvent sense in H = dom T ⊂ H , thatis, lim ε → R − λ ( T ε ) ζ = R − λ ( T ) P ζ, ∀ ζ ∈ H , ∀ λ > , where P is the orthogonal projection onto H . The next two results are included mainly because they help to elucidate theconnection between forms, operator actions and domains on the one hand, andminimalization of suitable functionals on the other hand; this sheds some light onthe role played by Γ-convergence in the convergence of self-adjoint operators.
Proposition 2.
Let b ≥ be a closed sesquilinear form on the Hilbert space H , T ≥ the self-adjoint operator associated with b and P be the orthogonal projectiononto H = dom T ⊂ H . Then ζ ∈ dom T and T ζ = P η iff ζ is a minimum point(also called minimizer) of the functional g : H → R , g ( ζ ) = b ( ζ ) − h η, ζ i − h ζ, η i . Proposition 3.
Let T : dom T → H be a positive self-adjoint operator, dom T = H , and b T : H → R the quadratic form generated by T . Then b T ( ζ ) = sup η ∈ dom T [ h T η, ζ i + h ζ, T η i − h T η, η i ]= sup η ∈ dom T (cid:2) b T ( η ) + h T η, ζ i + h ζ, T η i − h T η, η i (cid:3) , for all ζ ∈ H and b T ( ζ ) = ∞ if ζ ∈ H \ H . Finally we recall Theorem 13.5 in [6]:
Theorem 4.
Let b ε , b ≥ β > be sesquilinear forms on the Hilbert space H and T ε , T ≥ β the corresponding associated self-adjoint operators, and let dom T = H ⊂ H . Then the following statements are equivalent: i) b ε WΓ −→ b . ii) R ( T ε ) converges weakly to R ( T ) P , where P is the orthogonal projectiononto H . Norm Resolvent Convergence.
Before turning to an application of Theo-rem 3 in the next section, we introduce additional conditions in order to get normresolvent convergence of operators from Γ-convergence. This condition will be usedto recover a spectral convergence proved in [8] and, from the technical point ofview, can be considered our first contribution.
Proposition 4.
Let b ε , b ≥ β > −∞ be closed sesquilinear forms and T ε , T ≥ β the corresponding associated self-adjoint operators, and let dom T = H ⊂ H .Assume that the following three conditions hold: a) b ε SΓ −→ b and b ε WΓ −→ b . b) The resolvent operator R − λ ( T ) is compact in H for some real number λ > | β | . c) There exists a Hilbert space K , compactly embedded in H , so that if thesequence ( ψ ε ) is bounded in H and ( b ε ( ψ ε )) is also bounded, then ( ψ ε ) is abounded subset of K .Then, T ε converges in norm resolvent sense to T in H as ε → . C´ESAR R. DE OLIVEIRA
Proof.
We must show that R − λ ( T ε ) converges in operator norm to R − λ ( T ) P , where P is the orthogonal projection onto H ; to simplify the notation the projection P will be ignored.If R − λ ( T ε ) does not converge in norm to R − λ ( T ), there exist δ > η ε , k η ε k = 1, for a subsequence (we tacitly keep the same notation after takingsubsequences) of indices ε → k R − λ ( T ε ) η ε − R − λ ( T ) η ε k ≥ δ , ∀ ε > . We will argue to get a contradiction with this inequality, so proving the proposition.Denote ζ ε := R − λ ( T ε ) η ε . By the reflexivity of H one can suppose that η ε ⇀ η , forsome η ∈ H , and since R − λ ( T ) is compact we have R − λ ( T ) η ε → R − λ ( T ) η in H .The general inequalities k R − λ ( T ε ) η ε k ≤ | β − λ | and k T ε R − λ ( T ε ) η ε k ≤ k η ε k = 1 , ∀ ε, imply that | b ε ( ζ ε ) | = |h T ε ζ ε , ζ ε i| ≤ k T ε ζ ε k k ζ ε k ≤ | β − λ | , ∀ ε, and so it follows by c) that ( R − λ ( T ε ) η ε ) is a bounded sequence in K , and since thisspace is compactly embedded in H there exists a (strongly) convergent subsequenceso that R − λ ( T ε ) η ε → ζ for some ζ ∈ H . Next we will employ the Γ-convergence to show that ζ = R − λ ( T ) η .By Proposition 2, for each ε fixed, ζ ε is the minimizer in H of the functional g ε ( φ ) = b ε ( φ ) + λ k φ k − h η ε , φ i − h φ, η ε i , whereas ζ = R − λ ( T ) η is the unique minimizer of g ( φ ) = b ( φ ) + λ k φ k − h η, φ i − h φ, η i . Since λ > | β | it follows that g ε is weakly equicoercive (see the discussion after Re-mark 4) and, by Theorem 1, ζ ε ⇀ ζ . By this weak convergence and Theorem 3 iv),for all µ ≥ λ , one has b ( ζ ) + µ k ζ k − h η, ζ i − h ζ, η i = lim ε → [ b ε ( ζ ε ) + µ k ζ ε k − h η, ζ ε i − h ζ ε , η i ] . Theorem 3 and again ζ ε ⇀ ζ imply b ( ζ ) + λ k ζ k ≤ lim inf ε → [ b ε ( ζ ε ) + λ k ζ ε k ] , whereas the lower semicontinuity of the norm with respect to weak convergencegives ( µ − λ ) k ζ k ≤ lim inf ε → ( µ − λ ) k ζ ε k , µ > λ. The last three relations imply k ζ k = lim ε → k ζ ε k , and together with ζ ε ⇀ ζ oneobtains the strong convergence ζ ε → ζ , that is, R − λ ( T ε ) η ε → R − λ ( T ) η, which contradicts the existence of δ > (cid:3) UANTUM RESOLVENT AND Γ CONVERGENCES 9 Singular Limit of Dirichlet Tubes
There are many occasions in which particles or waves are restricted to propagatein thin domains along one-dimensional structures, as a graph. Optical fibers, carbonnanotubes and the motion of valence electrons in aromatic molecules are goodexamples. One natural theoretical consideration is to neglect the small transversalsections and model these systems by the true one-dimensional versions by meansof effective parameters and potentials. Besides the necessity of finding effectivemodels, one has also to somehow decouple the transversal and the longitudinalvariables. Such confinements can be realized by strong potentials (see [14, 23]and references therein) or boundary conditions, and the graph can be imbedded inspaces of different dimensions.In this section we apply the Γ-convergence in Hilbert spaces to study the limitoperator of the Dirichlet Laplacian in a sequence of tubes in R that is squeezed toa curve. This kind of problem (mainly in R ) has been considered in some papers(for instance, [10, 8, 9, 11, 13]), and here we show how part of the constructionin [8] can be carried out to curves of infinite length. As already mentioned, sinceour main interest is in quantum mechanics, in principle one should use complexHilbert spaces and adapt the results of Γ-convergence to this more general setting[5] before presenting applications, but since the Laplacian is a real operator, andits eigenfunctions can always be supposed to be real valued, one can reduce thearguments to the real setting.In a second step we propose a procedure to deal with more general vectors thanthat generated by the first eigenvector of the restriction of the Laplacian to thetube cross section. In simpler situations, like tubes in R with vanishing “twisting”(see Definition 3) or in R , it is possible to get a suitable separation of variables, itis not necessary to employ that procedure and the limit operator does not dependon the subspace considered. However, in the most general case the limit operatorwill depend on the subspace in the cross section due to an “additional memory ofextra dimensions” that can be present in R . This will appear explicitly in theexpressions of effective potentials ahead.3.1. Tube and Hamiltonian.
Given a connected open set Ω ⊂ R , denote by − ∆ Ω the usual negative Laplacian operator with Dirichlet boundary conditions,that is, the Friedrichs extensions of − ∆ with domain dom ( − ∆) = C ∞ (Ω). Moreprecisely, − ∆ Ω is the self-adjoint operator acting in L (Ω) associated with thepositive sesquilinear form b Ω ( ψ, ϕ ) = h∇ x ψ, ∇ x ϕ i , dom b Ω = H (Ω);the inner product is in the space L (Ω) and ∇ x is the usual gradient in cartesiancoordinates ( x, y, z ). The operator − ∆ Ω describes the energy of a quantum freeparticle in Ω. We are interested in Ω representing the following kind of tubes, whichwill be described in some details. Let γ : R → R be a C curve parametrized byits (signed) arc length s , and introduce T ( s ) = ˙ γ ( s ) , N ( s ) = 1 κ ( s ) ˙ T ( s ) , B ( s ) = T ( s ) × N ( s ) , with κ ( s ) = k ¨ γ ( s ) k being the curvature of γ ; the dot over a function always in-dicates derivative with respect to s . These quantities are the well-known tangent,normal and binormal (orthonormal) vectors of γ , respectively, which constitute a distinguished Frenet frame for the curve [24]. If κ vanishes in suitable intervals oneconsiders a constant Frenet frame and in many cases it is possible to join distin-guished and constant Frenet frames, for instance if κ > I and vanishing in R \ I [22]; this will be implicitly used when we deal with thebroken-line limit in Section 4. Here it is assumed that such a global Frenet frameexists, and so it changes along the curve γ according to the Serret-Frenet equations ˙ T ˙ N ˙ B = κ − κ τ − τ TNB , where τ ( s ) is the torsion of the curve γ ( s ). Although in principle closed curves canbe allowed we will exclude this case since our main interest is in curves of infinitelength. Nevertheless, the case of closed curves would lead to a bounded tube anddiscrete spectrum of the associated Laplacian − ∆ εα (see below) and would fit in asmall variation of the discussion in Subsection 3.5.Next an open, bounded and connected subset ∅ 6 = S ⊂ R will be transversallylinked to the reference curve γ , so that S will be the cross section of the tube.However, S will also be rotated with respect to the Frenet frame as one movesalong the reference curve γ , and with rotation angle given by a C function α ( s ).Given ε >
0, the tube so obtained (i.e., by moving S along the curve γ togetherwith the rotation α ( s )) is given byΩ εα := { ( x, y, z ) ∈ R : ( x, y, z ) = f εα ( s, y , y ) , s ∈ R , ( y , y ) ∈ S } , with f εα ( s, y , y ) = γ ( s ) + εy N α ( s ) + εy B α ( s ), and N α ( s ) = cos α ( s ) N ( s ) − sin α ( s ) B ( s ) B α ( s ) = sin α ( s ) N ( s ) + cos α ( s ) B ( s ) . The tube is then defined by the map f εα : R × S → Ω εα , and we will be interestedin the singular limit case ε →
0, that is, when the tube is squeezed to the curve γ and what happens to the Dirichlet Laplacian − ∆ Ω εα in this process (see [13]for the corresponding construction in R n ). This will result in the one-dimensionalquantum energy operator that arises after the confinement onto γ ; on basis ofProposition 8.1 in [14], it is expected that this will be the relevant operator alsoin the case of holonomic constraints (at least from the dynamical point of view forfinite times); see also Theorem 3 in [9].Note that the tube is completely determined by the curvature κ ( s ) and torsion τ ( s ) of the curve γ ( s ), together with the cross-section S and the rotation func-tion α ( s ). Below some conditions will be imposed on f εα so that it becomes a C -diffeomorphism. It will be assumed that γ has no self-intersection and that itscurvature is a bounded function of s , that is, k κ k ∞ < ∞ , and, for simplicity, unlessexplicitly specified that k τ k ∞ , k ˙ α k ∞ < ∞ .As usual in this context we rescale and change variables in order to work withthe fixed domain R × S ; the price we have to pay is a nontrivial Riemannian metric G = G εα , which is induced by the embedding f εα , that is, G = ( G ij ), G ij = e i · e j = G ji , 1 ≤ i, j ≤
3, with e = ∂f εα ∂s , e = ∂f εα ∂y , e = ∂f εα ∂y . UANTUM RESOLVENT AND Γ CONVERGENCES 11
Direct computations give G εα = β ε + ε ( ρ ε + σ ε ) ρ ε σ ε ρ ε ε σ ε ε , with β ε ( s, y , y ) = 1 − εκ ( s )( y cos α ( s ) + y sin α ( s )), ρ ε ( s, y , y ) = − ε y ( τ ( s ) − ˙ α ( s )) and σ ε ( s, y , y ) = ε y ( τ ( s ) − ˙ α ( s )). Its determinant is | det G εα | = ε β ε , so that f εα is a local diffeomorphism provided β ε does not vanish on R × S , which willoccur if κ is bounded and ε small enough (recall that S is a bounded set), so that β ε >
0. By requiring that f εα is injective (that is, the tube is not self-intersecting)one gets a global diffeomorphism.Coming back to the beginning of this subsection, we pass the sesquilinear formin usual coordinates ( x, y, z ), b Ω εα ( ψ, ϕ ) = h∇ x ψ, ∇ x ϕ i , dom b Ω εα = H (Ω εα ) , to coordinates ( s, y , y ) of R × S and express the inner product and gradientsappropriately. For ψ ∈ H (Ω εα ) set ψ ( s, y , y ) := ψ ( f εα ( s, y , y )). If ∇ denotes thegradient in the ( s, y , y ) coordinates, then by the chain rule ∇ x ψ = J − ∇ ψ where J is the 3 × T, N, B ), J = e e e = β ε ε ( τ − ˙ α )( y sin α − y cos α ) ε ( τ − ˙ α )( y sin α + y cos α )0 − ε cos α ε sin α ε sin α ε cos α . Noting that JJ t = G , det J = | det G | / = ε β ε , and introducing the notation h ψ, ϕ i G = Z R × S ψ ( s, y , y ) ϕ ( s, y , y ) ε β ε ( s ) dsdy dy , it follows that h∇ x ψ, ∇ x ϕ i = h J − ∇ ψ, J − ∇ ϕ i G = h∇ ψ, G − ∇ ϕ i G and the operator − ∆ εα can be described as the operator associated with the positivesesquilinear formdom ˜ b ε = H ( R × S, G ) , ˜ b ε ( ψ, φ ) := h∇ ψ, G − ∇ ϕ i G , and the ε dependence is now in the Riemannian metric G . More precisely, theabove change of variables is implemented by the unitary transformation U : L (Ω εα ) → L ( R × S, G ) , U ψ = ψ ◦ f εα . Note, however, that usually we will continue denoting
U ψ simply by ψ . Explicitlythe quadratic form is given by (with dy = dy dy and ∇ ⊥ ψ = ( ∂ y ψ, ∂ y ψ ), so that ∇ = ( ∂ s , ∇ ⊥ ))˜ b ε ( ψ ) = (cid:13)(cid:13) J − ∇ ψ (cid:13)(cid:13) G = ε Z R × S dsdy (cid:20) β ε |∇ ψ · (1 , y ( τ − ˙ α ) , y ( τ − ˙ α )) | + β ε ε |∇ ⊥ ψ | (cid:21) = ε Z R × S dsdy (cid:20) β ε |∇ ψ · (1 , Ry ( τ − ˙ α )) | + β ε ε |∇ ⊥ ψ | (cid:21) , where R = (cid:18) (cid:19) . On functions ψ ∈ C ∞ ( R × S ) a calculation shows that thecorresponding operator has the following action U ( − ∆ εα ) U ∗ ψ = − ε β ε div β ε G − ∇ ψ, and mixed derivatives will not be present iff τ ( s ) − ˙ α ( s ) = 0, since this is thecondition for G − be a diagonal matrix: G − = β − ε ε −
00 0 ε − . This hypothesis simplifies the operator expression and it is the main reason thiscase attracted more attention [11, 12]; more recently the general case has also beenconsidered and the spectral-geometric effects have been reviewed in [25]. With re-spect to the limit ε → γ ,i.e., ε →
0, it is still necessary to perform two kinds of “regularizations” in ˜ b ε in or-der to extract a meaningful limit; these are common approaches to balance singularproblems, particularly due to the presence of regions that scale in different man-ners, and so to put them in a tractable form [7]. The first one is physically relatedto the uncertainty principle in quantum mechanics and is in fact a renormalization.Let u n ∈ H ( S ) and λ n ∈ R , n ≥
0, be the normalized eigenfunctions and corre-sponding eigenvalues of the (negative) Laplacian restricted to the cross section S (since S is bounded the Dirichlet Laplacian on S has compact resolvent), and wesuppose that λ < λ < λ · · · and that all eigenvalues λ n are simple; later on wewill underline where this assumption is used (see, in particular, Subsection 3.3.1).When the tube is squeezed there are divergent energies due to terms of theform λ n /ε , and one needs a rule to get rid of these energies related to transverseoscillations in the tube. Since in quantum mechanics these quadratic forms ˜ b ε correspond to expectation values of total energy, one subtracts such diverging termsfrom ˜ b ε . However, in principle it is not clear which expression one should use inthe subtraction process and we will consider the following possibilities˜ b ε ( ψ ) − λ n ε k ψ k G = ˜ b ε ( ψ ) − λ n Z R × S dsdy β ε ( s, y ) | ψ ( s, y ) | . UANTUM RESOLVENT AND Γ CONVERGENCES 13
The second regularization is simply a division by the global factor ε so definingthe family of quadratic forms we will work with: b εn ( ψ ) := ε − (cid:18) ˜ b ε ( ψ ) − λ n ε k ψ k G (cid:19) = Z R × S dsdy (cid:20) β ε |∇ ψ · (1 , Ry ( τ − ˙ α )) | + β ε ε (cid:0) |∇ ⊥ ψ | − λ n | ψ | (cid:1)(cid:21) , dom b εn = H ( R × S ). In [8] only b ε was considered, and ahead we propose aprocedure to deal with b εn , for all n ≥
0. After such regularizations we finallyobtain the operators H εn , associated with these forms, for which we will investigatethe limit ε →
0. Let H ε denote the Hilbert space L ( R × S, β ε ), that is, the innerproduct is given by h ψ, ϕ i ε := Z R × S ψ ( s, y ) ϕ ( x, y ) β ε ( s, y ) dsdy. Definition 2.
The operator H εn is the self-adjoint operator associated with thesesquilinear form b εn (see [1] , page 101), whose domain dom H εn is dense in dom b εn and b εn ( ψ, ϕ ) = h ψ, H εn ϕ i ε , ∀ ψ ∈ dom b εn , ∀ ϕ ∈ dom H εn . In case the functions κ, τ, ˙ α are bounded and S has a smooth boundary then, byelliptic regularity [26] , dom H εn = H ( R × S ) ∩ H ( R × S ) . Remark 5.
In the above construction of H ε and H εn it was very important theuniform bound of β ε , which implies that for all ε > small enough there exist < a ε ≤ a ε < ∞ with a ε k · k ≤ k · k ε ≤ a ε k · k , (similar inequalities also hold for the associated quadratic forms b εn ) so that allHilbert spaces H ε coincide algebraically with L ( R × S ) and also have equivalentnorms. Furthermore, it is possible to assume that both a ε → and a ε → hold as ε → , since the sequence of functions β ε → uniformly. These properties implythe equality of the quadratic form domains for all ε > and permit us to speakof Γ -convergence of b εn and resolvent convergence of H εn in L ( R × S ) even thoughthey act in, strictly speaking, different Hilbert spaces. Such facts will be freely usedahead. Definition 3.
Given Ω εα , let C n ( S ) := R S dy |∇ ⊥ u n ( y ) · Ry | , where, as before, u n is the ( n + 1) th normalized eigenfunction of the negative Laplacian on S . The tube Ω εα is said to be quantum twisted if for some n the function A n ( s ) := ( τ ( s ) − ˙ α ( s )) C n ( S ) = 0 . Note that the quantities C n ( S ) are parameters that depend only on the crosssection S , and that A n is zero if either the cross section rotation compensatesthe torsion of the curve (i.e., τ − ˙ α = 0) or if the eigenfunction u n is radial (i.e., C n ( S ) = 0); A n has a geometrical nature and A was first considered [8]. In [25]there is an interesting list of equivalent formulations of the condition τ − ˙ α = 0.It should be mentioned that the case of twisting has been addressed also in thecase of infinite curve γ in [22], and that there are some studies realized in R n for n ≥ Confinement: Statements.
In this subsection we discuss results about thelimit ε → H εn . We begin with H ε and will follow closely [8], where Γ-conver-gence was used to study the limit operator upon confinement and convergence ofeigenvalues in case of bounded curves γ (so bounded tubes). We will point out thenecessary modifications in their approach in order to prove Theorem 5 below, whichis in the setting of L ( R × S ) and no restriction on the curve length, although ourresults on spectral convergence are limited to the standard consequences of strongoperator resolvent convergence (not stated here; see, for instance, Corollary 10.2.2in [1]); in any event, ahead we shall recover their spectral convergence by means ofour Proposition 4.For each n ≥
0, denote by L n (resp. h n ) the Hilbert subspace of L ( R × S )(resp. H ( R × S )) of vectors of the form ψ ( s, y ) = w ( s ) u n ( y ), w ∈ L ( R ) (resp. w ∈ H ( R ) = H ( R )), so thatL ( R × S ) = L ⊕ L ⊕ L ⊕ · · · , H ( R × S ) = h ⊕ h ⊕ h ⊕ · · · , and each h n is a dense subspace of L n . Note that h n is related to b εn . In Theorem 5we deal with the limit of b ε and generalize some results of [8] for elements of h ; thenwe propose a procedure to deal with the relation between the limit of b εn and h n ,for all n .Now define the real potentials V eff n ( s ) := A n ( s ) − κ ( s ) = ( τ ( s ) − ˙ α ( s )) C n − κ ( s ) , n ≥ , and the corresponding Schr¨odinger operators( H n ψ )( s ) = − d ds ψ ( s ) + V eff n ( s ) ψ ( s ) , whose domains are dom H n = H ( R ) , for all n ≥ , in case of bounded functions κ, τ, ˙ α . Here the curvature κ is always assumed to be bounded, but if either thetorsion τ or the ˙ α is not bounded, the domain must be discussed on an almostcase-by-case basis. The subspace L n can be identified with dom H n = L ( R ) via wu n w . Let b n be the sesquilinear form generated by H n , that is, b n ( ψ ) = Z R ds (cid:16) | ˙ ψ ( s ) | + V eff n ( s ) | ψ ( s ) | (cid:17) whose dom b n = H ( R ) (for bounded κ, τ, ˙ α ) can be identified with h n ; hence b n ( ψ ) = ∞ if ψ ∈ L n \ h n . Theorem 5.
The sequence of self-adjoint operators H ε converges in the strongresolvent sense to H in L as ε → . Remark 6.
We have then obtained an explicit form of the effective operator thatdescribes the confinement of a free quantum particle (in the subspace L ) from Ω εα to the curve γ . The action of the operator H is the same as that in [8] and, asalready anticipated, the arguments below for its proof are based on Γ -convergenceof quadratic forms and mainly consist of indications of the necessary modificationsto take into account unbounded curves, that is, the results of Section 2 and the UANTUM RESOLVENT AND Γ CONVERGENCES 15 important additional verification of weak Γ -convergence of the involved quadraticforms. It is possible to get some intuition about what should be expected for the conver-gence of H εn , n ≥
1, by considering the form b εn evaluated at vectors of h n , i.e., wu n , w ∈ H ( R ), and some formal arguments. A direct substitution gives an integralwith four terms b εn ( wu n ) = Z R × S dsdy h β ε | ˙ w | | u n | + | w | (cid:16) β ε |∇ ⊥ u n · Ry ( τ − ˙ α ) | + β ε ε (cid:0) |∇ ⊥ u | − λ | u | (cid:1) (cid:17) + 1 β ε wu n w ∇ ⊥ u n · Ry ) i , and if we approximate β ε ≈
1, except in the third term, we find that the lastterm vanishes due to the Dirichlet boundary condition and thus (recall that u n isnormalized in L ( S )) b εn ( wu n ) ≈ Z R ds (cid:2) | ˙ w ( s ) | + C n ( S )( τ ( s ) − ˙ α ( s )) | w ( s ) | (cid:3) + Z R ds | w ( s ) | Z S dy β ε ε (cid:0) |∇ ⊥ u n ( y ) | − λ n | u n ( y ) | (cid:1) (cid:1) , so that the first term A n ( s ) in the effective potential V eff n ( s ) is clearly visible.However, the remaining term K εn ( u, s ) := Z S dy β ε ( s, y ) ε (cid:0) |∇ ⊥ u ( y ) | − λ n | u ( y ) | (cid:1) is related to the curvature and requires much more work; this was a major contri-bution of [8] in case n = 0 through the study of minima and strong Γ-convergence.It is shown in [8] that K ε attains a minimum − κ / > −∞ since λ is the bot-tom of the spectrum of the Laplacian restricted to the cross section S ; however, aminimum is not expected to occur in case of K εn , n ≥
1, since, if we take again theapproximation β ε ≈
1, there will be vectors u ∈ h j , j < n , such that K εn ( u, s ) ≈ ε ( λ j − λ n ) → −∞ , ε → , and this unboundedness from below pushes u away of the natural range of applica-bility of quadratic forms and Γ-convergence. We then impose that if b εn ( ψ ) → ±∞ ,then the vector ψ is simply excluded from the domain of the limit sesquilinear form,and so also from the domain of the limit operator. Since Theorem 5 tells us howto deal with vectors in L , by taking the above discussion into account we proposeto study the convergence of b ε restricted to the orthogonal complement of L inL ( R × S ) and, more generally, to study the Γ-convergence of b εn restricted to E n := (L ⊕ L ⊕ · · · ⊕ L n − ) ⊥ , n ≥ . In what follows these domain restrictions will also apply to the resolvent conver-gence of the restriction to E n of the associated self-adjoint operators and will be usedwithout additional warnings. Under such procedure we shall obtain the followingresult (recall that it is assumed that all eigenvalues λ n of the Laplacian restrictedto the cross section S are simple). Theorem 6.
The sequence of self-adjoint operators obtained by the restriction of H εn to dom H εn ∩ E n converges, in the strong resolvent sense in L n , to H n as ε → . Remark 7.
Since quadratic forms can conveniently take the value + ∞ , here wehave rather different nomenclatures for singular convergences, that is, the sequenceof forms b εn converges in E n whereas the matching sequence of operators H εn con-verges in L n . Remark 8.
Note that this procedure leads to a kind of decoupling among the sub-spaces h n , which is supported by some cases of plane waveguides treated in [10] where the decoupling is found through an explicitly separation of variables; see thepresence of Kronecker deltas in the expressions for resolvents in Theorem 1 andLemma 3 in [10] . In 3D such separation occurs if there is no twisting [11] . Remark 9.
In the first approximation, in the vicinity of the overall spectral thresh-old (i.e., by considering λ ), the effective Hamiltonian describes bound states. How-ever, at higher thresholds (i.e., λ n , n ≥ ), the effective Hamiltonians are usuallyrelated to resonances in thin tubes (see [21, 20] ), and so this physical content isanother motivation for the consideration of b n , n ≥ . Remark 10.
A novelty with respect to similar situations in R [10, 9] is that theeffective operator H n describing the particle confined to the curve γ ( s ) dependson n , that is, we have different quantum dynamics for different subspaces L n . Asmentioned in the Introduction, this is a counterpart of some situations found in [15, 14] , since there the effective dynamics may depend on the initial condition. Since the above procedure was strongly based on the mentioned estimate K εn ( u, s ) ≈ ε ( λ j − λ n ) → −∞ , j < n, as well as the rather natural expectation that it is “spontaneously” possible anenergy transfer only from higher to lower energy states, we present a rigorousversion of such estimate. Proposition 5.
For normalized u ( s, y ) = w ( s ) u j ( y ) , w ∈ H ( R ) , one has lim ε → ε b εn ( wu j ) = ( λ j − λ n ) . In particular, for small ε and a.e. s , K εn ( u, s ) ≈ ε ( λ j − λ n ) → −∞ if j < n as ε → .Proof. Again a direct substitution and taking into account the Dirichlet boundarycondition yield b εn ( wu j ) = Z R × S dsdy h β ε | ˙ w | | u j | + | w | (cid:16) β ε |∇ ⊥ u j · Ry ( τ − ˙ α ) | + β ε ε (cid:0) |∇ ⊥ u j | − λ | u j | (cid:1) (cid:17)i , and the only term that may not vanish as ε → ε b εn ( wu j ) is the last one, so itis enough to analyze K εn ( u j , s ) := Z S dy β ε ( s, y ) ε (cid:0) |∇ ⊥ u j ( y ) | − λ n | u j ( y ) | (cid:1) . Write β ε = 1 − ξ · y , with ξ = εκ ( s ) z α and z α = (cos α, sin α ). Since u j ∈ H ( S )and u j satisfies the Dirichlet boundary condition, upon integrating by parts one UANTUM RESOLVENT AND Γ CONVERGENCES 17 gets, for a.e. s , K εn ( u j , s ) = − ε Z S dy u j ( ∇ ⊥ · ( β ε ∇ ⊥ u j ) + λ n β ε u j ))= − ε Z S dy u j ( ∇ ⊥ β ε · ∇ ⊥ u j + β ε (∆ ⊥ u j + λ n u j ))= − ε Z S dy u j ( − ξ · ∇ ⊥ u j + β ε ( − λ j + λ n ) u j )= 1 ε κ ( s ) z α · Z S dy u j ∇ ⊥ u j + ( λ j − λ n ) ε Z S dy β ε | u j | . Exchanging the roles of u j and u j yields K εn ( u j , s ) = 1 ε κ ( s ) z α · Z S dy u j ∇ ⊥ u j + ( λ j − λ n ) ε Z S dy β ε | u j | , and by adding such expressions K εn ( u j , s ) = 12 ε κ ( s ) z α · Z S dy ∇ ⊥ | u j | + ( λ j − λ n ) ε Z S dy β ε | u j | . The Dirichlet boundary condition implies R S dy ∇ ⊥ | u j | = 0. Since by dominatedconvergence R S dy β ε | u j | → k u j k = 1 as ε →
0, it follows that ε K n,j ( s ) → ( λ j − λ n ) for a.e. s ∈ R . Again by dominated convergencelim ε → ε b εn ( wu j ) = ( λ j − λ n ) . The proof is complete. (cid:3)
The reader may protest at this point that the above procedure of consideringrestriction to E n when dealing with b n , n >
0, should in fact be deduced insteadof assumed. However, we think that such deduction is certainly beyond the rangeof Γ-convergence, since by beginning with the form b εn the vectors ψ ∈ h j , j > n ,will not belong to the domain of the limit form b n since lim ε → b εn ( ψ ) = + ∞ , whilefor vectors ψ ∈ h j , j < n , the lim ε → b εn ( ψ ) = −∞ indicates that they wouldbe outside the usual scope of limit of forms and Γ-convergence. Further, we havegot reasonable expectations that the domain of the limit form b n should consists ofonly vectors in h n ⊂ L n . By taking into account the proposed procedure of suitabledomain restrictions, in the next subsection we will support such expectations byproving the above theorems; we stress that there is no mathematical issue in theformulation of the above procedure (e.g., if ψ ∈ dom H εn ∩ E n , then H εn ψ ∈ E n ) andthat it rests only on physical interpretations.3.3. Confinement: Convergence.
In this subsection the proofs of Theorems 5and 6 are presented. There are two main steps; the first one is to check the strongΓ-convergence (recall the restriction to the subspace E n ) b εn SΓ −→ b n , and the second one is a verification that if ψ ε ⇀ ψ in E n ⊂ L ( R × S ), then b n ( ψ ) ≤ lim inf ε → b εn ( ψ ε ) . The theorems will then follow by Theorem 3, including the additional information b εn WΓ −→ b n in E n . The proof of the first step is just a verification that the proof ofthe corresponding result in [8] can be explicitly carried out for unbounded tubes and for b εn with n ≥
0. Then we shall address the proof of the second step and atthe end of this subsection we briefly discuss how the spectral convergences in [8]can be recovered from our Proposition 4.3.3.1.
First Step.
I will be very objective and just indicate the adaptations to theproofs in [8]; I will also use a notation similar to the one used in that work. Thefirst point to be considered is a variation of Proposition 4.1 in [8], that is, the studyof the quantities λ n ( ξ ) := inf = v ∈H S ) v ∈ [ u , ··· ,un − ⊥ R S (1 − ξ · y ) |∇ ⊥ v ( y ) | R S (1 − ξ · y ) | v ( y ) | , n > , and γ ε,n ( s ) := 1 ε ( λ n ( ξ ) − λ n ) , ξ = εκ ( s ) z α ( s );recall that z α = (cos α, sin α ). In case n = 0 the above infimum is taken over0 = v ∈ H ( S ); [ u , · · · , u n ] denotes the subspace in H ( S ) spanned by the firsteigenvectors u , · · · , u n of the (negative) Laplacian restricted to the cross section S and recall that here its eigenvalues λ < λ < λ · · · are supposed to be simple.The prime goal is to show that γ ε,n ( s ) → − κ ( s )uniformly on R (Proposition 4.1 in [8]; don’t confuse γ ε,n ( s ) with the referencecurve γ ( s )). In order to prove this we consider the unique solution u ξ,n ∈ H ( S ) ofthe problem − ∆ ⊥ u ξ,n − λ n u ξ,n = − ξ · ∇ ⊥ u n , u ξ,n ∈ [ u , · · · , u n ] ⊥ , which exists by Fredholm alternative since [ u , · · · , u n ] ⊥ is invariant under the op-erator ( − ∆ ⊥ ), the real number λ n belongs to the resolvent set of the restriction( − ∆ ⊥ ) | [ u , ··· ,u n ] ⊥ (again because the eigenvalues are simple) and ξ ·∇ ⊥ u n is orthog-onal to u n . Note that the simplicity of eigenvalues was again invoked to guaranteethe unicity of solution and that u ξ,n is real since its complex conjugate is also asolution of the same problem.Since we can take real-valued eigenfunctions u n , the Lemma 4.3 in [8] will readinf v ∈H S ) v ∈ [ u , ··· ,un − ⊥ Z S dy (cid:2) |∇ ⊥ v | − λ n | v | + ( ξ · ∇ ⊥ u n )( v + ¯ v ) (cid:3) = − ξ , with the infimum reached precisely for u ξ,n , also a real-valued function. With suchremarks the proof of Lemma 4.3 in [8] also works for all n ≥
0. The same wordscan be repeated for the proof of the above mentioned Proposition 4.1, the onlynecessary additional remark is that the function ϕ in their equation (4.12) canalso be taken real, since in our context it always appears in the form ( ϕ + ¯ ϕ ).With such modifications we obtain suitable versions of that proposition, that is, γ ε,n ( s ) → − κ ( s ) uniformly on R as ε → n ≥
0. With such tools at handthe proof of b εn SΓ −→ b n in E n , for all n ≥
0, is exactly that done in Section 4.3 of [8].This completes our first main step.
UANTUM RESOLVENT AND Γ CONVERGENCES 19
Second Step.
Now we are going to complete the proofs of Theorems 5 and 6.However we consider a simple modification of the forms b εn , that is, we shall consider b εn,c ( ψ ) := b εn ( ψ ) + c k ψ k ε , dom b εn,c = dom b εn ⊂ E n , for some c ≥ k κ k ∞ so that we work with the more natural set of positive forms.This implies that if H εn is the self-adjoint operator associated with b εn , then H εn,c := H εn + c is the operator associated with b εn,c and vice versa; so strong resolventconvergence H εn → H n in L n is equivalent to strong resolvent convergence H εn,c → H n,c := H n + c in L n as ε → ε > ψ ∈ dom b εn,c we have the important lower bound (by thedefinitions of the forms and γ ε,n ; see also equation 4.19 in [8]) b εn,c ( ψ ) ≥ Z R × S dsdy (cid:18) β ε (cid:12)(cid:12)(cid:12) ˙ ψ + ∇ ⊥ ψ · Ry ( τ − ˙ α ) (cid:12)(cid:12)(cid:12) + β ε ( c − γ ε,n ) | ψ | (cid:19) . Our interest in considering the modified forms b εn,c is also that, for ε small enough, b εn,c ( ψ ) ≥ (cid:18) c − k κ k ∞ (cid:19) k ψ k ≥ k κ k ∞ k ψ k and so both b εn,c and H εn,c are positive; this will be important when dealing withweak convergence ahead. Furthermore, the arguments in First Step above andTheorem 2, with f playing the role of the norm, which is strongly continuous, alsoshow that b εn,c SΓ −→ b n,c in E n , where b n,c ( ψ ) := Z R ds (cid:18) | ˙ ψ ( s ) | + (cid:20) ( τ ( s ) − ˙ α ( s )) C n + c − κ ( s ) (cid:21) | ψ ( s ) | (cid:19) is the form generated by H n,c , with dom b n,c = H ( R ) identified with h n .Now we have the task of checking that if ψ ε ⇀ ψ in E n ⊂ L ( R × S ), thenlim inf ε → b εn,c ( ψ ε ) ≥ b n,c ( ψ ) , in other words, the second main step mentioned above. By Theorem 3 i) we willthen conclude the strong resolvent convergence H εn,c → H n,c in L n , which willcomplete the desired proofs. So assume the weak convergence ψ ε ⇀ ψ . If ψ ε doesnot belong to H ( R × S ) ∩ E n , then b εn,c ( ψ ε ) = ∞ , for all ε >
0; so we can assumethat ( ψ ε ) ⊂ H ( R × S ) ∩ E n and, up to subsequences, thatlim inf ε → b εn,c ( ψ ε ) = lim ε → b εn,c ( ψ ε ) . Since ( ψ ε ) is a weakly convergent sequence, it is bounded in L ( R × S ) and wecan also suppose that sup ε b εn,c ( ψ ε ) < ∞ ; hence, as on page 804 of [8], it followsthat ( ψ ε ) is a bounded sequence in H ( R × S ). Since Hilbert spaces are reflexive,( ψ ε ) has a subsequence, again denoted by ( ψ ε ), so that ψ ε ⇀ φ in H ( R × S ). Sincealso ψ ε ⇀ ψ in L ( R × S ) it follows that ψ = φ . Therefore˙ ψ ε + ∇ ⊥ ψ ε · Ry ( τ − ˙ α ) ⇀ ˙ ψ + ∇ ⊥ ψ · Ry ( τ − ˙ α )and the weak lower semicontinuity of the L -norm (again the importance of intro-ducing the parameter c above), the above lower bound b εn,c ( ψ ) ≥ / k κ k ∞ k ψ k , together with the uniform convergences β ε → , γ ε,n → − κ , imply thatlim ε → b εn,c ( ψ ε ) ≥ G ( ψ ):= Z R × S dsdy (cid:18)(cid:12)(cid:12)(cid:12) ˙ ψ + ∇ ⊥ ψ · Ry ( τ − ˙ α ) (cid:12)(cid:12)(cid:12) + (cid:18) c − κ (cid:19) | ψ | (cid:19) . If ψ ∈ dom b n,c , that is, ψ = w ( s ) u n ( y ), w ∈ H ( R ), then a direct substitutioninfers that G ( wu n ) = b n,c ( wu n ) and solim ε → b εn,c ( ψ ε ) ≥ b n,c ( ψ )in this case. Now we will show that, for ψ with a nonzero component in thecomplement of dom b n,c , necessarily lim ε → b εn,c ( ψ ε ) = ∞ . In fact, if ψ does notbelong to dom b n,c then k P n +1 ψ k > P n +1 is the orthogonal projection onto E n +1 . Because ψ ε ⇀ ψ in E n ∩ H ( R × S ) it follows that P n +1 ψ ε ⇀ P n +1 ψ , andsince the L -norm is weakly l.sc. we findlim inf ε → k P n +1 ψ ε k ≥ k P n +1 ψ k > . Hence for ε small enough the function ψ ε has a nonzero component P n +1 ψ ε in E n +1 and the L ( R × S )-norm of such components are uniformly bounded from zero by k P n +1 ψ k . Now, sup ε k ψ ε k ε < ∞ and recalling that β ε ( s, y ) = 1 − ξ · y, ξ = εκ ( s ) z α , one has b εn,c ( ψ ε ) = Z R × S dsdy h β ε |∇ ψ ε · (1 , Ry ( τ − ˙ α )) | + β ε ε (cid:0) |∇ ⊥ ψ ε | − λ n | ψ ε | (cid:1) i + c k ψ ε k ε ≥ Z R × S dsdy β ε ε (cid:0) |∇ ⊥ ψ ε | − λ n | ψ ε | (cid:1) = 1 ε Z R × S dsdy (cid:0) |∇ ⊥ ψ ε | − λ n | ψ ε | (cid:1) − ε Z R × S dsdy ( ξ · y ) (cid:0) |∇ ⊥ ψ ε | − λ n | ψ ε | (cid:1) . Let us estimate the remanning two integrals above; for φ ∈ H ( S ) ∩ [ u , · · · , u n − ] ⊥ denote by φ ( n ) the component of φ in [ u n ] and by Q n +1 the orthogonal projectiononto [ u , · · · , u n ] ⊥ in H ( S ). The first integral is positive and divergent as 1 /ε to UANTUM RESOLVENT AND Γ CONVERGENCES 21 + ∞ since for ε small enough Z R × S dsdy ε (cid:0) |∇ ⊥ ψ ε | − λ n | ψ ε | (cid:1) = 1 ε Z R ds (cid:16) k∇ ⊥ ψ ε ( s ) k ( S ) − λ n k ψ ε ( s ) k ( S ) (cid:17) = 1 ε Z R ds (cid:16) k ψ ε ( s ) k H ( S ) − ( λ n + 1) k ψ ε ( s ) k ( S ) (cid:17) = 1 ε Z R ds (cid:16) k Q n +1 ψ ε ( s ) k H ( S ) + k ψ ( n ) ε ( s ) k H ( S ) − ( λ n + 1) k ψ ε ( s ) k ( S ) (cid:17) = 1 ε Z R ds (cid:16) k∇ ⊥ Q n +1 ψ ε ( s ) k ( S ) + k Q n +1 ψ ε ( s ) k ( S ) + k∇ ⊥ ψ ( n ) ε ( s ) k ( S ) + k ψ ( n ) ε ( s ) k ( S ) − ( λ n + 1) k ψ ε ( s ) k ( S ) (cid:17) ≥ ε Z R ds (cid:16) λ n +1 k Q n +1 ψ ε ( s ) k ( S ) + λ n k ψ ( n ) ε ( s ) k ( S ) − λ n k ψ ε ( s ) k ( S ) (cid:17) = 1 ε Z R × S ds ( λ n +1 − λ n ) k Q n +1 ψ ε k ( S ) = ( λ n +1 − λ n ) ε k P n +1 ψ ε k ≥ ( λ n +1 − λ n ) ε k P n +1 ψ k , and, by hypothesis, λ n +1 > λ n . The absolute value of the second integral divergesat most as 1 /ε ; indeed, since ( ψ ε ) is a bounded sequence in H ( R × S ) and we have1 ε (cid:12)(cid:12)(cid:12)(cid:12)Z R × S dsdy ( ξ · y ) (cid:0) |∇ ⊥ ψ ε | − λ n | ψ ε | (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k κ k ∞ ε Z R × S dsdy | z α · y | (cid:0) |∇ ⊥ ψ ε | + λ n | ψ ε | (cid:1) ≤ k κ k ∞ ε sup | z α · y | × max { λ n , /λ n }k ψ ε k H . Therefore, if ψ does not belong to dom b n,c , thenlim inf ε → b εn,c ( ψ ε ) = + ∞ . We have then verified the statement i) of Theorem 3, and so Theorem 6 follows byTheorem 3v); Theorem 5 is just a particular case with n = 0.3.4. Spectral Possibilities.
There is a competition between the curvature andthe twisting terms in the effective potential V eff n ( s ) = ( τ ( s ) − ˙ α ( s )) C n ( S ) − κ ( s ) , n ≥ , since the curvature gives an attractive term and the twisting a repulsive one.This effective potential is the net result of a memory of higher dimensions thattakes into account the geometry of the confining region. In planar (2D) cases only the curvature term is present and (if its not zero) a bound state does always exist.In the spatial (3D) case, by tuning up the tubes, and so the functions that defineeffective potentials, one finds a huge amount of spectral possibilities for the effectiveSchr¨odinger operator in the reference curve γ ( s ). Below some of them are selected;it will be assumed that C n ( S ) = 0 and that τ, ˙ α are not necessarily bounded; this isonly related to the domain of the forms and involved operators, differently from theessential technical condition of bounded curvature. It is worth mentioning that forlower bounded potentials V (in particular for V eff n above) that belong to L ( R k )the operators − ∆ + V are essentially self-adjoint when defined on C ∞ ( R k ); see, forinstance, Section 6.3 in [1].(1) No twisting.
In this case A n ( s ) = 0 and the 3D situation is quite similarto the 2D one; the effective potential V eff n ( s ) = − / κ ( s ) is purely at-tractive and the spectrum of H n has at least one negative eigenvalue. SeeSubsection 11.4.4 in [1].(2) Periodic.
By choosing bounded κ, τ, ˙ α so that V eff n ( s ) becomes periodicthe resulting effective operators H n have purely absolutely continuous spec-tra and with a band-gap structure [27]. Such periodicity may come fromdifferent combinations; for instance, the tube curvature and torsion couldbe periodic (with the same period) and α ( s ) a constant function, or thetube could be straight so that κ ( s ) = 0 = τ ( s ) but the cross section S rotates at a periodic speed ˙ α ( s ). See also [28].(3) Purely discrete.
The operators H n will have this kind of spectrum iflim | s |→∞ V eff n ( s ) = ∞ (see Section 11.5 in [1]); this happens iff the torsion τ or ˙ α , as well as their difference, diverge at both ±∞ . In particular H n will have discrete spectrum in case this limit operator is obtained from astraight tube with growing rotation speed of the cross section such thatlim | s |→∞ ˙ α ( s ) = ∞ .(4) Quasiperiodic.
For one of the simplest situations select ˙ α and κ periodicfunctions with (minimum) periods t α > t κ >
0, respectively; if t α /t κ is an irrational number we are in the case of quasiperiodic potentials. Inthis case there are many spectral possibilities that usually are very sensitiveto details of the potential. Of course one may also take V eff n in the moregeneral class of almost periodic functions; see, for instance, [29].(5) Singular continuous.
An appealing possibility is the choice of decayingpotentials V eff n in the class studied by Pearson [30], which leads to singularcontinuous spectrum for H n . See also explicit examples in [31].The reader can play with his/her imagination in order to consider tubes thatgive rise to previously selected spectral types.3.5. Bounded Tubes.
Now we say something about the particular case of boundedtubes; the goal is to recover the spectral results of [8]. Since the cross section S isa bounded set, the boundedness of the tubeΩ εα,L := (cid:8) ( x, y, z ) ∈ R : ( x, y, z ) = f εα ( s, y , y ) , s ∈ [0 , L ] , ( y , y ) ∈ S (cid:9) , is a consequence of a bounded generating curve γ ( s ) defined, say, on a compactset s ∈ [0 , L ] instead of on the whole line R as before. In this case the negativeLaplacian operator − ∆ Ω εα,L has compact resolvent and its spectrum is composedonly of eigenvalues λ εj , j ∈ N ; denote by ψ εj the normalized eigenfunction associated UANTUM RESOLVENT AND Γ CONVERGENCES 23 with λ εj . Let b εL and H ε ( L ) be the the corresponding sesquilinear form and self-adjoint operator, after the “regularizations” and acting in subspaces of L ([0 , L ] × S ), suitably adapted from Subsection 3.1 (the same quantities as in [8]). Theorem 7.
For each j ∈ N one has lim ε → (cid:18) λ εj − λ ε − µ j (cid:19) = 0 , where µ j are the eigenvalues of the of the Schr¨odinger operator dom H ( L ) = H (0 , L ) ∩ H (0 , L ) , ( H ( L ) ψ )( s ) = − ¨ ψ ( s ) + (cid:18) ( τ ( s ) − ˙ α ( s )) C ( S ) − α ( s ) (cid:19) ψ ( s ) . Furthermore, there are subsequences of f εα ( ψ εj ) that converge to w j ( s ) u ( y ) in L ([0 , L ] × S ) as ε → , where w j are the normalized eigenfunctions corresponding to µ j .Proof. The proof will be an application of Proposition 4, with T = H ( L ), T ε = H ε ( L ), H = { w ( s ) u ( y ) : w ∈ H (0 , L ) } . Let b L be the form generated by H ( L ).Previously discussed results in the case of unbounded tubes apply also here andthey show that b εL SΓ −→ b L , b εL WΓ −→ b L , that is, item a) of Proposition 4 holds in this setting. Since H ( L ) has compactresolvent, item b) in that proposition follows at once.Finally, for each ε > ψ ε ) is bounded in L ( R × S ) and b εL ( ψ ε ) is bounded, imply that (see page 804 of [8]) ( ψ ε ) is a bounded sequence in K = H ([0 , L ] × S ). By Rellich-Kondrachov Theorem the space K is compactlyembedded in L ([0 , L ] × S ) (due to the boundedness of [0 , L ] × S ), and so item c)of Proposition 4 holds. By that proposition H ε ( L ) converges in the norm resolventsense to H ( L ) in H , and it is well known that the spectral assertions in Theorem 7follow by this kind of convergence, that is, the convergence of eigenvalues of H ε ( L )to eigenvalues of H ( L ) as well as the assertion about convergence of eigenfunctions.Taking into account that in the construction of H ε ( L ) there was the “regularization”subtraction of ( λ /ε ) k ψ k ε from the original form of the Laplacian − ∆ Ω εα,L , theconclusions of Theorem 7 follow. (cid:3) Remark 11.
Theorem 7 makes clear the mechanism behind the spectral approxi-mations in case of bounded tubes, that is, the powerful norm resolvent convergence is in action!
Remark 12.
Although we expect that for Theorem 6 the norm resolvent conver-gence takes place, we were not able to prove it; at the moment, to get norm con-vergence we need a combination of Γ -convergence and compactness of the tube (asin Theorem 7). A very simple example indicates how subtle those properties can becombined and that our expectations might be wrong.Consider the sequence of multiplication operators T n ψ ( x ) = xψ ( x ) /n and T = 0 .In the space L ( R ) , dominated convergence implies that T n converges to T in thestrong resolvent sense. Now, σ ( T n ) = R , for all n , while σ ( T ) = { } ; thus, T n doesnot converge in the norm resolvent sense to T . However, for the same operatoractions in L [0 , one gets that T n converges to T in the norm resolvent sense (dueto the compactness of [0 , ). Remark 13.
It is also possible to consider semi-infinite tubes, that is, s ∈ [0 , ∞ ] .In this case all previous constructions apply and the limit operator has the expectedaction but with Dirichlet boundary condition at zero. The details are similar to thearguments previously discussed here and in [8] and will be omitted. Broken-Line Limit
In this section we discuss the operators H n , defined on a spacial curve γ ( s ) withcompactly supported curvature, that approximate another singular limit, now givenby two infinite straight edges with one vertex at the origin. The angle between thestraight edges is θ and is kept fixed during the approximation process. In caseof planar curves this problem has been considered in [9, 10] and a variation of itin [32], and those authors had at hand explicitly expressions for the resolvents ofthe Hamiltonians as integral kernels.This geometrical broken line is a simple instance of a quantum graph and themain question is about the boundary conditions that is selected at the vertex inthe convergence process; that is expected to be the physical boundary conditions.We refer to the above cited references for more physical and mathematical details.Note that in this work we have restricted ourselves to first confine the quantumsystem from the tube to the curve, and then take the broken-line limit. In theplanar curve cases both limits are taken together (as in [9, 10, 32]) and with noreference to Γ-convergence; but since it may involve sequences of operators thatare not uniformly bounded from below [9], it is not clear that in our 3D settingwe could address both limits together by using Γ-convergence; this seems to be aninteresting open problem.Of course a novelty here is the possibility of quantum twisting in the effectivepotentials V eff n ( s ) = ( τ ( s ) − ˙ α ( s )) C n − κ ( s ) , n ≥ , since in the plane cases [9, 10] only the curvature term − κ ( s ) / n + 1)th sectorspanned by the eigenvector u n of the Laplacian restricted to the cross section S .Thus the limit operator depends on n and since this additional term A n ( s ) ispositive we have a wide range of possibilities in the 3D case, that is, not just anattractive potential as in 2D. From the technical point of view we will follow closelythe proof of Lemma 1 in [10], which uses results of [33]. However, differently thanthe planar situation, the condition h V eff n i := Z R ds V eff n ( s ) = 0may not hold in 3D, but we will see that the same proof can be adapted to the case h V eff n i = 0 by using results of [34]; the boundary conditions at the vertex dependexplicitly on the curvature and twisting.We will be rather economical in the proofs below, since we do not intend to justrepeat whole parts of published works; we are sure that from the statements belowand references to papers and specific equations, the interested reader will have nospecial difficulties in filling out the missing details. UANTUM RESOLVENT AND Γ CONVERGENCES 25
Assume that the curvature, torsion and the speed of the rotation angle ˙ α arecompactly supported in ( − ,
1) and scale them as κ δ ( s ) := 1 δ κ (cid:16) sδ (cid:17) , τ δ ( s ) := 1 δ τ (cid:16) sδ (cid:17) , ˙ α δ ( s ) := 1 δ ˙ α (cid:16) sδ (cid:17) , and the continuations of the half-lines to the left and to the right of that supportjoint at the origin with an angle θ ; this angle is exactly the integral θ = Z R ds κ δ ( s ) = Z R ds κ ( s ) , ∀ δ > . Of course, as above, the curve γ is supposed to be smooth and without self-intersection. Here we consider only the above scales. Our concern now is to studythe limit δ → H n ( δ ) ψ )( s ) = − ¨ ψ ( s ) + V eff n,δ ( s ) ψ ( s ) , dom H n ( δ ) = H ( R ) , n ≥ , where V eff n,δ ( s ) := ( τ δ ( s ) − ˙ α δ ( s )) C n − κ δ ( s ) . It turns out that this limit δ → R k ( H n ), Im k >
0, as explained on page 8 of [10]. The operator H n is said to have a resonance at zero if there exists ψ r ∈ L ∞ ( R ), ψ / ∈ L ( R ),such that H n ψ r = 0 in the sense of distributions; in this case ψ r can be chosenreal and is unique (as a subspace). Since all V eff n have compact support, one has R R ds e as | V eff n ( s ) | < ∞ for some a >
0, which is a technical condition necessary forwhat follows [10, 34, 33]. Now we consider two complementary cases: h V eff n i 6 = 0and h V eff n i = 0.4.1. h V eff n i 6 = 0 . Assume that this condition holds. In this case we may directlyapply Lemma 1 in [10], which employs results of [33], to obtain:
Proposition 6. (a) If H n has no resonance at zero, then H n,δ converges in thenorm resolvent sense, as δ → , to the one-dimensional Laplacian − ∆ D withDirichlet boundary condition at the origin, that is, dom ( − ∆ D ) = (cid:8) ψ ∈ H ( R ) ∩ H ( R \ { } ) : ψ (0) = 0 (cid:9) , ( − ∆ D ψ )( s ) = − ¨ ψ ( s ) . (b) If H n has a resonance at zero, then H n,δ converges in the norm resolventsense, as δ → , to the one-dimensional Laplacian − ∆ r given by dom ( − ∆ r ) = (cid:8) ψ ∈ H ( R \ { } ) : ( c n + c n ) ψ (0 + ) = ( c n − c n ) ψ (0 − ) , ( c n − c n ) ˙ ψ (0 + ) = ( c n + c n ) ˙ ψ (0 − ) (cid:9) , ( − ∆ r ψ )( s ) = − ¨ ψ ( s ) , where c n = 12 h V eff n i Z R × R dsdy V eff n ( s ) | s − y | V eff n ( y ) ψ r ( y ) ,c n = − Z R ds sV eff n ( s ) ψ r ( s ) . Moreover, c n and c n do not vanish simultaneously. h V eff n i = 0 . Assume that this condition holds. Now we cannot apply directlyLemma 1 in [10], but by invoking results of [34] we can check that the proof of suchLemma 1 may be replicated to conclude:
Proposition 7. (a) If H n has no resonance at zero, then H n,δ converges in thenorm resolvent sense, as δ → , to the one-dimensional Laplacian − ∆ D withDirichlet boundary condition at the origin. (b) If H n has a resonance at zero, then H n,δ converges in the norm resolventsense, as δ → , to the one-dimensional Laplacian − ∆ r , as in Proposition 6 (b) ,but now c n = 12 W Z R dsdxdy V eff n ( s ) | s − x | V eff n ( x ) | x − y | V eff n ( y ) ψ r ( y ) ,c n = − Z R ds sV eff n ( s ) ψ r ( s ) ,W = Z R dsdy V eff n ( s ) | s − y | V eff n ( y ) > . Moreover, c n and c n do not vanish simultaneously. Note the different expressions for the parameter c n from the case h V eff n i 6 = 0;in both Propositions 6 and 7, the expressions for c n , c n were obtained by workingwith relations in references [33] and [34], respectively. In order to replicate the proofof the above mentioned Lemma 1, it is enough to check some key properties thatcan be found spread along reference [34]; there is a complete parallelism betweenboth cases, although the expressions defining the involved quantities are different(that was a chief contribution of [34]). In what follows we indicate what are suchproperties, where their versions in case h V eff n i = 0 can be found in [34] and we usethe notation of [10, 34] without explaining the meaning of some of the symbolsemployed (e.g., t j , M j , φ , · · · ). Unfortunately a short explanation of the involvedsymbols will not be very helpful to the understanding of the large amount of involvedtechnicalities; at any rate, they are not necessary to state the above results, theycan be easily found in the references and the equations in [10, 34] we shall use inthe proof below will be explicitly indicated. Proof.
Introduce the functions v = | V eff n | / , u = | V eff n | / (sgn V eff n ) , so that V eff n = vu and h V eff n i = ( v, u ) (inner product in L ( R )). The propertiesneeded for the proof of Proposition 7(a) appear in equation (25) of [10], that is,( v, t u ) = 0 , (( · ) v, t u ) = ( v, t u ( · )) = 0 , ( v, t u ) = − . The first and the third ones can be found in equation (3.83) of [34], while the secondone is obtained by combining equations (2.8) and (3.98) of that work.
UANTUM RESOLVENT AND Γ CONVERGENCES 27
For the proof of Proposition 7(b) one need to check equations (17) and (18)of [10]; equation (18) reads(( · ) v, t − u ( · )) = 2( c n ) ( c n ) + ( c n ) , (( · ) v, t u ) = 2 c n c n ( c n ) + ( c n ) , ( v, t u ) = − c n ) ( c n ) + ( c n ) ;these relations are found in equations (4.16), (4.15) and (3.91) of [34], respectively.Now equation (17) of [10] reads t − u = 0 , t ∗− v = 0 , ( v, t u ) = 0 . The third relation follows from equation (3.90) of [34], and their equation (3.93)implies (recall we are using their notation) t ∗− v = (sgn V eff n ) t − (sgn V eff n ) v = (sgn V eff n ) t − u = 0 , that is, we have got the second relation by accepting that the first one holds. Nowwe show how to derive the first one from [34]. By equations (3.45) and (3.5) in [34]it is found that t − = − c P ˆ Q = − c P ( − ˆ P )= − c P (cid:18) − c M P (cid:19) = − c P + c c M P, with P ( · ) = ( v, · ) u and (see also equations (3.2) and (3.3) in [34]) P ( · ) = ( ˆ φ , · ) φ , ˆ φ = (sgn V eff n ) M φ . The proof finishes as soon as we check that P u = 0 and P u = 0. By thehypothesis on the potential we have
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