Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy
SStanding waves for the NLS on the double-bridge graphand a rational-irrational dichotomy
Diego Noja a , Sergio Rolando a , Simone Secchi a a Dipartimento di Matematica e Applicazioni, Universit`a di Milano Bicocca, via R. Cozzi 55, 20125 Milano,Italy
Abstract
We study a boundary value problem related to the search of standing waves for thenonlinear Schr¨odinger equation (NLS) on graphs. Precisely we are interested in char-acterizing the standing waves of NLS posed on the double-bridge graph , in which twosemi-infinite half-lines are attached at a circle at di ff erent vertices. At the two ver-tices the so-called Kirchho ff boundary conditions are imposed. The configuration ofthe graph is characterized by two lengths, L and L , and we are interested in the ex-istence and properties of standing waves of given frequency ω . For every ω > L / L ; they can be extended to every ω ∈ R . We study, for ω <
0, the solutionsperiodic on the circle but with nontrivial components on the half-lines. The problemturns out to be equivalent to a nonlinear boundary value problem in which the boundarycondition depends on the spectral parameter ω . After classifying the solutions with ra-tional L / L , we turn to L / L irrational showing that there exist standing waves onlyin correspondence to a countable set of frequencies ω n . Moreover we show that thefrequency sequence { ω n } n ≥ has a cluster point at −∞ and it admits at least a finite limitpoint, in general non-zero. Finally, any negative real number can be a limit point of aset of admitted frequencies up to the choice of a suitable irrational geometry L / L forthe graph. These results depend on basic properties of diophantine approximation ofreal numbers. Keywords:
Quantum graphs, non-linear Schr¨odinger equation, standing waves
1. Introduction and main results
The analysis of nonlinear equations on graphs, especially nonlinear Schr¨odinger equa-tion (NLS), is a new and rapidly growing research subject, which already produced awealth of interesting results (for review and references see [9]). Roughly speaking a
Email addresses: [email protected] (Diego Noja), [email protected] (SergioRolando), [email protected] (Simone Secchi)
Preprint submitted to arXiv.org October 14, 2018 a r X i v : . [ m a t h . A P ] J un etric graph is a structure built by edges connected at vertices. Some of the edges maybe of infinite length. On the edges a di ff erential operator is given, with suitable bound-ary condition at vertices which makes it self-adjoint. This generates a dynamics (Wave,Heat, Schr¨odinger, Dirac or other). The attractive feature of these mathematical mod-els is the complexity allowed by the graph structure, joined with the one dimensionalcharacter of the equations. While they are certainly an oversimplification in many realproblems coming from Physics in which geometry and transversal directions are notnegligible, they however appear indicative of several dynamically interesting phenom-ena non typical or not expected in more standard frameworks. This is already true at thelevel of the linear Schr¨odinger equation, the so called Quantum Graphs theory, wherean enormous literature exists ([7] and reference therein). The most studied topic inthe context of nonlinear Schr¨odinger equation is certainly existence and characteriza-tion of standing waves [19, 20, 26]. More particularly several results are known aboutground states (standing waves of minimal energy at fixed mass, i.e. L norm) as regardexistence, non existence and stability properties, depending on various characteristicsof the graph [1, 2, 3, 4, 9]. A distinguished role is played by topology, by the vertexconditions and by the possible presence of external potentials.In this paper we are interested in a special example which reveals an unsuspected finestructure of the set of standing waves when the metric properties of the graph are takenin account, and a relation with nonstandard boundary value problem and their solu-tions.Namely we consider a metric graph G made up of two half lines joined by two boundededges, i.e., a so-called double-bridge graph (see Fig.1). We may also think at G as aring with two half lines attached in two distinct vertices. The half lines will be bothidentified with the interval [0 , + ∞ ), while the bounded edges will be represented bytwo bounded intervals of lengths L > L ≥ L , precisely [0 , L ] and [ L , L ] with L = L + L . ∞ ∞ L L Figure 1: The double-bridge graph.
A function ψ on G is a cartesian product ψ ( x , ..., x ) = ( ψ ( x ) , . . . , ψ ( x )) with x j ∈ I j for j = , . . . ,
4, where I = [0 , L ], I = [ L , L ] and I = I = [0 , + ∞ ).Then a Schr¨odinger operator H G on G is defined as H G ψ ( x , . . . , x ) = (cid:0) − ψ (cid:48)(cid:48) ( x ) , . . . , − ψ (cid:48)(cid:48) ( x ) (cid:1) , x j ∈ I j , (1)with domain D (cid:0) H G (cid:1) given by the functions ψ on G whose components satisfy ψ j ∈ ( I j ) together with the so-called Kirchho ff boundary conditions , i.e., ψ (0) = ψ ( L ) = ψ (0) , ψ ( L ) = ψ ( L ) = ψ (0) , (2) ψ (cid:48) (0) − ψ (cid:48) ( L ) + ψ (cid:48) (0) = ψ (cid:48) ( L ) − ψ (cid:48) ( L ) − ψ (cid:48) (0) = . (3)As it is well known, the operator H G is self-adjoint on the domain D ( H G ), and it gener-ates a unitary Schr¨odinger dynamics. Essential information about its spectrum is givenin Appendix.We perturb this linear dynamics with a focusing cubic term, namely we consider thefollowing nonlinear Schr¨odinger equation on G i d ψ t dt = H G ψ t − | ψ t | ψ t (4)where the nonlinear term | ψ t | ψ t is a shortened notation for ( | ψ , t | ψ , t , . . . , | ψ , t | ψ , t ).Hence Eq. (4) is a system of scalar NLS equation on the intervals I j coupled throughthe Kirchho ff boundary conditions (2)-(3) included in the domain of H G .On rather general grounds it can be shown that this problem enjoys well-posednessboth in strong sense and in the energy space (see in particular [9, Section 2.6]).We want to study standing waves of Eq. (4), i.e., its solutions of the form ψ t = e − i ω t u ( x , ω )where ω ∈ R and u is a purely spatial function on G . In the sequel for the sake of brevitywe will often omit the explicit dependence on ω . Writing the equation component-wise,we get the following scalar problem: − u (cid:48)(cid:48) j − u j = ω u j , u j ∈ H ( I j ) u (0) = u ( L ) = u (0) , u ( L ) = u ( L ) = u (0) u (cid:48) (0) − u (cid:48) ( L ) + u (cid:48) (0) = , u (cid:48) ( L ) − u (cid:48) ( L ) − u (cid:48) (0) = . (5)In [2, 3] it is shown, among many other things, that the focusing NLS on a doublebridge graph has no ground state, i.e. no standing wave exists that minimizes theenergy at fixed L -norm (see also [4] for the critical power NLS). In the recent [5] in-formation on positive bound states which are not ground states is given. In this paperwe are interested, instead, in studying non positive standing waves profiles.We discuss first the case ω >
0, taking also the opportunity to fix notations and to re-call some elementary but useful facts. It is well known that non vanishing L solutionsof the stationary focusing NLS on the half-line do not exist. So any solution of ourproblem is supported on the circle. This further condition forces Dirichlet boundaryconditions at the two vertices and makes the above problem (5) overdetermined: a so-lution u belongs to H ([0 , L ]) necessarily (see definition (14) below), and moreoverit has to satisfy Dirichlet boundary conditions at 0 , L and L . Periodic solutions ofstationary NLS on the interval are Jacobi snoidal, cnoidal and dnoidal functions (for atreatise on the Jacobian elliptic functions, we refer e.g. to [24, 21]). Only cnoidal anddnoidal functions satisfy the focusing NLS on the circle, and the dnoidal functions donot vanish anywhere and so we rule out them.3ore precisely, the cnoidal function v ( y ) : = cn( y ; k ) with parameter k solves the equa-tion − v (cid:48)(cid:48) ( y ) − k v ( y ) = (1 − k ) v ( y ) . (6)Up to translations it is the only periodic solution of (6) oscillating around zero and itsminimal period is given by T ( k ) = K ( k ) = (cid:90) dt (cid:113) (1 − t )(1 − k t ) (7)where K ( k ) is the so called complete elliptic integral of first kind. There results cn(0; k ) =
1, and cn( T / k ) = , T ]. The scaling u ( x ) = √ kpv ( px ) , p = (cid:114) ω − k , k ∈ (0 , / , ω > u ( x ) = (cid:114) ω k − k cn (cid:32) (cid:114) ω − k x ; k (cid:33) (9)solves for every k ∈ (0 , / √
2) the equation − u (cid:48)(cid:48) − u = ω u . (10)It is a periodic solution of (10) oscillating around zero and its minimal period is givenby T ω ( k ) = (cid:114) − k ω K ( k ) . (11)Assuming periodicity on [0 , L ] ( u ∈ H ([0 , L ])) gives a countable family of periodiccnoidal functions u n with parameter k n defined by the condition that the length of theinterval is an integer multiple of the period, nT ω ( k n ) = L . A translation of a quarter ofperiod along the circle allows to satisfy the Dirichlet conditions at 0 and L . Up to nowwe have a sequence of parameters k n and functions u ± n ,ω : u ± n ,ω ( x ) = (cid:115) ω k n − k n cn (cid:32) (cid:114) ω − k n ( x ± T ω ( k n ) / k n (cid:33) , nT ω ( k n ) = L . (12)Now it is clear that the further Dirichlet condition at L can be satisfied if and only ifthere exists m < n such that mT ω ( k n ) / = L , i.e L / L = m / (2 n ) is a rational number.When we add this further condition, an infinite strict subset of the above families ofcnoidal functions u ± n ,ω satisfies the complete problem (5) for every positive ω , namelythe ones with n ∈ N q , where we denote by p , q the unique coprime naturals suchthat L / L = p / (2 q ) (cf. Remark 6.1). These solutions are supported on the circle anddisappear when the length L is not a rational multiple of the length L . Moreover, asexpected and shown in the Appendix, they bifurcate from the linear eigenvectors of thedouble bridge quantum graph in the limit of small amplitude.4 similar argument shows that for ω = L / L ∈ Q and form a sequence of suitably rescaled and translated cnoidal functions (cf. [8] fordetails).The situation is completely di ff erent when we consider solutions with ω <
0. In thefirst place the above families, which we indicate again as u ± n ,ω , can be continued toevery ω < u ± n ,ω ( x ) = (cid:115) | ω | k n k n − (cid:115) | ω | k n − x ± T ω ( k n ) / k n , k n ∈ (1 / √ , . (13)So there is an infinite number of global bifurcation branches { ( ω, u ± hq ,ω : ω < } , h ∈ N , originating in correspondence of the linear eigenvalues λ h , extending throughthe range ( −∞ , λ h ) and compactly supported on the graph. We stress again that thisinfinite family of global bifurcation branches exists only when the ratio L / L is rational.These solutions are the only ones with u = u =
0. However, many more solutionsare expected to arise, since for ω < ff condition is also satisfied. So, at ω =
0, from any branch of solutionsoriginating from the linear eigenvalues, a secondary bifurcation branch arises, with nontrivial component on the tails (see Fig.2). This phenomenon, in the simpler exampleof a tadpole graph (a circle with a single half-line attached), was noticed and studiedin [8, 23], where several bifurcations and in particular birth of edge solitons and theirstability is studied.Again, such a mechanism of attaching two half-solitons to a shifted cnoidal solutionworks for every ω < L / L is rational, making the problem nontrivial forirrational ratios.In our main results, we look for solutions of system (5) and show that they actuallyexist for every real value of the ratio L / L . As a matter of fact, a rather complexclassification of the general solutions to system (5) arises for ω <
0, requiring in generalcnoidal solutions with di ff erent parameters k and k on the two di ff erent pieces of thering, but, precisely in view of this complexity, the study of the complete geography ofstanding waves branches for ω < ff erent paper.The more restricted subject of this paper is the complete description of standing wavesof NLS on the double bridge graph which exhibit the following special features:(P ) u , u are nontrivial,(P ) u , u are the restriction to I , I of some u ∈ H ([0 , L ])where H ([0 , L ]) = (cid:110) u ∈ H ([0 , L ]) : u (0) = u ( L ) , u (cid:48) (0) = u (cid:48) ( L ) (cid:111) (14)is the second Sobolev space of periodic functions. As already remarked, condition (P )implies ω < u j ( x ) = ± (cid:112) | ω | sech (cid:16)(cid:112) | ω | ( x + a j ) (cid:17) , a j ∈ R , j = , . (15)5ondition (P ) implies u (cid:48) (0) − u (cid:48) ( L ) = u (cid:48) ( L ) − u (cid:48) ( L ) = a j = ff conditions. Hence we are led to study the solutions ( ω, u ) of thefollowing problem: − u (cid:48)(cid:48) − u = ω u , u ∈ H per ([0 , L ]) , ω < u (0) = ± u ( L ) = √ | ω | ( P ± )where the sign ± distinguishes the cases of u and u with the same sign (which we mayassume positive, thanks to the odd parity of the equation) or with di ff erent signs. Weremark that ( P ± ) is a nonlinear boundary value problem in which the spectral parameter ω appears explicitly in the boundary conditions. This makes the problem interesting initself, as spectral parameter dependent (also said ”energy dependent”) boundary valueproblems occur frequently in applications. Indeed, they often arise in the passage froma complete system to a reduced system in which the remaining part is eliminated andits e ff ect embodied in a nonstandard boundary condition (see for example [6, 17] andreference therein). This is also our case, where the soliton-like nature of the solutionon the half-line forces an ω -dependent value of the solution at vertices.Properties of standing waves of the focusing NLS on the double bridge graph satisfyingboth (P ) and (P ), or equivalently the solutions to problem ( P ± ), are described inthe following four main theorems. We anticipate that (see proof of Lemma 2.1) theycannot be of dnoidal type. To state the main results, we preliminarily define a function S : (1 / √ , → R by setting S ( k ) = √ k − K ( k ) , (16)Notice that S is strictly increasing, continuous and such that S (cid:16) (1 / √ , (cid:17) = (0 , + ∞ ).Moreover from now on we denote by [ · ] the floor function, or integer part ([ x ] is thegreatest integer smaller than or equal to the argument x ).The first two results give the classification of standing waves. Surprisingly enough,they constitute a countable set if L / L (cid:60) Q (Theorem 1.1). If L / L ∈ Q , this setof solutions essentially persists, besides the aforementioned solutions made up of twohalf-solitons attached to a shifted cnoidal solution. Theorem 1.1.
Suppose that L / L ∈ R \ Q . The solutions ( ω, u ) of problem ( P ± ) are acountable family. More precisely, there exist two sequences { ω + n } n ≥ and { ω − n } n ≥ suchthat the solutions of ( P ± ) are (cid:8) ( ω ± n , u ± n ) : n ∈ N (cid:9) withu ± n ( x ) : = (cid:115) | ω ± n | k n k n − (cid:115) | ω ± n | k n − (cid:0) x − s ± n (cid:1) ; k n , k n : = S − L (cid:112) | ω ± n | n , (17) s + n : = L n r n , s − n : = L n (cid:32) | r n | − (cid:33) sgn ( r n ) , r n : = L L n − (cid:34) L L n + (cid:35) . (18)Note that the last equality of (17) means that u ± n has period L / n . The explicit construc-tion of the sequences { ω + n } and { ω − n } is the subject of Lemmas 2.6 and 2.7 in Section 2.They are described as the solution of the equations (cid:112) | ω + n | = nLG ( ξ n ) and (cid:112) | ω − n | = nLG (1 − ξ n ) , n ∈ N ξ n : = | r n | and G is a certain monotone function (namely G = S ◦ ϕ − , see (38)for definition of ϕ ). Theorem 1.2.
Assume that L / L = p / q ∈ Q with p , q coprime. The set of the solutionsto ( P + ) is { ( ω, ˜ u ± n ,ω ) : ω < , n ∈ N q } ∪ { ( ω + n , u + n ) : n ∈ N , n (cid:60) N q , np / q + / (cid:60) N } ,where ω + n , u + n are the same of Theorem 1.1 and ˜ u ± n ,ω ( x ) : = (cid:115) | ω | k n ,ω k n ,ω − (cid:115) | ω | k n ,ω − (cid:0) x ± γ n ,ω (cid:1) ; k n ,ω , k n ,ω : = S − (cid:32) L √| ω | n (cid:33) ,γ n ,ω = (cid:115) k n ,ω − | ω | (cid:90) (cid:114) k n ,ω − k n ,ω dt (cid:113)(cid:0) − t (cid:1) (1 − k n ,ω (1 − t )) . (19) The set of the solutions to ( P − ) is { ( ω − n , u − n ) : n ∈ N , n (cid:60) N q } if q is odd, and { ( ω, ˜ u ± n ,ω ) : ω < , n ∈ (2 N − q / } ∪ { ( ω − n , u − n ) : n ∈ N , n (cid:60) N q / } if q is even, where ω − n , u − n arethe same of Theorem 1.1. Some comments about Theorem 1.2 are in order. First, the functions ˜ u ± n ,ω have period L / n and constitute the already mentioned secondary bifurcation branches with non triv-ial components on the tails, arising at ω = γ n ,ω tends to a quarter of the period,i.e. L / (4 n ), as ω → γ n ,ω ).Second, the solutions ( ω + n , u + n ) in Theorem 1.2 are a countable family, since L / L = p / q ∈ Q implies ξ n + q = ξ n for all n ∈ N and therefore {| ω + n | / } is a diverging se-quence made up of q subsequences { ( mq + G ( ξ ) / L } m ≥ , . . . , { ( mq + q ) G ( ξ q ) / L } m ≥ .We also point out that the solutions ( ω + n , u + n ) do not belong to any branch { ( ω, ˜ u ± n ,ω ) } ,since ( ω + n , u + n ) satisfies (28) and Remark 2.3 holds. Similarly for ( ω − n , u − n ).Finally, the content of Theorem 1.2 can also, and maybe better, be explained in terms ofbifurcation diagrams. As before, it is convenient to denote by p , q the unique coprimenaturals such that L / L = p / (2 q ), in such a way that q = q if q is odd and q = q / q is even (cf. Remark 6.1); note also that n (cid:60) N q , np / q + / (cid:60) N just means n (cid:60) N q .From every bifurcation branch { ( ω, u ± hq ,ω ) : ω < } originating from the eigenvalues λ h , h ∈ N , a secondary bifurcation branch { ( ω, ˜ u ± hq ,ω ) : ω < } bifurcates at ω = P + ) for all h if q is odd, and ( P + ) or ( P − ) according as h iseven or odd if q is odd. Away from these secondary branches, we find isolated solutionsto problems ( P ± ) coming from the countable families { ( ω ± n , u ± n ) : n ∈ N } , namely theones with n (cid:60) N q . Observe that we have solutions that oscillate any number of timeson [0 , L ]. This situation is portrayed in Fig.2.For the sake of completeness, we also represent in Fig.3 the sets of the solutions to ( P ± )for L / L (cid:60) Q , as they appear according to Theorems 1.1 and 1.4. Remark 1.3.
The translation parameters γ n ,ω and s ± n of Theorems 1.1 and 1.2 are es-sentially the same, and coincide with the ones needed to match the continuity conditionwith the half solitons at the vertices. More precisely, with the notations of Theorem 1.1one has γ n ,ω ± n = | s ± n | for all n (see Lemmas 2.2 and 2.4).7 k u k L λ λ u ± q ,ω ˜ u ± q ,ω u ± q ,ω ˜ u ± q ,ω u ± q − u ± q +1 u ± q − u ± q +1 ω ± ω ± ... ω ± q − ω ± q +1 u ± u ± bbbbbbbbbbbbbbbbbb Figure 2: Bifurcation diagram for L / L = p / q = p / (2 q ) with p , q and p , q coprime. The functions˜ u ± hq ,ω solve ( P + ) for all h if q is odd, and solve ( P + ) or ( P − ) according as h is even or odd if q is odd. All ofthem have period L / ( hq o ), the same of u ± hq ,ω and the eigenfunctions related to λ h . The functions u ± n , n (cid:60) N q , solve ( P ± ) respectively, and have period L / n . The frequencies ω ± , ..., ω ± q need not to be ordered asin the figure. The next theorem gives some information about the sequence of frequencies of standingwaves pertaining to any irrational geometry: the set of frequencies is unbounded frombelow and moreover it has at least a finite limit point which, whatever the irrational L / L is, is located in a precise interval (see also Remarks 3.6 and 3.7 below). Theorem 1.4.
Assume that L / L ∈ R \ Q . Then the sequences { ω ± n } n ≥ of Theorem 1.1are unbounded from below and have at least a finite cluster point, respectively fallingin the intervals I ± defined byI + = − L K (cid:32) √ (cid:33) , and I − = − L K (cid:32) √ (cid:33) , . Finally, the last theorem answers in the a ffi rmative the following inverse problem: canany fixed negative real number be a limit point of standing wave frequencies providedwe can choose in a suitable way the ratio L / L ? Theorem 1.5.
For every ω ≤ there exists a number L / L : = α ∈ (0 , such that thefrequency sequence { ω + n } n ≥ of Theorem 1.1 has a subsequence converging to ω . The detailed proofs of the previous theorems fill Sections 2, 3 and 4. We only noticethat a relevant part of the proofs is played by properties of diophantine approximationof real numbers. Some of them are elementary or well known (as in the case of Dirich-let theorem and Weyl equidistribution theorem) and some other are less. In particular adetailed analysis of the possible cluster points of ω ± n would require information about8 k u k L bb bb bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb Figure 3: The appearance of each of the sets of solutions { ( ω + n , u + n ) } n ∈ N and { ( ω − n , u − n ) } n ∈ N for L / L ∈ R \ Q ,according to Theorems 1.1 and 1.4. the so called inhomogeneous diophantine approximation constants of real numbers (seeRemark 3.6), which are strictly related to the properties of sequence ξ n (see for exam-ple the classical treatise [10], [18] and the more recent [27]), about which only fewprecise results are known. However we stress the fact that the analysis here presentedis essentially elementary and self contained.Once again, we stress that we are not classifying the totality of standing waves of thedouble bridge graph, but the subfamily with a periodical component on the ring, whichin turn is in 1 − P ± ).From this point of view, the rather surprising structure of the obtained frequencies,constitutes the nonlinear spectrum of problems ( P ± ). This seems a new result with anindependent interest.In the e ff ort of giving further information on the frequency sequences of Theorem 1.1and to guess possible directions for rigorous analysis, in the last Section 5, some nu-merical results about the sequence { ω + n } are given, with the aid of a simple code run by Wolfram MATHEMATICA 10.4.1 . The numerics highlights several phenomena. In thefirst place the appearance of a single or also several cluster points for { ω + n } in the inter-val I + , depending on the choice of di ff erent ratios α . Secondarily, for several choicesof algebraic ratios α the indices corresponding to the subsequences of { ω + n } convergingin I + are recognized as distinguished and well known sequences, for example relatedto Fibonacci or Chebyshev sequences. We do not have at present any clue about thissecond seemingly curious behavior. We however notice that in principle this is a purenumber theoretic property of diophantine approximation constants; it is perhaps note-worthy its appearance in the boundary value problem here studied.For the convenience of the reader, we collect here some notation. • N stands for the set of positive integers (0 excluded).9 [ t ] is the integer part of t ∈ R , while { t } = t − [ t ] is the fractional part of t ∈ R .Finally we notice once and for all that that L / L ∈ R \ Q is equivalent to L / L ∈ R \ Q ,since L L = L / L + L / L .
2. Proof of Theorems 1.1 and 1.2
With the aim of proving Theorems 1.1 and 1.2, we first solve the auxiliary problem − u (cid:48)(cid:48) − u = ω uu ∈ H ([0 , L ]) , ω < . (20)Clearly, the solutions of problem ( P ± ) are the solutions of (20) satisfying the boundarycondition u (0) = ± u ( L ) = √ | ω | . Lemma 2.1.
The set of the solutions ( ω, u ) to problem (20) assuming the value √ | ω | is the family (cid:110) ( ω, c n ,ω ( · ; a )) : n ∈ N , ω < , a ∈ (cid:104) − L n , L n (cid:17)(cid:111) , wherec n ,ω ( x ; a ) = (cid:115) | ω | k n ,ω k n ,ω − (cid:115) | ω | k n ,ω − x + a ); k n ,ω , (21) k n ,ω = S − (cid:32) L √| ω | n (cid:33) . (22)P roof . The periodic solutions of the equation − u (cid:48)(cid:48) − u = ω u with ω ∈ R are wellknown and can be expressed in terms of the Jacobian elliptic functions (cf. the discus-sion in Section 1, and see [8] and references therein). In particular, for ω <
0, suchsolutions are the functions c ω ( x ; k , a ) = ± (cid:114) | ω | k k − (cid:114) | ω | k − x + a ); k (23)with k ∈ (cid:16) / √ , (cid:17) and a ∈ R free parameters, and d ω ( x ; k , a ) = ± (cid:114) | ω | − k dn (cid:114) | ω | − k ( x + a ); k (24)with k ∈ [0 ,
1) and a ∈ R free parameters. The negative sign in (24) is ruled out,because the corresponding maps only take negative values. Moreover, the dnoidalfunction dn oscillates between √ − k and 1 and therefore the positive funtions d ω oscillate between (cid:112) | ω | (1 − k ) / √ − k and √ | ω | / √ − k < √ | ω | , which impliesthat they cannot assume the value √ | ω | . So the whole family (24) is ruled out. On theother hand, the function cn oscillates between − c ω contains √ | ω | < (cid:112) | ω | k / √ k − k ∈ (1 / √ ,
1) . The period T of c ω depends on k (and ω ) and is given by T = S ( k ) √| ω | . c ω belongs to H per ([0 , L ]) if and only if L is an integer multiple of T , i.e., k = S − ( L √| ω | / n ) for some n ∈ N . Therefore the solutions to (20) assuming the value √ | ω | are the functions c n ,ω ( x ; a ) = c ω ( x ; k n ,ω , a ) with n ∈ N , ω < a ∈ R .Since c n ,ω ( · ; a ) = − c n ,ω ( · ; a − L n ), the negative sign in (23) can be removed in orderto avoid duplicate solutions. Finally, the parameter a can be limited to the interval (cid:104) − T , T (cid:17) = (cid:104) − L n , L n (cid:17) by periodicity.Notice that, according to the proof, the function (21) has period L / n for every n , ω, a .For n ∈ N and ω <
0, define the auxiliary function c n ,ω ( x ) : = c n ,ω ( x ; 0)(cf. Fig.4). Observe that c n ,ω ( R ) = − (cid:115) | ω | k n ,ω k n ,ω − , (cid:115) | ω | k n ,ω k n ,ω − and 0 < (cid:112) | ω | < (cid:115) | ω | k n ,ω k n ,ω − , since k n ,ω ∈ (1 / √ , √ | ω | has 2 n preimages in (cid:104) − L n , L − L n (cid:105) under c n ,ω ,which we denote by x ( n ,ω )1 < x ( n ,ω )2 < ... < x ( n ,ω )2 n . Similarly, − √ | ω | has 2 n preimages in (cid:104) − L n , L − L n (cid:105) under c n ,ω as well, which wedenote by y ( n ,ω )1 < y ( n ,ω )2 < ... < y ( n ,ω )2 n . For future reference, we also set γ n ,ω : = − x ( n ,ω )1 (cid:16) = x ( n ,ω )2 > (cid:17) , (25)in such a way that for j = , . . . , n one has x ( n ,ω )2 j − = ( j − Ln − γ n ,ω , x ( n ,ω )2 j = ( j − Ln + γ n ,ω (26)and y ( n ,ω )2 j − = (2 j − L n − γ n ,ω , y ( n ,ω )2 j = (2 j − L n + γ n ,ω . (27)Note that definition (25) is equivalent to (19), as we will show at a later stage (see (41)). Lemma 2.2.
A solution ( ω, c n ,ω ( · ; a )) of problem (20) solves problem ( P + ) if and onlyif a = ± γ n ,ω and nL / L ∈ N , ora = ± γ n ,ω and L L n − (cid:34) L L n + (cid:35) = ∓ nL γ n ,ω (28) (with obvious relation between the signs of the right hand sides), i.e.,a = − s + n and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = nL γ n ,ω where s + n is the defined in (18). igure 4: The function c n ,ω on [ − L n , L ], with n = ω = − L = Remark 2.3. (28) and the case with nL / L ∈ N exclude each other. Indeed, nL / L ∈ N implies | nL / L − [ nL / L + / | = γ n ,ω > roof . Denote m n = (cid:104) L L n + (cid:105) for brevity. If a = − γ n ,ω , one has c n ,ω (0; a ) = c n ,ω (cid:0) − γ n ,ω (cid:1) = c n ,ω (cid:0) − γ n ,ω ; 0 (cid:1) = c n ,ω ( x ( n ,ω )1 ) = (cid:112) | ω | . Moreover there exists m ∈ N such that c n ,ω ( L ; a ) = c n ,ω ( L − γ n ,ω ) = c n ,ω ( mL / n − γ n ,ω ) = c n ,ω ( x ( n ,ω )1 ) = (cid:112) | ω | if nL / L ∈ N , and one has c n ,ω ( L ; a ) = c n ,ω ( L − γ n ,ω ) = c n ,ω (2 γ n ,ω + m n L / n − γ n ,ω ) = c n ,ω ( x ( n ,ω )2 ) = (cid:112) | ω | if (28) holds. Hence ( ω, c n ,ω ( · ; a )) solves problem ( P + ). The conclusion similarlyensues if a = γ n ,ω .Now assume that ( ω, c n ,ω ( · ; a )) solves problem ( P + ). Since c n ,ω (0; a ) = c n ,ω ( a ) and − L n ≤ a < L n , the condition c n ,ω (0; a ) = √ | ω | meanseither a = − γ n ,ω or a = γ n ,ω . (29)In the first case, we have c n ,ω ( L ; a ) = c n ,ω ( L − γ n ,ω ) = c n ,ω ( L + x ( n ,ω )1 ) with x ( n ,ω )1 < L + x ( n ,ω )1 < L ≤ L < L − L n (recall that 0 < L ≤ L /
2, since L ≤ L ), so that the condition c n ,ω ( L ; a ) = √ | ω | implies L + x ( n ,ω )1 ∈ { x ( n ,ω )2 , x ( n ,ω )3 , . . . , x ( n ,ω )2 n } , i.e., L ∈ (cid:110) x ( n ,ω )2 − x ( n ,ω )1 , x ( n ,ω )3 − x ( n ,ω )1 , . . . , x ( n ,ω )2 n − x ( n ,ω )1 (cid:111) . (30)Recalling (26), for j = , . . . , n one has x ( n ,ω )2 j − = ( j − Ln + x ( n ,ω )1 and x ( n ,ω )2 j = ( j − Ln − x ( n ,ω )1 = ( j − Ln + γ n ,ω + x ( n ,ω )1 ,
12o that (30) means that there exists m ∈ { , ..., n − } such that L = m Ln or L = m Ln + γ n ,ω . (31)If the first of such cases occurs, then m ≥ L (cid:44)
0) and the proof is complete.If the second case holds true, we get nL / L = m + n γ n ,ω / L and therefore nL / L − / < m < nL / L + /
2, since γ n ,ω < L / (4 n ). This implies nL / L + / (cid:60) N and m = [ nL / L + / L = m Ln or L L n = m − nL γ n ,ω instead of (31), and the conclusion follows as above. Lemma 2.4.
A solution ( ω, c n ,ω ( · ; a )) of problem (20) solves problem ( P − ) if and onlyif a = ± γ n ,ω and nL / L + / ∈ N , ora = ± γ n ,ω and L L n − (cid:34) L L n + (cid:35) = ± (cid:32) − nL γ n ,ω (cid:33) (32) (with obvious relation between the signs of the right hand sides), i.e.,a = − s − n and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − nL γ n ,ω where s − n is the defined in (18). Remark 2.5.
Condition (32) and the case with nL / L + / ∈ N exclude each other.Indeed, (32) with nL / L + / ∈ N implies γ n ,ω = γ n ,ω = L / (2 n ), which isimpossible since 0 < γ n ,ω < L / (4 n ).P roof . Denote m n = (cid:104) L L n + (cid:105) for brevity. If a = γ n ,ω , one has c n ,ω (0; a ) = √ | ω | as inthe proof of Lemma 2.2. Moreover there exists m ∈ N such that c n ,ω ( L ; a ) = c n ,ω ( L − γ n ,ω ) = c n ,ω (cid:18) mLn − L n − γ n ,ω (cid:19) = c n ,ω ( − y ( n ,ω )2 ) = − (cid:112) | ω | if nL / L ∈ N (note that c n ,ω is even), and one has c n ,ω ( L ; a ) = c n ,ω ( L − γ n ,ω ) = c n ,ω (cid:32) m n Ln − n + γ n ,ω (cid:33) = c n ,ω ( − y ( n ,ω )1 ) = − (cid:112) | ω | if (28) holds. This implies that ( ω, c n ,ω ( · ; a )) solves problem ( P + ). A similar computa-tion yields the same result if a = γ n ,ω .Now assume that ( ω, c n ,ω ( · ; a )) solves problem ( P − ). Since − L n ≤ a < L n , condition c n ,ω (0; a ) = √ | ω | means a = − γ n ,ω or a = γ n ,ω . (33)13n the first case, we have c n ,ω ( L ; a ) = c n ,ω ( L + x ( n ,ω )1 ) with x ( n ,ω )1 < L + x ( n ,ω )1 < L ≤ L < L − L n (recall that 0 < L ≤ L /
2, since L ≤ L ), so that condition c n ,ω ( L ; a ) = − √ | ω | implies L + x ( n ,ω )1 ∈ { y ( n ,ω )1 , y ( n ,ω )2 , ..., y ( n ,ω )2 n } , i.e., L ∈ (cid:110) y ( n ,ω )1 − x ( n ,ω )1 , y ( n ,ω )2 − x ( n ,ω )1 , ..., y ( n ,ω )2 n − x ( n ,ω )1 (cid:111) . (34)Recalling (27), for j = , . . . , n one has y ( n ,ω )2 j − = (2 j − L n + x ( n ,ω )1 , y ( n ,ω )2 j = (2 j − L n + γ n ,ω = (2 j − L n + x ( n ,ω )1 + γ n ,ω , so that (34) means that there exists j ∈ { , ..., n } such that L = j − n L or L = j − n L + γ n ,ω . In the first case, it follows that nL / L + / ∈ N and this completes the proof. In thesecond case, we get nL / L + / = j + n γ n ,ω / L . Since 0 < γ n ,ω < L / (4 n ), this implies j < nL / L + / < j + / j = [ nL / L + / L L n − (cid:34) L L n + (cid:35) = − (cid:32) − nL γ n ,ω (cid:33) < . In the second case of (33), we have c n ,ω ( L ; a ) = c n ,ω ( L + x ( n ,ω )2 ) with x ( n ,ω )2 < L + x ( n ,ω )2 < L + L n ≤ L − L n , so that the condition c n ,ω ( L ; a ) = − √ | ω | implies L + x ( n ,ω )2 ∈ { y ( n ,ω )1 , y ( n ,ω )2 , ..., y ( n ,ω )2 n } ,i.e., L ∈ (cid:110) y ( n ,ω )1 − x ( n ,ω )2 , y ( n ,ω )2 − x ( n ,ω )2 , ..., y ( n ,ω )2 n − x ( n ,ω )2 (cid:111) . (35)Recalling (27), for j = , . . . , n one has y ( n ,ω )2 j − = (2 j − L n − γ n ,ω = (2 j − L n − γ n ,ω + x ( n ,ω )2 , y ( n ,ω )2 j = (2 j − L n + x ( n ,ω )2 , so that (35) means that there exists j ∈ { , ..., n } such that L = j − n L or L = j − n L − γ n ,ω . (36)The first of such cases is the same of above, while in the second one there exists j ∈{ , ..., n } such that L L n + = j − nL γ n ,ω . (37)14ince 0 < γ n ,ω < L n , this implies j − / < nL / L + / < j and therefore j = [ nL / L + / +
1, so that (37) gives L L n − (cid:34) L L n + (cid:35) = − nL γ n ,ω > . This ends the proof.For future reference, we define a function ϕ : (1 / √ , → R by setting ϕ ( k ) = (cid:90) (cid:113) k − k dt (cid:113) (cid:0) − t (cid:1) (cid:0) − k (1 − t ) (cid:1)(cid:90) dt (cid:113) (cid:0) − t (cid:1) (cid:0) − k t (cid:1) . (38)Note that the denominator is the elliptic integral K ( k ). Such a function ϕ is continuousand strictly decreasing, since the denominator K ( k ) is positive and strictly increasingand the numerator is positive and strictly decreasing. Indeed, one has ddk (cid:90) (cid:113) k − k dt (cid:113) (cid:0) − t (cid:1) (cid:0) − k + k t (cid:1) == (cid:90) (cid:113) k − k k √ − t (cid:0) − k (1 − t ) (cid:1) / dt − k (cid:113) (cid:0) k − (cid:1) (cid:0) − k (cid:1) , where k √ k − √ − k has a maximum point on (1 / √ ,
1) for k = (9 + √ / k ∈ (1 / √ ,
1) we have (cid:90) (cid:113) k − k k √ − t (cid:0) − k (1 − t ) (cid:1) / dt ≤ k (cid:113) − k − k (cid:16) − k (1 − k − k ) (cid:17) / − (cid:114) k − k = √ − k k − (cid:114) − k ≤ t and the left-hand side of (39) is decreasing in k ), so ddk (cid:90) (cid:113) k − k dt (cid:113) (cid:0) − t (cid:1) (cid:0) − k + k t (cid:1) ≤ − + √ (cid:113) √ − < . Moreover one has the two limitslim k → − ϕ ( k ) = (cid:90) dt (cid:113) (cid:0) − t (cid:1) t (cid:90) dt − t = , lim k → (cid:18) √ (cid:19) + ϕ ( k ) = (cid:90) dt (cid:113) (cid:0) − t (cid:1) (cid:0) + t (cid:1)(cid:90) dt (cid:113) (cid:0) − t (cid:1) (cid:0) − t (cid:1) = , (cid:82) π/ d θ √ + sin θ by the changes of variable t = sin θ and t = cos θ respectively.Hence ϕ (cid:16) (1 / √ , (cid:17) = (0 , Lemma 2.6.
For every n ∈ N , the equation (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = nL γ n ,ω (40) has one solution ω < if nL / L (cid:60) N and nL / L + / (cid:60) N , has no solution otherwise. P roof . As γ n ,ω = x ( n ,ω )2 is the unique value in (cid:16) , L n (cid:17) such that c n ,ω ( γ n ,ω ) = √ | ω | , i.e.,cn (cid:115) | ω | k n ,ω − γ n ,ω ; k n ,ω = (cid:115) k n ,ω − k n ,ω , one has that γ n ,ω = (cid:115) k n ,ω − | ω | arc cn (cid:115) k n ,ω − k n ,ω ; k n ,ω = (cid:115) k n ,ω − | ω | (cid:90) (cid:114) k n ,ω − k n ,ω dt (cid:113)(cid:0) − t (cid:1) (cid:16) − k n ,ω (1 − t ) (cid:17) (41)(see [24] for the inverse function arc cn of cn and its representation as an elliptic in-tegral). Since (22) and (16) imply (cid:113) k n ,ω − | ω | = L nK ( k n ,ω ) , from equality (41) we deducethat γ n ,ω = L n ϕ (cid:0) k n ,ω (cid:1) for all n ∈ N and ω <
0. Then, recalling the definition (22) of k n ,ω , equation (40) is equivalent to2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ϕ (cid:18) S − (cid:18) Ln (cid:112) | ω | (cid:19)(cid:19) . (42)Recalling that S is strictly increasing, continuous and such that S (cid:16) (1 / √ , (cid:17) = (0 , + ∞ ),the right hand side of (42) defines a continuous and strictly increasing function of ω from ( −∞ ,
0) onto (0 , t − < [ t ] ≤ t for all t ∈ R , we have − ≤ (cid:32) L L n − (cid:34) L L n + (cid:35)(cid:33) < , where the first sign is an equality if and only if nL / L + / ∈ N , and the secondmember vanishes if and only if nL / L ∈ N . In these cases, equation (42) is impossible.Otherwise, we have 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ (0 , ω <
0, which is given by ω = − n L (cid:34)(cid:16) S ◦ ϕ − (cid:17) (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33)(cid:35) . (43)16 emma 2.7. For every n ∈ N , the equation (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − nL γ n ,ω . (44) has one solution ω < if nL / L (cid:60) N and nL / L + / (cid:60) N , has no solution otherwise. P roof . We argue as in the proof of Lemma 2.6. Since γ n ,ω = L n ϕ (cid:0) k n ,ω (cid:1) for all n ∈ N and all ω <
0, equation (44) is equivalent to2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − ϕ (cid:18) S − (cid:18) Ln (cid:112) | ω | (cid:19)(cid:19) where right hand side defines a continuous and strictly decreasing function of ω from( −∞ ,
0) onto (0 , ,
1) if nL / L (cid:60) N and nL / L + / (cid:60) N , and equals 0 or 1 otherwise. In this latter case, the equation has no solution. In theformer case, it has a unique solution ω <
0, given by ω = − n L (cid:34)(cid:16) S ◦ ϕ − (cid:17) (cid:32) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33)(cid:35) . (45)P roof (P roof of T heorems and L / L (cid:60) Q , we point outthat conditions nL / L (cid:60) N and nL / L + / (cid:60) N are always true and the fact that theset of solutions is countable (not finite) follows from Theorem 1.4, where we showthat the sequences { ω ± n } are unbounded below. As to the case L / L = p / q ∈ Q with p , q coprime, we observe that condition nL / L ∈ N is equivalent to n ∈ N q , whereascondition nL / L + / ∈ N is impossible if q is odd and amounts to n ∈ (2 N − q / q is even.
3. Proof of Theorem 1.4
This section is devoted to the proof of Theorem 1.4, so assume L / L ∈ R \ Q and let { ω ± n } n ≥ be the sequences of Theorem 1.1.According to the proof of Lemmas 2.6 and 2.7 (see in particular (43) and (45)), for all n ≥ (cid:112) | ω + n | = nLG ( ξ n ) and (cid:112) | ω − n | = nLG (1 − ξ n ) , where G : = S ◦ ϕ − and ξ n : = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L L n − (cid:34) L L n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Clearly, with a view to proving Theorem 1.4, we can equivalently study the limit pointsof { (cid:112) | ω ± n |} . In doing this, we will exploit some well known results from the metrictheory of Diophantine approximations, for which we refer to [10, 18, 27].Denote α = L L ξ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n − (cid:34) α n + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:40) α n + (cid:41) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ (0 , . Here and in the following, { t } = t − [ t ] denotes the fractional part of t ∈ R . Note thatthe cases ξ n = ξ n = α (cid:60) Q . Lemma 3.1.
The sequence { (cid:112) | ω + n |} is unbounded. P roof . By contradiction, assume that there exists a constant c > (cid:112) | ω + n | ≤ c for all n ∈ N . Since S is increasing and ϕ is decreasing and positive, G − is decreasingand positive and thus we get ξ n = G − (cid:18) Ln (cid:112) | ω + n | (cid:19) ≥ G − ( Lc ) > n ∈ N . (46)On the other hand, since α (cid:60) Q , by the Dirichlet’s approximation theorem (see e.g.[27, Theorem 1A and Corollary 1B]) there exist infinitely many rational numbers m / n such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − mn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < n . (47)This amounts to m ∈ (cid:16) α n − n , α n + n (cid:17) where the right hand side interval has length2 / n and is centered in the irrational number α , so that necessarily m = (cid:104) α n + (cid:105) if n ≥
2. The set of the denominators of the rationals m / n must be infinite (otherwise,(47) implies that the set of the numerators is also finite) and we may arrange them in adivergent sequence { n j } such that n j ≥
2. Hence for all j we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − n j (cid:34) α n j + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < n j and therefore ξ n j = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n j − (cid:34) α n j + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < n j . This is a contradiction, since { ξ n j } is bounded away from zero by (46).In order to investigate the existence of finite cluster points for { (cid:112) | ω + n |} , we observe thatthey can only come from subsequences of { ξ n } converging to 1. Indeed, recalling theproperties of the functions S and ϕ , the function G is continuous and strictly decreasingfrom (0 ,
1) onto (0 , + ∞ ) and therefore ξ n j → (cid:96) ∈ [0 ,
1) implies (cid:113) | ω + n j | = n j G ( ξ n j ) / L → + ∞ . On the other hand, if ξ n j →
1, then G ( ξ n j ) → (cid:113) | ω + n j | de-pends on the rate of the infinitesimal G ( ξ n j ). Note that such a case actually occurs, sincethe Weyl criterion for uniformly distributed sequences (see e.g. [10, page 66]) assuresthat the sequence (cid:110)(cid:110) α n + (cid:111)(cid:111) n ≥ is dense in [0 ,
1] and therefore it admits subsequencesconverging both to 0 and to 1, to each of which there correspond a subsequence of { ξ n } converging to 1. 18 emma 3.2. Let { ξ n j } be any subsequence of { ξ n } such that ξ n j → . Then (cid:113) | ω + n j | ∼ L K (cid:32) √ (cid:33) n j (cid:16) − ξ n j (cid:17) as j → ∞ . Here and in the following, ∼ denotes the asymptotic equivalence of functions ( f ∼ g ⇔ f = g + o ( g )).P roof . We want to estimate the rate at which G ( t ) = S ( ϕ − ( t )) → t → − . Wehave that ϕ − ( t ) → (1 / √ + as t → − , whence G ( t ) = S ( ϕ − ( t )) ∼ √ K (cid:32) √ (cid:33) (cid:32) ϕ − ( t ) − √ (cid:33) / as t → − . (48)Now denote ϕ ( k ) = H ( k ) / K ( k ), with H ( k ) given by the numerator of definition (38).As k → (1 / √ + , both H ( k ) and K ( k ) converge to K (cid:16) / √ (cid:17) and we have K (cid:48) ( k ) = (cid:90) kt dt (cid:113) (1 − t )(1 − k t ) → (cid:90) t dt (cid:113) (1 − t )(2 − t ) ∈ R \ { } and H (cid:48) ( k ) = (cid:90) (cid:113) k − k k √ − t (cid:0) − k (1 − t ) (cid:1) / dt − k (cid:113)(cid:0) − k (cid:1) (cid:16) √ k + (cid:17) √ (cid:16) k − / √ (cid:17) , where (cid:90) (cid:113) k − k k √ − t dt (cid:0) − k (1 − t ) (cid:1) / → (cid:90) √ − t dt (cid:0) + t (cid:1) / ∈ R \ { } and 1 k (cid:113)(cid:0) − k (cid:1) (cid:16) √ k + (cid:17) √ → √ . Hence ϕ (cid:48) ( k ) = H (cid:48) ( k ) K ( k ) − K (cid:48) ( k ) H ( k ) K ( k ) ∼ − √ K (cid:18) √ (cid:19) (cid:18) k − √ (cid:19) / as k → (cid:32) √ (cid:33) + and therefore lim k → (1 / √ + − ϕ ( k ) (cid:18) k − √ (cid:19) / = lim k → (1 / √ + − ϕ (cid:48) ( k ) (cid:18) k − √ (cid:19) − / = √ K (cid:18) √ (cid:19) . This implies lim t → − ϕ − ( t ) − √ (1 − t ) = lim k → (1 / √ + k − √ (1 − ϕ ( k )) = K (cid:18) √ (cid:19) √ , ϕ − ( t ) − √ ∼ K (cid:18) √ (cid:19) √ − t ) as t → − . (49)The result then follows from (48) and (49).According to the last lemma, the problem of the finite cluster points of { (cid:112) | ω + n |} is re-duced to the one of the convergent subsequences of { n (1 − ξ n ) } . Notice that1 − ξ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { α n } − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all n ∈ N . (50) Lemma 3.3.
There exist infinitely many indices n such that ≤ (cid:112) | ω + n | ≤ L K (cid:32) √ (cid:33) . P roof . As α is irrational, there exist infinitely many n , m ∈ Z such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:32) α n + − m (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) <
14 and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n + − m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n + − m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = min k ∈ Z (cid:12)(cid:12)(cid:12) α n + − k (cid:12)(cid:12)(cid:12) , i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n + − m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , where (cid:107) t (cid:107) : = min k ∈ Z | t − k | denotes the distance from Z . Hence the first inequality saysthat there exist infinitely many n ∈ Z such that n (cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13) < . Since (cid:13)(cid:13)(cid:13) α ( − n ) + (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) α n − (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13) , we may assume that such integers n are positive, so that weconclude that n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) <
14 for infinitely many n ∈ N . (51)Now observe that, for every n ∈ N there exists m ∈ N such that m − < α n < m or m < α n < m + . In the first case, one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = α n + − m , [ α n ] = m − , { α n } > (cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13) = α n + − [ α n ] − = { α n } − >
0. In the second case, we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = m + − (cid:32) α n + (cid:33) , [ α n ] = m , { α n } < (cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13) = [ α n ] − α n + = − { α n } >
0. So, in any case, we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α n + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { α n } − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all n ∈ N (52)and the conclusion follows from (51), (50) and Lemma 3.2. Lemma 3.4.
The sequence { (cid:112) | ω − n |} is unbounded. P roof . Assuming by contradiction that { (cid:112) | ω − n |} is bounded, as in the proof of Lemma3.1 we get that 1 − ξ n = G − ( L (cid:112) | ω − n | / n ) is bounded away from zero. But this is false,since, as in the proof of Lemma 3.3, there exist infinitely many n ∈ N such that1 − ξ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { α n } − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n . The possible finite cluster points of { (cid:112) | ω − n |} can only come from subsequences of { ξ n } converging to 0, since ξ n j → (cid:96) ∈ (0 ,
1] implies (cid:113) | ω − n j | = n j G (1 − ξ n j ) / L → + ∞ ,whereas, if ξ n j →
0, the behaviour of (cid:113) | ω − n j | depends on the rate of the infinitesimal G (1 − ξ n j ). Note that the Weyl criterion for uniformly distributed sequences (see e.g.[10, page 66]) assures that the sequence (cid:110)(cid:110) α n + (cid:111)(cid:111) is dense in [0 ,
1] and thus admitssubsequences converging to 1 /
2, to each of which there corresponds a subsequence { ξ n j } of { ξ n } converging to 0. As in Lemma 3.2, for such a subsequence we get that (cid:113) | ω − n j | ∼ L K (cid:32) √ (cid:33) n j ξ n j as j → ∞ . (53) Lemma 3.5.
There exist infinitely many indices n such that ≤ (cid:112) | ω − n | ≤ L √ K (cid:32) √ (cid:33) . (54)P roof . Since α is irrational, by the Hurwitz approximation theorem (see e.g. [18,Theorem 1.5]) there exist infinitely many rational numbers m / n such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − mn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < √ n . (55)Hence, as in the proof of Lemma 3.1, there exists a diverging sequence of indexes ( n j )such that for all j we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − n j (cid:34) α n j + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < √ n j and therefore n j ξ n j = n j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − n j (cid:34) α n j + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < √ . The conclusion then follows from (53). 21 roof (P roof of T heorem Remark 3.6.
In Diophantine Analysis, one introduces the quantities M ± ( α, β ) = lim inf n →∞ n (cid:107)± α n − β (cid:107) , n ∈ N , called one side inhomogeneous diophantine approximation constants ( homogeneous for β = α approximate a real β . From our point of view • if M + ( α, − ) =
0, the sequence ω + n accumulates to zero; • if M + ( α, − ) (cid:44)
0, the sequence ω + n has a non trivial limit point and it does notaccumulate to zero.The main classical result about the inhomogeneous diophantine approximation con-stant is the following (Minkowski 1901, Kintchine 1935, Cassels 1954): For every β (cid:60) Z + α Z one has M − ( α, β ) = M + ( α, β ) ≤ . It is possible to construct a real number α = L / L such that M + ( α, − /
2) is exactlyknown, below the threshold 1 / M + ( √ e , − / = / α = L / L such that M + ( α, − / = Remark 3.7.
The right hand side of (54) depends on the Hurwitz approximation the-orem (in particular on (55)), which holds true for every irrational α . If we excludeclasses of irrationals, estimate (55) can be refined by replacing √ I − becomes smaller, giving a more accurate localization ofthe cluster point of Theorem 1.4. For example, excluding the irrational ( √ − / √ √
4. Proof of Theorem 1.5
Thanks to Lemma 3.2, we can prove Theorem 1.5 by showing that for every (cid:96) ≥ α ∈ (0 ,
1) such that the sequence { n (1 − ξ n ) } has a subsequence converging to (cid:96) .As (cid:96) is arbitrary and according to (50) and the proof of Lemma 3.3 (see in particular(52)), this is equivalent to show that˜ ξ n : = n (cid:32) { n α } − (cid:33) (56)has a subsequence converging to (cid:96) .It is natural to construct the number α by looking at its expansion in a fixed base, let ussay 2. Hence we need to construct a number α = . b b b . . . b j ∈ { , } . The idea of the proof is to observe that, in the binary system, amultiplication by a power of 2 simply moves the “binary point” to the right.To simplify the construction, let us assume that our limit (cid:96) is an integer. To representsuch an integer in the binary system we need, say, n − b , b , . . . , b n . When we construct the number2 n { n α } , the “block” of digits from the position n + n are moved on the lefthand side of the binary point. We choose these digits so that they represent (cid:96) + n − ,and we set b n + = . . . = b n = . Now we pick n = n and we repeat the construction. We begin with a block of n arbitrary digits as before, where b n + plays the same rˆole as b . We insert a block of n digits that represent (cid:96) + n − and a block of n null digits. It is easy to realize that ξ nj = n j { n j α } − n j − → (cid:96) as j → + ∞ .If (cid:96) is a real number, the first step of the previous case “fixes” its integer part with n digits. In the second step, we insert a longer “block” of arbitrary digits, whose length isthe number of digits of the integer part of (cid:96) plus one. After this arbitrary block, we puta block consisting of the integer part of (cid:96) glued to the first digit of its fractional part,and another block of zeroes of the same length. We repeat again to “fix” the seconddigit of the fractional part of (cid:96) , and so on.We remark that now the length of each new block of non-trivial numbers increases,since the binary expansion of (cid:96) may contain infinitely many digits to “fix”. This proce-dure produces a sequence n < n < · · · of integers such that ξ n j = n j { n j α }− n j − → (cid:96) as j → + ∞ .More precisely, let us write α = b + b + . . . where each b j ∈ { , } . Hence2 n α = b n − + b n − + · · · + b n + b n + + b n + + · · · so that { n α } = b n + + b n + + · · · and 2 n { n α } = b n + n − + b n + n − + · · · + b n + b n + + b n + + · · · Therefore2 n { n α } − n − = ( b n + −
1) 2 n − + b n + n − + · · · + b n + b n + + b n + + · · · Now we choose the digits b n + , . . . , b n in such a way that( b n + −
1) 2 n − + b n + n − + · · · + b n = (cid:96) by taking n − (cid:96) , and of course b n + =
1. Call n this integer n , and repeat.23 emark 4.1. A careful inspection of the previous proof shows that we have indeedmuch more freedom in the construction. We have constructed α = . b . . . b n b n + . . . b n b n + . . . b n b n + . . . b n . . . However, after 3 n we could of course insert as much “junk” (namely arbitrary digits)as we wish, before fixing n . As we said above, we are just gluing blocks of digitsof (cid:96) “sliding o ff ” to the right. At each step, the number ξ nj approximates the binaryexpansion of (cid:96) with higher precision.
5. Numerics
Motivated by the proof of Lemma 3.3 (in particular by (51)) and with a view to study-ing numerically the behaviour of the sequence { ω + n } in the interval I + (see Theorem1.4), we used Wolfram MATHEMATICA 10.4.1 on a personal computer to analyze thesequence of integers n such that | ˜ ξ n | < ξ n is defined in (56) and satisfies | ˜ ξ n | = n (cid:107) α n + / (cid:107) .The code is almost trivial (here we use α = / √ In[1]:= alpha = 1/Sqrt[5]In[2]:= n=1In[3]:= While[n < 1000000, If[n Abs[FractionalPart[n a] - 1/2] <1/4,Print[n, " ,", N[n (FractionalPart[n a] - 1/2), 12]]]; n++]
This is the output corresponding to α = / √ n ˜ ξ n n ˜ ξ n − . − . − . − . − . ξ ∞ ≈ − . . The output corresponding to α = / √ n ˜ ξ n n ˜ ξ n . . − . − . . . − . − . . . − . ff erent phenomenon seems to arise: there are actually two sequencesthat produce cluster points ˜ ξ ∞ , ≈ . ξ ∞ , ≈ − . α = / (1 + √ ˜ ξ n n ˜ ξ n − . − . . . . . − . − . − . − . . . . . − . − . − . − . . . . . − . ξ ∞ , = − ˜ ξ ∞ , .We then looked up the sequences of these integers n in the Online Encyclopaedia ofInteger Sequences . The sequence corresponding to α = / √ n (cid:55)→ F (6 n + − F (6 n + , where F is the Fibonacci sequence. In the case α = / √
3, a first sequence was recog-nized as A001570, namely numbers n such that n is a centered hexagonal, also knownas Chebyshev T-sequence with Diophantine property. An explicit formula is known: n (cid:55)→ (2 + √ n − + (2 − √ n − . The second sequence has been recognized as A011945, the area of triangles with in-tegral side lengths m − m , m + α = / (1 + √
5) wasrecognized: even integers are the so-called “even Fibonacci numbers”, while odd inte-gers are defined recursively by n = n = , n i + = n i − + n i − . On the other hand experimental numerics has not shown known patterns in correspon-dence to trascendental numbers.
6. Appendix: Spectrum of H G and bifurcation from eigenvalues The essential spectrum of the free Schr¨odinger operator on the double bridge graphcoincides with [0 , + ∞ ), see [7]. It also admits a countable set of embedded eigenvalues,which we now compute for completeness. http://oeis.org D (cid:0) H G (cid:1) , the eigenvalue problem for the self-adjointoperator H G writes componentwise as follows: − ψ (cid:48)(cid:48) j = λψ j , ψ j ∈ H ( I j ) , λ ∈ R , j = , . . . , ψ (0) = ψ ( L ) = ψ (0) , ψ ( L ) = ψ ( L ) = ψ (0) ψ (cid:48) (0) − ψ (cid:48) ( L ) + ψ (cid:48) (0) = , ψ (cid:48) ( L ) − ψ (cid:48) ( L ) − ψ (cid:48) (0) = . (57)We set µ : = √| λ | for brevity and split the problem into three cases, according to λ < λ = λ > λ <
0, then (57) is equivalent to ψ j (cid:16) x j (cid:17) = a j e µ x j + b j e − µ x j for j = , ψ j (cid:16) x j (cid:17) = b j e − µ x j for j = , a + b = a e µ L + b e − µ L = b a e µ L + b e − µ L = a e µ L + b e − µ L = b a − b − a j e µ L + b j e − µ L − b = a e µ L − b e − µ L − a e µ L + b e − µ L + b = . (58)After some computation, the determinant of this linear system turns out to be ∆ = (cid:16) e − µ ( L − L ) + e µ ( L − L ) − e µ L − e µ L + e − µ ( L − L ) + (cid:17) , which defines a strictly decreasing function of L . Since L ≥ L , we have ∆ ≤ (cid:16) e − µ L + e µ L − e µ L − e µ L + (cid:17) = − e − µ L (cid:16) e µ L + (cid:17) (cid:16) e µ L − (cid:17) < a = a = b = b = b = b = λ =
0, then (57) is equivalent to ψ = ψ = ψ j (cid:16) x j (cid:17) = a j + b j x j for j = , a = a + b L = a + b L = a + b L = b − b = , which readily implies a = a = b = b = λ >
0, then (57) is equivalent to ψ = ψ = ψ j (cid:16) x j (cid:17) = a j cos (cid:16) µ x j (cid:17) + b j sin (cid:16) µ x j (cid:17) for j = , a = a cos ( µ L ) + b sin ( µ L ) = b sin ( µ L ) = a cos ( µ L ) + b sin ( µ L ) = b + a sin ( µ L ) − b cos ( µ L ) = b cos ( µ L ) + a sin ( µ L ) − b cos ( µ L ) = . If b =
0, the second and fifth equations give a = b = b (cid:44)
0, the third equation gives sin ( µ L ) = a = b = b , so that the system is equivalent to a = a = , b = b (cid:44) , sin ( µ L ) = , sin ( µ L ) = , cos ( µ L ) = . k , h ∈ N such that µ L = k π and µ L = h π , i.e., L L = k h and µ = h π L . Remark 6.1.
For every positive rational number p / q , there exists a unique pair ofcoprime integers p , q ∈ N such that p / q = p / (2 q ). Indeed, assuming p , q ∈ N coprime, we can take ( p , q ) = (2 p , q ) if q (cid:60) N and ( p , q ) = ( p , q /
2) if q ∈ N . Onthe other hand, p / (2 q ) = p (cid:48) / (2 q (cid:48) ) with p , q ∈ N coprime and p (cid:48) , q (cid:48) ∈ N coprimereadily implies p = p (cid:48) and q = q (cid:48) .As a conclusion, taking into account Remark 6.1, we obtain the following proposition. Proposition 6.2.
If L / L ∈ R \ Q , then the operator H G has no eigenvalues. If L / L ∈ Q , let p , q ∈ N be the unique coprime integers such that L / L = p / (2 q ) . Then theeigenvalues of H G are λ n = n π q L = n π p L , n ∈ N , with corresponding eigenspaces E λ n = span (cid:110)(cid:16) sin (cid:16) n π q L · (cid:17) , sin (cid:16) n π q L · (cid:17) , , (cid:17)(cid:111) . We stress the fact that for L / L ∈ R \ Q the lost eigenvalues λ of the operator H G becomeresonances, i.e. poles of the meromorphic continuation of the resolvent ( H G − k ) − through the real axis to C − (notice the change of the spectral parameter k = λ ). Thesubject of these so called topological resonances is studied in several recent papers,see for example [12, 13, 14, 16, 11].As a final remark we want to show in a direct way that the compactly supported solu-tions of cnoidal type of the NLS equation on the double-bridge graph bifurcate, whenthe parameter ω → λ n , from the linear eigenvectors E λ n of the double bridge linearquantum graph discussed above.Consider the solution (12) having the same period Lnq of the eigenfunctions in E λ n ,namely u ± n ,ω ( x ) = (cid:115) k n ω − k n cn (cid:32) (cid:114) ω − k n (cid:32) x ± T ω ( k n )4 (cid:33) ; k n (cid:33) , (59) nq T ω ( k n ) = L , n p T ω ( k n ) = L . (60)Define W ( k ) = √ − k K ( k ) in such a way that T ω ( k n ) = W ( k n ) / √ ω and thus W ( k n ) = L √ ω nq = π (cid:114) ωλ n . (61)Notice that the function W is strictly decreasing for k ∈ (cid:16) , / √ (cid:17) and satisfies W ( k ) = (cid:16) − k + o (cid:16) k (cid:17)(cid:17) (cid:18) π + π k + o (cid:16) k (cid:17)(cid:19) = π − π k + o (cid:16) k (cid:17) as k → . t → ( π/ − W − ( t ) √ π/ − t = lim k → k √ π/ − W ( k ) = (cid:114) π and therefore W − ( t ) ∼ (cid:114) π (cid:18) π − t (cid:19) as t → (cid:18) π (cid:19) − . (62)Putting ω = ω ε = λ n − ε in (60) we obtain u ± n ,ω ε ( x ) = (cid:115) k n ( λ n − ε )1 − k n cn (cid:115) λ n − ε − k n (cid:32) x ± L nq (cid:33) ; k n . As ε →
0, we have that π (cid:113) ωλ n → (cid:16) π (cid:17) − and therefore, using (61) and (62), we get that k n = W − (cid:32) π (cid:114) ωλ n (cid:33) ∼ (cid:114) (cid:115) − (cid:114) − ελ n ∼ (cid:114) ε λ n as ε → (cid:115) k n ( λ n − ε )1 − k n ∼ (cid:114) ε ε → . On the other hand, as ε → k n → (cid:115) λ n − ε − k n (cid:32) x ± L nq (cid:33) → (cid:112) λ n (cid:32) x ± L nq (cid:33) . Hence cn (cid:18) (cid:113) λ n − ε − k n (cid:16) x ± L nq (cid:17) ; k n (cid:19) tends pointwise tocn (cid:32) (cid:112) λ n (cid:32) x ± L nq (cid:33) ; 0 (cid:33) = cos (cid:32) (cid:112) λ n (cid:32) x ± L nq (cid:33)(cid:33) = cos (cid:32) n π q L x ± π (cid:33) = ∓ sin (cid:32) n π q L x (cid:33) . As a conclusion, we deduce that for every x and n one has for ε → u ± n ,ω ε ( x ) = (cid:114) ε + o (cid:16) √ ε (cid:17) (cid:32) ∓ sin (cid:32) n π q L x (cid:33) + o (1) (cid:33) = ∓ (cid:114) ε (cid:32) n π q L x (cid:33) + o (cid:16) √ ε (cid:17) . ReferencesReferences [1] R. Adami, C. Cacciapuoti, D. Finco, D. Noja,
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