DISPERSION FOR THE SCHRÖDINGER EQUATION ON THE LINE WITH MULTIPLE DIRAC DELTA POTENTIALS AND ON DELTA TREES

Transcription

1 DISPERSION FOR THE SCHRÖDINGER EQUATION ON THE LINE WITH MULTIPLE DIRAC DELTA POTENTIALS AND ON DELTA TREES VALERIA BANICA AND LIVIU I IGNAT Abstract In this paper we consider the time dependent one-dimensional Schrödinger equation with multiple Dirac delta potentials of different positive strengths We prove that the classical dispersion property holds The result is obtained in a more general setting of a Laplace operator on a tree with δ-coupling conditions at the vertices The proof relies on a careful analysis of the properties of the resolvent of the associated Hamiltonian With respect to the analysis done in [12] for Kirchoff conditions here the resolvent is no longer in the framework of Wiener algebra of almost periodic functions and its expression is harder to analyze 1 Introduction Let us first consider the linear Schrödinger equation on R: { iut +u 1 xx = 0 x Rt R u0x = u 0 x x R The linear semigroup e it has two important properties the conservation of the L 2 -norm: 2 e it u 0 L 2 R = u 0 L 2 R and a dispersive estimate of the form: 3 e it u 0 L R C t u 0 L 1 R t 0 It is now well-known that from these two inequalities space-time estimates follow known as Strichartz estimates [34][19]: 4 e it u 0 L q t RLr x R C u 0 L 2 R where q r are so-called admissible pairs: 5 2 q + 1 r = 1 2 qr 2 These dispersive estimates have been successfully applied to obtain wellposedness results for the nonlinear Schrödinger equation see [13] [35] and the reference therein VB is partially supported by the French ANR projects RAS ANR-08-JCJC and SchEq ANR-12-JS L I is partially supported by PN II-RU-TE 4/2010 and PN-II-ID-PCE of CNCSIS UEFISCDI Romania MTM C02-00 MICINN Spain and ERC Advanced Grant FP NUMERIWAVES 1

2 2 V BANICA AND L I IGNAT In the present paper we analyze the above dispersion property in two particular settings The first one concerns the semigroup exp ith α where H α is a perturbation of the Laplace operator with n Dirac delta potentials with positive strengths {α k } n k=1 6 H α = + n α j δx x j j=1 The spectral properties of the Laplacian with multiple Dirac delta potentials on R n have been extensively studied We only refer to [9] and to the references within The time dependent propagator of the linear Schrödinger equation has also been considered in the case of one Dirac delta potential [17] [33] [7] or one point interactions [8] [6] or two symmatric Dirac delta potentials [16] [28] Concerning the nonlinear Schrödinger equation with a Dirac delta potential standing wave and bound states have been analyzed [15] [16] [32] as well as the time dynamics of solitons [21] [23] [22] Our main result in the case of H α is the following one Theorem 11 The solution of the linear Schrödinger equation on the line with multiple delta interactions with positive strengths satisfies the dispersion inequality 7 e ithα u 0 L R C t u 0 L 1 R t 0 We are considering here the case when all α k k = 1n are positive This guarantees that operator H α is positive definite and its discrete spectrum is empty The case with one single delta interaction without sign condition on the streght has been analyzed in [7] where similar dispersive estimates has been proved but for e ithα P c P c begin the projection of H α on the continuous spectrum The case of two point interactions has been considered in [28] In both previous works given the particular structure of the operator H α the authors obtain explicit representations of the resolvent and then of e ithα However in the general case of multiple delta interactions an explicit representation is not easy to obtain; even in [10] [9 Ch II2] the resolvent is obtained in terms of the inverse of some matrix that depends on {α k } n k=1 and on {x k} n k=1 The setting might be seen as the special case of the equation posed on a simple graph with n vertices with only two edges starting from any vertex and with delta connection conditions at each vertex x 0 = x n+1 = { iut tx+u xx tx = 0 x x k 1 x k k = 1n 8 u x tx k + u x tx k = α k ux k t > 0 k = 1n Our second framework refers to the Dirac s delta Hamiltonian H Γ α on tree Γ with a finite number of vertices with the external edges thats that have only one internal vertex as an endpoint formed by infinite strips We consider the linear Schrödinger equation in the case of a tree Γ with positive

3 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 3 delta conditions of non necessarily equal strength at the vertices { iut = H 9 αu Γ x Γt R u0x = u 0 x x Γ The presentation of operator H Γ α will be given in full details in Sec 2 Let us just say here that H Γ α acts as xx on each edge of the tree and that at vertices the δ-coupling must be fulfilled: continuity condition for the functions on the graph and a δ transmission condition at the level of their first derivative at all internal vertices v: e E v n uv = αvuv The following questions have been considered for equation 9 The study of bound states on a star shaped tree with delta conditions have been analyzed in a series of papers [1] [5] [3] [2] Dispersion for the linear Schrödinger equation on a star shaped tree with delta conditions has been proved in [4] where the main result concerns the time evolution of a fast soliton in the spirit of [4] The main result of this paper is the following Theorem 12 The solution of the linear Schrödinger equation on a tree with delta connection conditions with positive strengths satisfies the dispersion inequality 10 e ithγ αu 0 L Γ C t u 0 L 1 Γ t 0 We mention here that dispersion was previously proved in [12] see also [24] for the case of Kirchoff s connection condition on trees ie αv = 0 for all internal vertices of the tree The case of δ and δ coupling on a star shaped tree has been considered in [4] The extension of the results presented here to the case of δ - coupling or to general strengths remains to be analyzed More coupling conditions are discussed in Section 2 The proof of Theorem 12 uses elements from [11] [12] [18] in an appropriate way related to the delta connection conditions on the tree The starting point consists in writing the solution in terms of the resolvent of the Laplacian which in turn is determined by recursion on the number of vertices With respect to the previous works with Kirchoff conditions the novelty in here is that we are not any longer in the framework of the almost periodic Wiener algebra of functions and that the expression of the resolvent is harder to analyse The linear solution e ithγ α u0 will be shown to be a combination of oscillatory integrals that becomes more and more involved as the number of vertices of the tree grows We do not have any more that e ithγ α u0 is a summable superposition of solutions of the linear Schrödinger equation on the line as for Kirchoff conditions in [12] Theorem 11 follow from Theorem 12 by considering the particular case of a tree Γ with all the internal vertices having degree two Denoting by H either H α or H Γ α the arguments used in the context of NSE on R can also be used here to obtain the following as a typical result

4 4 V BANICA AND L I IGNAT Theorem 13 Let p 04 For any u 0 L 2 Γ there exists a unique solution u CRL 2 Γ L q loc RLr Γ qradmissible of the nonlinear Schrödinger equation { iut +Hu± u p u = 0 t 0 11 u0 = u 0 t = 0 Moreover the L 2 Γ-norm of u is conserved along the time ut L 2 Γ = u 0 L 2 Γ The proof is standard once the dispersion property is obtained and it follows as in [13] p 109 Theorem 461 The paper is organized as follows In the next section we introduce the framework of the Laplacian analysis on a graph In 3 we give the proof of Theorem 12 2 Preliminaries on graphs and δ-coupling In this section we present some generalities about metric graphs and introduce the Dirac s delta Hamiltonian Hα Γ on such structure More general type of self-adjoint operators A B have been considered in [27] [26] We collect here some basic facts on metric graphs and on some operators that could be defined on such structure [31] [29] [30] [27] [20] [14] Let Γ = VE be a graph where V is a set of vertices and E the set of edges For each v V we denote by E v = {e E : v e} the set of edges branching from v We assume that V is connex and the degree of each vertex v of Γ is finite: dv = E v < The edges could be of finite length and then their ends are vertices of V or they have infinite length and then we assume that each infinite edge is a ray with a single vertex belonging to V see [31] for more details on graphs with infinite edges The vertices are called internal if dv 2 or external if dv = 1 In this paper we will assume that there are not external vertices We fix an orientation of Γ and for each oriented edge e we denote by Ie the initial vertex and by Te the terminal one Of course in the case of infinite edges we have only initial vertices We identify every edge e of Γ with an interval I e where I e = [0l e ] if the edge is finite and I e = [0 if the edge is infinite This identification introduces a coordinate x e along the edge e In this way Γ is a metric space and is often named metric graph [31] Let v be a vertex of V and e be an edge in E v We set for finite edges e { 0 if v = Ie jve = l e if v = Te and for infinite edges jve = 0 if v = Ie

5 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 5 We identify any function u on Γ with a collection {u e } e E of functions u e defined on the edges e of Γ Each u e can be considered as a function on the interval I e In fact we use the same notation u e for both the function on the edge e and the function on the interval I e identified with e For a function u : Γ C u = {u e } e E we denote by fu : Γ C the family {fu e } e E where fu e : e C A function u = {u e } e E is continuous if and only if u e is continuous on I e for every e E and moreover is continuous at the vertices of Γ: u e jve = u e jve ee E v v V The space L p Γ 1 p < consists of all functions u = {u e } e E on Γ that belong to L p I e for each edge e E and u p L p Γ = e E u e p L p I e < Similarly the space L Γ consists of all functions that belong to L I e for each edge e E and u L Γ = sup u e L I e < e E The Sobolev space H m Γ m 1 an integer consists in all continuous functions on Γ that belong to H m I e for each e E and u 2 H m Γ = e E u e 2 H m e < The above spaces are Hilbert spaces with the inner products uv L 2 Γ = u e v e L 2 I e = u e xv e xdx e E e E I e and uv H m Γ = e E u e v e H m I e = e E m k=0 Ie d k u e dx k d k v e dx k dx We now define the normal exterior derivative of a function u = {u e } e E at the endpoints of the edges For each e E and v an endpoint of e we consider the normal derivative of the restriction of u to the edge e of E v evaluated at jve to be defined by: u e n e jve = { u e x 0+ if jve = 0 u e x l e if jve = l e We now introduce Hα Γ It generalizes the classical Dirac s delta interactions with strength parameters 6 The Dirac s delta Hamiltonian is defined on the domain 12 DH Γ α = { u H 2 Γ e E v ue n e jve = αvuv v V Operator H Γ α acts as following for any u = {u e } e E H Γ αux = u e xxx x I e e E }

6 6 V BANICA AND L I IGNAT The quadratic form associated to Hα Γ is defined on H 1 Γ and it is given by Eα Γ u = u e x x 2 dx+ αv uv 2 e E I e v V Observe that in our case since all the strength parameters are nonnegative the quadratic form Eα Γ is nonnegative The case when all strengths vanish corresponds to the Kirchhoff coupling analyzed in [12] Finally let us mention that there are other coupling conditions see [26] which allow to define a Laplace operator on a metric graph To be more precise let us consider an operator that acts on functions on the graph Γ as the second derivative d2 dx and its domain consists in all functions u 2 that belong to the Sobolev space H 2 e on each edge e of Γ and satisfy the following boundary condition at the vertices: 13 Avuv +Bvu v = 0 for each vertex v Here uv and u v are correspondingly the vector of values of u at v attained from directions of different edges converging at v and the vector of derivatives at v in the outgoing directions For each vertex v of the tree we assumethat matrices Av and Bv are of size dv and satisfy thefollowing two conditions 1 the joint matrix Av Bv has maximal rank ie dv 2 AvBv T = BvAv T Under those assumptions it has been proved in [26] that the considered operator denoted by A B is self-adjoint The case considered in this paper the δ-coupling corresponds to the matrices Av = αv Bv = More examples of matrices satisfying the above conditions are given in [26 25] 3 Proof of Theorem 12 We shall use a description of the solution of the linear Schrödinger equation in terms of the resolvent For > 0 let R be the resolvent of the Laplacian on a tree R u 0 = H Γ α + 2 I 1 u 0 Before starting let us choose an orientation on tree Γ Let us choose an internal vertex O This will be the root of the tree and the initial vertex for all the edges that branch from it This procedure introduces an orientation for all the edges staring from O For the other endpoints of the edges belongingtoe O werepeattheaboveprocedureandinductively weconstruct an orientation on Γ

7 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 7 O Figure 1 The orientation on Γ 31 The structure of the resolvent In order to obtain the expression of the resolvent second-order equations R u 0 = 2 R u 0 u 0 must be solved on each edge of the tree together with coupling conditions at each vertex Then on each edge parametrized by I e for x I e 14 R u 0 x = c e e x + c e e x + t ex with t e x = 1 u 0 ye x y dy 2 I e Since R u 0 belongs to L 2 Γ the coefficients c s are zero on the infinite edges e E parametrized by [0 If we denote by I the set of internal edges we have 2 I + E coefficients The delta conditions of continuity of R u 0 and of transmission of R u 0 at the vertices of the tree give the system of equations on the coefficients We have the same number of equations as the number of unknowns We denote D Γp the matrix of the system where p stands for the number of vertices of the tree and by T Γp the column of the free terms in the system Therefore the resolvent R u 0 x on an edge I e is 15 R u 0 x = detmce detd Γp ex + detm detd Γp e x + t ex where M ce and M ce are obtained from D Γp by replacing the column corresponding to the unknown c e and respectively c e by the column of the free terms T Γp 32 The expression of detd Γp In view of the form 14 of the resolvent we obtain on an edge I e 16 R u 0 0 = c e + c e + t e0 R u 0 0 = c e c e +t e 0 c e

8 8 V BANICA AND L I IGNAT and in case I e is parametrized by [0a] with a < 17 R u 0 a = c e e a + c e e a + t e0 R u 0 a = c e e a c e e a t e a 321 The star-shaped tree case In the case of a single vertex and n 1 2 edges I j 1 j n 1 parametrized by [0 we have only the coefficient c j on each edge I j since all c j vanish The delta conditions are continuity of the resolvent at the vertex together with the fact that the sum of the first derivatives must be equal to α times the value of the resolvent at the vertex R u 0 j 0 = R u R u 0 j 0 = α 1R u j n 1 From 16 we obtain as matrix for the system of c s D Γ1 = 1 +α 1 +α 1 +α 1 +α 1 and as a free term column T Γ1 = t 2 0 t 1 0 t n1 0 t n0 α 1 t 1 0 +α 1 + t j 0 +α 1 2 j n 1 +α 1 By developing detd Γ1 with respect to its last column we obtain by recursion that detd Γ1 = n 1 +α 1 +α 1 Thus detd Γ1 does no vanish on the imaginary axis and R u 0 can be analytically continued in a region containing the imaginary axis We introduce here the matrix D Γ1 which is the matrix of the coefficients of the resolvent if on the last edge I n1 we should have c n1 e x instead of c n1 e x This changes only then 1 n 1 -entry of D Γ1 in +α 1 instead of +α 1 D Γ1 = 1 +α 1 +α 1 +α 1 +α 1 +α 1

9 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 9 Moreover the free term column remains the same for this new system We have again by recursion det D Γ1 = n 1 2 +α 1 +α The general tree case Any tree with p vertices p 2 can be seen as a tree 1 with p 1 vertices to which we add a new vertex on one of its infinite edges and n p 1 new infinite edges from it Let us denote by N the number of edges of 1 By this transformation I N becomes an internal edge parametrized by [0a p 1 ] and we have in addition I N+j as external edges for 1 j n p 1 We denote α p the positive strength of the δ condition in the new p th vertex The matrix of the new system unknowns of the 1 system together with an extra-unknown on the new internal line I N aswellasn p 1unknownsonthenewn p 1externaledgesisdenotedby D Γp Notice that if we write the system of unknowns of by changing the order of the unknowns ie permuting columns or the order of the conditions at vertices ie permuting lines then the determinant remains unchanged or it changes sign and the ratio det D Γp detd Γp remains unchanged We shall prove the following Lemma Lemma 31 We have the recursion formulae detd Γ1 = n 1 +α 1 +α 1 detd Γp = n p +α p +α p e a p 1 detd Γp 1 det D Γp detd Γp = n p 2+α p n p+α p det D Γ1 detd Γ1 = n 1 2 +α 1 n 1 +α 1 np 4+αp n p+α p 1 np 2+αp n p+α p 1 n p 2 +α p n p +α p e 2a det D Γp 1 p 1 detd Γp 1 e 2a p 1 det D Γp 1 detd Γp 1 e 2a det D p 1 Γp 1 detd Γp 1 Proof The part on Γ 1 was proved in subsection 321 For by writing the delta conditions at the end of I N together with the two conditions involving the coefficients on I N at the begining of I N we obtain the matrix D Γp as D Γp 1 1 +α p 1 e a p 1 e a p 1 1 +α p +α p e a p 1 e a p 1 +α p +α p +α p +α p +α p

10 10 V BANICA AND L I IGNAT and the free term column as T Γp = T Γp 1 t N+1 0 t N a p 1 t N+np 1 0 t N+np 2 0 α p t N a p 1 +α p + t N+j 0 +α p 1 j n p 1 By developing detd Γp with respect to the last n p lines we obtain an alternated sum of determinants of n p n p minors composed from the last n p lines of D Γp times the determinant of the matrix D Γp without the lines and columns the minor is made of On the last n p lines there are only n p +1 columns that does not identically vanish The only possibility to obtain a n p n p minor composed from the last n p lines of D Γp with determinant different from zero is to choose all last n p 1 columns together with a previous one This follows from the fact that if we eliminate from detd Γn both previous columns together with n p 2 columns among the last n p columns we obtain a block-diagonal type matrix with first diagonal block D Γp 1 with its last column replaced by zeros so its determinant vanishes Therefore detd Γp = detd Γp 1 deta np det D Γp 1 detb np where for m 1 A m and B m are the m m matrices e a p 1 1 A m = e a p 1 +α p +α p +α p +α p B m = e a p 1 +α p +α p e a p 1 1 +α p +α p +α p +α p +α p +α p We have deta 2 = 2 +α p e a p 1 +α p and by developing A m with respect to the first last column we obtain the recursion formula deta m = +α p e a p 1 +deta np 1 so deta m = m +α p +α p e a p 1

11 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 11 Similarly we obtain Therefore we find indeed detb m = m 2 +α p +α p e a p 1 detd Γp = n p +α p +α p e a p 1 detd Γp 1 In a similar way we get 1 n p 2 +α p n p +α p e 2a det D p 1 Γp 1 detd Γp 1 det D Γp = n p 2 +α p e a p 1 detd Γp 1 n p 4 +α p e a p 1 det +α p +α D Γp 1 p so det D Γp detd Γp = n p 2+α p n p+α p np 4+αp n p+α p 1 np 2+αp n p+α p and the proof of the Lemma is complete e 2a det D Γp 1 p 1 detd Γp 1 e 2a p 1 det D Γp 1 detd Γp 1 33 A lower bound for detd Γp iτ away from 0 Lemma 32 Function detd Γp is lower bounded by a positive constant on a strip containing the imaginary axis away from zero: det D Γp δ > 0 c Γp ǫ Γp > 0 0 < r Γp < 1st detd Γp > c Γp detd Γp < r Γ for all C with R < ǫ Γp and I > δ Proof We shall prove this Lemma by recursion on p For p = 1 Lemma 31 insures us that detd Γ1 = n 1 +α 1 +α 1 det D Γ1 detd Γ1 = n 1 2 +α 1 n 1 +α 1 We obtain a positive lower bound for detd Γ1 if we avoid that it approaches zero Therefore the existence of c Γp > 0 is obtained by considering ǫ Γ1 α 1 2n 1 Next we have n 1 2 +α 1 n 1 +α 1 < 1 0 < α1 R +n 2 so for any δ > 0 we get an appropriate 0 < r Γ1 < 1 by choosing ǫ Γ1 n δ 2 2α 1 Assume that we have proved this Lemma for p 1 We shall show now that it also holds for p Now from ratio information part in this Lemma for 1 we can choose ǫ Γp small enough to have for R < ǫ Γp and I > δ 1 n p 2 +α p n p +α p e 2a det D p 1 Γp 1 > c 0 > 0 detd Γp 1 Also from this Lemma for 1 we have the existence of two positive constants c Γp 1 and ǫ Γp 1 such that detd Γp 1 > c Γp 1 C R <

12 12 V BANICA AND L I IGNAT ǫ Γp 1 and I > δ Finally np+αp +α p is lower bounded by a positive constant for R small enough so eventually we get c Γp ǫ Γp > 0 detd Γp > c Γp C R < ǫ Γp I > δ We are left with showing that the ratio det D Γp detd Γp is of modulus less than one In view of the recursion formula on the ratio from Lemma 31 we first impose as a condition on ǫ Γp that r Γp 1 := e 2ǫ Γp a p 1 r Γp 1 < 1 and then we have to show that for z < r Γp 1 n p 2 +α p [n p 4 +α p ]z n p +α p [n p 2 +α p ]z < r for all complex with R < ǫ Γp and I > δ for ǫ Γp to be chosen and r Γp < 1 Denoting q = n p 2 +α p the above inequality is written in as q q 2z < q +2 qz q1 z+2z < q1 z+2 Expanding this last inequality we find that we have to prove that 0 < 2 1 z z 2 n p 2 2 +α p R Since n p 2 and z < r Γp 1 < 1 it is enough to have Also Rz < r Γp 1 < 1 so by choosing we get the existence of r Γp < 1 0 < 2 1 z z 2 α p R ǫ Γp 1 r2 1 δ 2 2α p 1 r Γp The multiplicity of the root = 0 of detd Γp Lemma 33 For all p 1 we have the following informations Pp 1 : det D Γp det 0 = 1 Pp 2 detd : DΓp 0 < 0 Γp detd Γp Proof Lemma31insuresusthat det D Γ1 detd Γ1 = n 1 2+α 1 n 1 +α 1 andinparticular det D Γ1 2α 1 = detd Γ1 n 1 +α 1 2 and the Lemma follows for p = 1 since α 1 > 0 We shall show the general case by recursion Let us denote by P p and Q p the numerator and respectively the denominator in recursion formula of the ratio from Lemma 31 P p = n p 2 +α p n p +α p n p 4 +α p n p +α p Q p = 1 n p 2 +α p n p +α p e 2a p 1 det D Γp 1 detd Γp 1 e 2a p 1 det D Γp 1 detd Γp 1

13 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 13 We have P p 0 = Q p 0 = 0 and in view of Pp 1 1 we compute P p 0 = Q p 0 = 2 αp +2a p 1 det DΓp 1 detd Γp 1 Therefore P 2 p 1 insures us that P p 0 = Q p 0 0 and we apply l Hôpital s rule to conclude P 1 p 0 Since P p Q p = 2 1 e 2a p 1 det D Γp 1 n p +α p detd Γp 1 we define P p and Q p by In particular P p = 2 n p +α p Pp Q p = det D Γp detd Γp = P p Q p Pp Q p = By using P 1 p 1 and P2 p 1 Moreover 2 n p +α p Qp P p Q p 0 = 2a p 1 + det DΓp 1 detd Γp 1 1 e 2a p 1 det D Γp 1 detd Γp 1 P p 0 = Q p 0 = α p 2 P p 0 = α p 2 Q p 0 0 and we can compute det DΓp 0 = P p 0 Q p 0 P p 0 Qp 0 = P p Q p 0 detd Γp Q p 0 2 Q p 0 = α p 2 By using again P 2 p 1 we obtain P2 p 2a p 1 det DΓp 1 detd Γp α p +2a p 1 det D Γp 1 detd Γp 1 0 Lemma 34 = 0 is a root of order p 1 of detd Γp Proof From Lemma 31 we have detd Γ1 = n 1 +α so detd Γ1 0 0 Lemma 31 provides us the expression detd Γp = n p +α p +α p e a p 1 detd Γp n p 2 +α p n p +α p so by recursion it is enough to show that = 0 is a simple root for 1 n p 2 +α p n p +α p e 2a p 1 det D Γp 1 detd Γp 1 e 2a p 1 det D Γp 1 detd Γp 1

14 14 V BANICA AND L I IGNAT This expression is precisely Q p from the proof of Lemma 33 and it was proved there that Q p Vanishing of the numerator at τ = 0 Recall that we have denoted by M ce respectively detm ce the matrix D Γp with the column corresponding to the unknown c e respectively c e DΓ e p respectively Dẽ replaced by the free terms column T Γp In particular detm ce respectively detm ce is the determinant of the matrix D Γp with the column corresponding to the unknown c e respectively c e replaced by T Γp Lemma 35 The following holds 18 T Γp 0 = e E t e 00D e 0+ e I t e 00Dẽ 0 Remark 36 From the shape of D Γp displayed in the proof of Lemma 31 we notice that the two junction columns with D Γp 1 corresponding to the coefficients of the resolvent on the connecting edge I N are D I N Γp = 00 1 t e a p α p e a p 1 +α p 1 +α p and Γp = 00 1 t e a p 1 00e a p 1 +α p 1 DĨN In particular these two columns are the same at = 0 Moreover D Γp contains p 1 such pair of columns D e 0 = Dẽ 0 for all e I Thus the last term in the right hand side of 18 could be D e 0 either Dẽ 0 e I Proof We will prove this identity inductively In the case p = 1 we use that T Γ1 isgiveninsection321 WechooseX 1 = t 1 00t 2 00t n1 00 and then D Γ1 0X 1 = T Γ1 0 which proves 18 when p = 1 Given now X p 1 such that D Γp 1 0X p 1 = T Γp 1 0 we construct X p as follows t X p = t X p 1 0t N+1 00t N+np 100 Using the recursion between D Γp and D Γp 1 used in the proof of Lemma 31 identity T Γp 1 t N+1 0 t N a p 1 T Γp = t N+np 10 t N+np 20 α p +α p t N a p 1 + +α p 1 j n t p 1 N+j0 and the fact that t e 00 = t e a e 0 for all e I we obtain that X p satisfies the system D Γp 0X p = T Γp 0 Writing this identity in terms of the columns of matrix D Γp 0 we obtain the desired identity Lemma 37 = 0 is a root of order p 1of detm ce and of detm ce for all edge e

15 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 15 Proof WeshallperformtheprooffordetM ce ; theresultfordetm ce will bethesame From theshapeof D Γp displayed inthe proofof Lemma 31 and Remark 36 we have p 1 pairs of columns that are equal at = 0 Moreover by Lemma 35 T Γp 0 is a linear combination of these columns evaluated at = 0 The derivative of a determinant is the sum of the determinants of the matrices obtained by differentiating one column When T Γp does not replace any of these 2p 1 columns it follows that the result of this Lemma holds since there are always two columns identically Then by the above argument we have already 19 k detm ce 0 = 0 0 k p 3 Assume now that T Γp replaces one of these 2p 1 columns For proving the Lemma we are left to show that p 2 detm ce 0 = 0 Using again the fact that D Γp contains p 1 pairs of columns that are the same two by two at = 0 we only need to show that deta Γp 0 = 0 where A Γp is D Γp with the column T Γp replacing one column of one pair and one column of each remaining p 2 pairs of columns is differentiated In particular A Γp 0 contains one column of each p 1 pairs unchanged Since by Lemma 35 we know that T Γp 0 is a linear combination of the columns corresponding to external edges and of the internal ones each one from the p 1 pairs the new determinant vanishes and the proof is finished Lemma 38 For all edges indices λ and e = 0 is a root of order p 2 for the coefficient f λe of t λ 0 in detm ce and the same holds for the coefficient f λe of t λ 0 in detm ce Proof This result follows from the discussion that led to 19: the matrix M ce has p 2 pairs of columns that are identical at = 0 Lemma 39 For all edge index e and all external edge index λ = 0 is a root of order p 1 for the coefficient f λe of t λ 0 in detm ce and the same holds for the coefficient f λe of t λ 0 in detm ce Proof The statement corresponds to the particular case of Lemma 37 where all the components of T Γp are taken to be zero except t λ 0 which is replaced by one Lemma 310 For all edge index e and all internal edge index λ = 0 is a root of order p 1 for fλe 1 + f2 λe where f1 λe is the coefficient of t λ 0 in detm ce and fλe 2 is the coefficient of t λa λ in detm ce Also the same holds for f λe 1 + f λe 2 where f λe 1 is the coefficient of t λ 0 in detm ce and f λe 2 is the coefficient of t λ a λ in detm ce Proof The proof goes the same as for Lemma 39

16 16 V BANICA AND L I IGNAT 36 The end of the proof From the previous subsections we conclude the following result Lemma 311 Function R fx can be analytically continued in a region containing the imaginary axis Proof The proof is an immediate consequence of decomposition 15: 20 R u 0 x = detmce detd Γp ex + detm detd Γp e x + t ex for x I e and Lemma 32 Lemma 33 and Lemma 37 Proof of Theorem 12 As a consequence of Lemma 311 we can use a spectral calculus argument to write the solution of the Schrödinger equation with initial data u 0 as 21 e ithγ αu 0 x = 1 iπ c e e itτ2 τr iτ u 0 xdτ In view of the definition of t e and with the notations from Lemma 39 and Lemma 310 we can also write the decomposition 20 as 22 τr iτ u 0 x = 1 u 0 e iτ x y dy + f λe iτ u 0 ye iτy dye iτx 2 I e detd Γp iτ λ E + f λe iτ u 0 ye iτy dye iτx detd Γp iτ λ E + u 0 y e iτy fλe 1 iτ f 2 λ I detd Γp iτ +eiτa λ y λe iτ dye iτx detd Γp iτ + u 0 y e iτy f1 λe iτ f 2 λ I detd Γp iτ +eiτa λ y λe iτ dye iτx detd Γp iτ Moreover in view of the results in Lemma 310 and Lemma 39 we gather the terms as follows 23 τr iτ u 0 x = 1 u 0 e iτ x y dy 2 I e + f λe iτ u 0 y λ E detd Γp iτ eiτx+y dy+ f λe iτ u 0 y λ E detd Γp iτ eiτy x dy + u 0 y f1 λe iτ+f2 λe iτ e iτx+y dy+ f λe 1 u 0 y iτ+ f λe 2 iτ e iτy x dy λ I detd Γp iτ λ I detd Γp iτ + e iτa λ y e iτy fλe 2 u 0 y iτ e iτx dy λ I detd Γp iτ + e iτa λ y e iτy f2 λe iτ u 0 y e iτx dy λ I detd Γp iτ

17 DISPERSION FOR SCHRÖDINGER WITH δ POTENTIALS AND FOR δ TREES 17 Let e be an external edge In view of Lemma 34 and Lemma 39 we obtain that the fraction is upper bounded near τ = 0 Outside f λe iτ detd Γp iτ a neighbourhood of τ = 0 we use Lemma 32 to infer that detd Γp iτ is positively lower bounded outside neighbourhoods of τ = 0 Moreover in view of the explicit entries of M ce iτ we see that f λe iτ is upper bounded for any τ R since all the entries of matrix D Γp iτ as well as the coefficients of t λ in T Γpiτ have absolute value less than one Summarizing we have obtained that f λe iτ detd Γp iτ L R The derivative of this fraction is upper-bounded near τ = 0 by limited development at τ = 0 Outside neighbourhoods of τ = 0 we have that τ f λe iτ and τ detd Γp iτ have upper bounds of type 1 τ This is because 2 in each term of τ f λe iτ and τ detd Γp iτ contains a derivative of an element of the line given by the δ-condition involving the derivatives in the root vertex O This vertex is the one which is an initial vertex for all n edges emerging from it: Ie = O e EO e If α denotes the strength of the δ-condition in O then this line of the matrix D Γp iτ is composed by 01 and ± iτ iτ+α where the minus sign appears only on the finite edges that stars from O and this line for the column matrix iτt Γp iτ is iτ α iτ +α t 10iτ + iτ t j 0iτ iτ +α 2 j n Finally as above f λe iτ and detd Γp iτ are upper bounded and from Lemma 32 we have that detd Γp iτ is positively lower bounded outside neighbourhoods of τ = 0 As a conclusion we infer that τ f λe iτ detd Γp iτ L1 R The same argument using Lemma 39 Lemma 310 and Lemma 38 can be performed to obtain that f λe iτ detd Γp iτ fλe 1 iτ+f2 λe iτ detd Γp iτ f λe 1 iτ+ f λe 2 iτ detd Γp iτ e iτa λ y e iτy fλe 2 iτ detd Γp iτ e iτa λ y e iτy f2 λe iτ detd Γp iτ are in L with derivative in L 1 Notice that when λ belongs to an internal edge it follows that the interval have finite length Therefore for the last fractions we use that e iτaλ y e iτy fλe 2 iτ vanishes or order p 1 at τ = 0 and repeat the argument used above The only difference with the previous cases is that we will obtain bounds in terms of parameter y Since y is now on an internal edge of finite length we obtain uniform bounds Therefore the dispersion estimate 10 of Theorem 12 follows from 21 by

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