Dispersion property for discrete Schrödinger equations on networks

Transcription

1 SCOALA NORMALA SUPEROARA BUCUREST Master s thesis Dispersion property for discrete Schrödinger equations on networks Author: Cristian Gavrus Supervisor: Liviu gnat June 4, 202

2 Summary This thesis treats the analysis of the Schrödinger equation on networks consisting of a tree with infinite edges in the last generation: (0. iu t (t, v + u(t, v = 0, t 0, v V, deg(v = 2, u(t, v = 0, t 0, v V, deg(v, u(0, v = u 0 (v, v V, deg(v = 2. This model corresponds to a system of discrete linear Schrödinger equations on each edge of the tree coupled at the vertives of the network by a discrete Kirchhoff s-type coupling. This problem was posed by my supervisor, Prof. Liviu gnat, and we have been working on this project motivated by earlier results of Liviu gnat, Valeria Banica and Diana Stan. The main result of our work (which we expect to publish soon is the following one, from which Strichartz estimates follow: Theorem 0.. For every u 0 l 2 (Γ, this system has a unique solution u C(R, l 2 (Γ. Additionally, there exists C > 0 for which C (0.2 u(t l (Γ ( + t u 0 / l (Γ u 0 l (Γ. would like to warmly thank my advisor, Liviu gnat from MAR, for all his support during these years.

3 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS. ntroduction Let us consider the linear Schrödinger equation (LSE: { iut + u (. xx = 0, x R, t 0, u(0, x = ϕ(x, x R. Linear equation (. is solved by u(t, x = S(tϕ, where S(t = e it is the free Schrödinger operator. The linear semigroup has two important properties. First, the conservation of the L 2 -norm: (.2 S(tϕ L 2 (R = ϕ L 2 (R and a dispersive estimate of the form: (. (S(tϕ(x (4π t /2 ϕ L (R, x R, t 0. More general estimates known as Strichartz s estimates has been obtained. These estimates have been successfully applied to obtain well-posedness results for the nonlinear Schrödinger equation (see [], [2] and the reference therein. n the case of the Schrödinger equation on trees similar properties have been obtained in [5] and [2]. n [5] the case of regular tree has been analyzed. The main idea has been the connexion between the Schrödinger equations on regular trees and the -D laminar Schrödinger equation considered in []. The extension of these result to the case of a general tree has been considered in [2]. The main goal of this article to consider problems similar to those above but in the discrete case. Let us recall some previous results in the discrete framework. Let us now consider the following system of difference equations { iut + d u = 0, j Z, t 0, (.4 u(0 = ϕ, where d is the discrete laplacian defined by (.5 ( d u(j = u j+ 2u j + u j, j Z. Concerning the long time behavior of the solutions of system (.4 in [] the authors have proved that a decay property similar to the one obtained for the continuous Schrödinger equation holds: (.6 u(t l (Z C( t + / ϕ l (Z, t 0. 2

4 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS The proof of (.6 consists in writing the solution u of (.4 as the convolution between a kernel K t and the initial data ϕ and then estimate K t by using Van der Corput s lemma. n [] the authors apply these estimates on the linear semigroup to prove that in some cases the semilinear discrete Schrödinger equation decay like the free solution in the corresponding l p (Z-norms. n a recent paper [9] the authors use some modifications of the stationary phase method to obtain improved l l p decay estimates for the linear discrete Schródinger equation and better than in [] results for the nonlinear case. More general models have been considered in [6] where the authors couple two discrete Schrödinger equations posed on two infinite strips: (.7 iu t (t, j + b 2 ( d u(t, j = 0 j, t 0, iv t (t, j + b 2 2 ( d v(t, j = 0 j, t 0, u(t, 0 = v(t, 0, t 0, b 2 (u(t, u(t, 0 = b 2 2 (v(t, 0 v(t,, t 0, u(0, j = ϕ(j, j, v(0, j = ϕ(j, j. n the above system u(t, 0 and v(t, 0 have been artificially introduced to couple the two equations on positive and negative integers. The third condition in the above system requires continuity along the interface j = 0 and the fourth one can be interpreted as the continuity of the flux along the interface. We will make evidence in the paper the connection between this result and the models considered here. n this paper we prove dispersion inequality similar to (.6 for the linear discrete Schrödinger equation defined on a network formed by a tree Γ = (V, E (connected graph without closed paths, V being the set of vertices and E of edges, with all the edges having length an integer number greater than two and with the external edges infinite. Once we fixed the tree Γ as in Fig. we discretize all its edges as in Fig. 2 obtaining a new graph Γ d = (V d, E d. Now we consider a discrete Laplace operator on each of the edges. The presentation of the discrete Laplace operator d will be given in the next section. Let us just say here that d acts as the usual discrete Laplacian (.5 at each internal node of each edge, and at the vertices of the tree we consider a coupling condition. We now discretize all the edges of the tree choosing the mesh size to be one. Since the lengths of the edges has been chosen to be integer numbers greater than two this is always possible and moreover there is at least one internal vertex on each edge of the considered tree. n this way we have vertices with degree at least three that come form the way that we have considered the tree and vertices with their degree of order two that come from the discretization. n Fig. we have a such structure. The discrete Laplacian of a function u on such structure Γ d is given as follow:

5 4 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS Figure. A tree consisting in two vertices, one finite edge and four infinite edges. (.8 ( d u(v = w E v (u(w u(v, where the sum is taken over all the neighbors w Γ d of the vertex v. We now can state the main results of this paper. Let us consider the the iu t (t, v + u(t, v = 0, t 0, v V, deg(v = 2, (.9 u(t, v = 0, t 0, v V, deg(v, u(0, v = u 0 (v, v V, deg(v = 2. This model corresponds to a system of discrete linear Schrödinger equations on each edge of the tree coupled at the vertives of the network by a discrete Kirchhoff s-type coupling. We will explain later this type of coupling. We point out that function u should be considered only at the internal nodes V d \ V. Another model of interest is the case of the combinatorial Laplacian. Here we have no coupling condition and the Schrödinger equation should be satisfies even at the vertices with degree greater than three.

6 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS 5 Figure 2. The discrete graph obtained from the previous one in Fig.. At the blue points we have coupling conditions, at the rest classical discrete Laplacian (.0 { iut (t, v + u(t, v = 0, t 0, v V, u(0, v = u 0 (v, v V. n some sense this model consider a dynamic coupling at the original vertices of the tree Γ. This two type of models has been already observed in the simpler case of a system formed by two discrete Schrödinger equations on two infinite half-lines coupled at the origin [6]. The main result of this paper is the following one: Theorem.. For every u 0 l 2 (Γ, system (.9 has a unique solution u C(R, l 2 (Γ. Additionally, there exists C > 0 for which C (. u(t l (Γ ( + t u 0 / l (Γ u 0 l (Γ. An immediate consequence (see [7] for an abstract result are the following space-time estimates for the solutions of the linear problem. Theorem.2. For every u 0 l 2 (Γ, the solution u of system (.9 satisfies (.2 u L q (R,l r (Γ C q,r u 0 l 2 (Γ

7 6 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS for any pair (q, r satisfying q ( 2 r. 2. Notations and Preliminaries n this section we present some generalities about graphs and introduce the discrete Laplace operator on such structure. Let Γ = (V, E be a graph where V is a set of vertices and E the set of edges. For each v V we denote E v = {e E : v e}. We assume that Γ is a connected locally finite graph, i.e. the degree of each vertex v of Γ is finite: d(v = 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 [8] for more details on graphs with infinite edges. We fix an orientation of Γ and for each oriented edge e, we denote by (e the initial vertex and by T (e the terminal one. Of course in the case of infinite edges we have only initial vertices. We assume that the considered tree has all the vertices, except the origin, with degree at least three. We identify every edge e of Γ with an interval e, where e = [0, N e + ], N e an integer, if the edge is finite and e = [0, if the edge is infinite. Let v be a vertex of V and e be an edge in E v. We set for finite edges e { 0 if v = (e, j(v, e = l e if v = T (e and j(v, e = 0, if v = (e for infinite edges. We now discretize all the edges of the tree choosing the mesh size to be one. Since the lengths of the edges has been chosen to be integer numbers greater than two this is always possible and moreover there is at least one internal vertex on each edge of the considered tree. n this way we have vertices with degree at least three that come form the way that we have considered the tree and vertices with their degree of order two that come from the discretization. We identify any function u on Γ with a collection {u e } e E of functions u e defined on the vertices of the edges e of Γ. Each u e can be considered as a function on the set e Z. n fact, we use the same notation u e for both the function on the edge e and the function on the interval e identified with e. For a function u : Γ C, u = {u e } e E, we denote by f(u : Γ C the family {f(u e } e E, where f(u e : e C. Each edge α E is parametrized by α, which is {, 2,... } or {, 2,... }. Each vertex v V is the initial point for m v edges: αi v ( i m v and (except for the origin the final point for a single edge, α0. v We indentify the funtions defined on the graph by the families (u α α E such that u α : α C with the convention that u e (j(v, e = u e (j(v, e, e, e E. v

8 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS 7 f the v vertex is: the final point of the α edge and the origin of the α,..., α m vertices, then we extend these functions by setting (2. u α ( + = u α i (0 = uα ( + u α ( +... u αm (, i m. m + n the origin of the graph we extend the functions by the same formula, but without the α term, and divide by m instead. We denote l 2 the space of functions (u α α E for which u 2 2 = u α (j 2 <. α E j α Then l 2 is a Hilbert space with the inner product u, v = u α (jv α (j. α E j α n the usual way, and by analogy with l 2, we also consider the l p spaces for p. We will use both of the following notations: u α (j = u α j. Of course, the functions u α obey the continuity and coupling conditions defined in (2... The spectrum and the resolvent of the discrete laplacian n this section we determine the spectrum and the resolvent of the considered discrete laplacian on the graph Γ. Theorem.. The spectrum of the discrete laplacian on Γ is σ( = [ 4, 0]. Proof. By the triangle inequality, we immediately get u 2 4 u 2 u l 2, so B(l 2 4. We see that is self-adjoint and 0 because u, u is a sum of squares and these show that σ( [ 4, 0]. t is known that the discrete laplacian on l 2 (N has the spectrum [ 4, 0]. This means that for every λ [ 4, 0] there is a sequence (u m m 0 such that for every m we have u m ( l 2 (N and u m 2 =, ( λu m 2 0. We now use it to construct a sequence vm( α l 2 (Γ, for λ [ 4, 0], in the following way: we chose an infinite edge α 0 and define v α 0 m (j = u m (j for j, and vm(k α = 0 otherwise, except for v α 0 m (0 = vα 0 m ( so that (. is verified ( t is the number of edges that t+ start from the same vertex as α 0. t is clear then that vm α 2 = and ( λvm α 2 0, which proves that σ( = [ 4, 0]. We now consider the resolvent R λ = ( λ B(l 2 for λ C \ [ 4, 0]. Theorem.2. For any λ C \ [ 4, 0] and f l 2 (Γ d, the resolvent R λ f is given by (. (R λ f α n = a α r n + b α r n 2 + S α n, α E, n α where r < < r 2 are the roots of the second order equation r 2 (2 + λr + = 0

9 8 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS and (.2 S α j = r r k α [ ] r j k r j+k fk α. Remark. We emphasize that on infinite edges a α = u α 0 and b α = 0. Proof. Denoting u = R λ f we have u α j+ (2 + λu α j + u α j = f α j, j α. This is a second order recurrence relation for which Sj α is a particular solution. The general solution is u α j = Tj α + Sj α, where T α is the general solution of the homogenous equation (. v α j+ (2 + λv α j + v α j = 0. The set of solutions of (. forms a two-dimensional vector space, so for any α, T α j = a α r j + b α r j 2 where r, r 2 are the solutions of r 2 (2 + λr + = 0 with r <. On the infinite edges we must have T α l 2 ( α, so in that case b α = 0. n the following proposition we prove a limiting absorption principle. n particular, it shows that R x R x+ is well-defined as an operator in B(l, l. Proposition.. Let x [ 4, 0] and ε > 0. For any f l 2 and any α E we have (.4 [R x iε R x+iε ] fn α = 2i m[aα r n +b α r2 n ]+ [ f α ( ] k m 2i r k r r n k r n+k, n α α where r < < r 2 are the roots of the equation r 2 (2 + λr + = 0 for λ = x iε. For any ϕ l and a, b R we have (.5 lim ε 0+ b a [R x iε R x+iε ] ϕ α n dx = b a [R x R x+ ] ϕ α n dx, n α where the last term in the RHS is given by (.6 2i [R x R x+ ] ϕ α n = m[a α r n + b α r n ] + [ ϕ α ( k m r n k r n+k ] r r k α such that r is the solution with m r 0 of the equation r 2 (2 + xr + = 0 (Remark: a α, b α means lim (x iε x 0 of these functions. Proof. The formulas (.4 and (.5 are only computation and then we use the dominated convergence theorem. Checking the necessary conditions to apply it is done identically as Step from the Lemma 4.4 of [6].

10 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS 9 We return to the equation 4. The main result (4. { i t u α (t, j + u α (t, j = 0, t 0 u α (0, j = ϕ(j, α E, j α. Theorem 4.. For any ϕ l 2, there exists a unique solution u = (u α α E C(R, l 2 of the system (4. which satisfies u(t l 2 = ϕ l 2. Moreover, if ϕ l then u can be represented as (4.2 u(t n = ( e it ϕ = e itλ [R n λ R λ+ ] ϕ n dλ. 2πi Proof. We now use the Hille-Yosida theorem, or it s particular case, Stone s theorem (theorems.7,.6 of [] to show that there exists a unique solution u C(R, l 2 of equation (6., given by (4. u(t = e it ϕ, where the operators e it are defined by power series or by functional calculus. They are unitary operators, so u(t l 2 = ϕ l 2. Before proving (4.2 let us recall that as a consequence of Cauchy s formula (see [4] Th. X.6., for any bounded and self-adjoint operator T, the resolution of the identity E is given by E(a, b = lim lim b δ R (x iε T R (x+iε T dx δ 0+ ε 0+ 2πi a+δ where the limit is in the strong-operator topology sense. Let us define (4.4 µ ϕ,n (a, b = E(a, bϕ n = lim lim δ 0+ ε 0+ 2πi b δ a+δ [R x iε R x+iε ] ϕ n dx,. Using functional calculus (see [0], Th. 2.2 and Th the solution of equation (4. satisfies (4.5 u(t n = ( e it ϕ = e itλ dµ n ϕ,n,. σ( From now on, the integral on = [ 4, 0] means = lim δ 0+. we get lim ε 0+ b a [R x iε R x+iε ] ϕ n dx = Thus, from (4.5 and (4.4 we get u(t n = ( e it ϕ = n 2πi which finishes the proof. b a [R x R x+ ] ϕ n dx. e itλ [R λ R λ+ ] ϕ n dλ, δ. Using Proposition 4+δ

11 0 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS Before we prove the main result, we state the key lemma which will be proved in the following sections. Lemma 4.. For any ϕ l and any t R, n Z the following inequalities are true: (4.6 e itλ a α r n dλ C ϕ l, + t e itλ b α r n dλ C ϕ l + t Proof of the main result: We use (.6 and (4.2 and observe that, in order to get the inequality (., it suffices to check that for all n and k (4.7 e itλ m a α r n dλ C ϕ l, + t e itλ m b α r n dλ C ϕ l + t (4.8 [ e itλ m r r ( r n k r n+k] dλ C. + t We recall that r is defined in Proposition., and u α 0,, b α means lim (x iε x 0 of u α 0 si b α as functions of λ. nequality (4.8 was proved in [], (4., in a more general case by using Van der Corput s lemma. t is sufficient to prove (4.7 without the imaginary part m because, by chosing t instead of t and conjugating both, we get the inequalities with additional m. So, by using the lemma stated before, the result follows. Let s note that what we have obtained thus far did not use the structure of the graph. To prove (4.6 we must use the structure of the graph to estimate u α 0, a α and b α using the continuity and coupling conditions ( A particular case n this subsection, in order to better understand the difficulties of determining the coefficients in Lemma 4., we first consider the particular case of a tree as in Fig.. n the considered case the set or vertices is given by V = {v 0, v }. From v 0 starts a finite edge α = on which we have the function u α = u : = {, 2,... N} C and two infinite edges u 2 and u. From v, which is the endpoint of α =, start two infinite edges: α {, 2}. We are in the case where λ, but the sum (.2 is also defined there for r = r. Looking at (., we see that we have to determine: u 0, u 2 0, u 2 0, u 0, a = a and b = b. Writing the coupling conditions at vertices v 0 and v it follows that (5. u 0 = a + b = u 2 0 = u 0 = u +u2 +u, u N+ = u 0 = u 2 0 = u N +u +u2.

12 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS u 2 ũ u u 2 Figure. A particular case. At the blue points we have coupling conditions, at the rest classical discrete Laplacian Using the representation formula given by Theorem.2 we obtain that u 2 0 u 0 = 0, (5.2 u 2 0 a b = 0, (r u (r u 0 + (r a + (r 2 b = (S + S 2 + S = K, r N+ a + r N+ 2 b u 0 = 0, u 0 u 2 0 = 0, r N a + r N 2 b + ( r 2 u 0 + ( r 2 u 2 0 = (S N + S + S 2 = K 6.

13 2 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS n matrix formulation we have Mx = K where x T = [x i ] = [u 2 0, u 0, a, b, u 0, u 2 0 ], K T = [0, 0, K, 0, 0, K 6 ] and r r r r M = 0 0 r N+ r2 N r N r2 N r r 2 2 The determinant marked in the corner comes from the case where there are only three infinite edges and it equals (r. Each x i is given by x i = D i / det M where D i is the determinant of the matrix obtained from M by inserting column K in the place of column i of M. Lemma 5.. For any i =,..., 6 and every α E there is a polynomial q α,i (x, y and another polynomial q α,i (x, y when α is finite such that (5. x i = α E S α q α,i (r, r (r (r + h(r + α finite S α q α,i (r, r (r (r + h(r, where h is a rational function that does not vanishes on the unit circle. Proof. Using that x i = D i / det M we have to give some representation for D i and to compute det M. Observe that [ det M = (r 2 r2 N+ 9 r 2N (2r 2]. Denoting ϕ(x = 9 x 2N (2x 2, it satisfies ϕ( = 0 and ϕ ( 0. Thus, there is a polynomial function g(x which does not vanish on T such that ϕ(x = (x + g(x. Denoting now h(r = r2 N+ g(r, we obtain that (5.4 det M = (r 2 (r + h(r. Let us now analyze the form of the determinants D i. We want to prove that for every i 6 and every α E there is a polynomial q α,i (x, y and another polynomial q α,i (x, y when α is finite such that [ (5.5 D i (r = (r S α q α,i (r, r + S α q α,i (r, ]. r α E α finite Recall that λ, r 2 = /r since they are the solutions of the equation x 2 (2 + λx + = 0. By inserting the column K in the position of column i in M and expanding the determinant we obtain that D i is a polynomial in the variables (r, r 2. Setting r 2 = /r, there exists

14 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS p N such that D i (r = P (r. We prove that, when r =, D i (r = 0. We distinguish two cases. f i / {, 4} then the third and fourth columns are equal. When i {, 4} we easily make a linear combination of columns that vanishes. For example, for i = we have: C + C 2 + C 4 + C 5 + C 6 = 0. This proves the existence of a polynomial Q(x such that D i (r = (r Q(r r p. By observing that K = (S + S 2 + S and K 6 = (SN + S + S 2 are the elements in K, and considering the expansion by that column, we get the desired formula for D i. Using now (5.4 and 5.5 we get the desired expression for x i. r p 6. The general case Like in the particular case, we now have to determine the coefficients in Lemma 4.. Starting from a graph with n vertices Γ n, we consider it as a graph with n vertices Γ n to which we truncate an infinte edge U at the N + point - where we add a new vertex v, from which m new edges start: u,..., u m. The edge U starts from vertex w, from which t edges start in total. n the case of Γ 2 we have a truncated edge, m = 2 new edges, and t = because edges start from the origin. For Γ n, we denote by V 0 the finite edges, by W 0 the infinite edges except for U, and consider U separately. The expression of the resolvent in (5. is: (6. u α n = u α 0 r n + S α n, α W 0 u β n = a β r n + b β r n 2 + S β n, β V 0 U n = U 0 r n + S U n. For Γ n, the former edge U, now restricted at {,..., N}, is denoted by V. The resolvent is: u α n = u α 0 r n + Sn, α α W 0 (6.2 u β n = a β r n + b β r n 2 + S β n, β V 0 V n = a r n + b r n 2 + S V n u i n = u i 0 r n + S i n, i m The continuity and coupling conditions (. form a system, like in the previous case. For the moment, we are only interested in the matrix of the system s coefficients, which we denote by M n for Γ n.

15 4 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS The matrix M n characterises the system corresponding to Γ n, but written in such a way (by permuting the lines and the order of the variables that U 0 is the last variable and the last two equations correspond to the vertex w in (. (see the matrix below or the particular case. To get the relation between M n and M n we see that, except for a - which formerly coresponeded to U 0, we have m + new variables: [b, u 0,... u m 0 ]. By using the system obtained for Γ n and the new coupling conditions, we get the system for Γ n given by the matrix: ( a b u 0 u u m 0 u m M n r t+ r t 2 t t M n = r N+ r2 N r N r2 N r m r m m+ m The matrix marked in the up-left corner is M n (the first line just lists the variables. (An exception occurs when the vertex w is the origin and the term t+ is replaced by. t We denote by M n the matrix M n in which we have inserted in the last position r 2 m+ m instead of r m+ m. We denote these determinants by D n and D n. Lemma 6.. We have D n D n = m r 2 m 2r 2 m+ m+ ( r r 2 N+ (6.4 ( N+ r r 2 m r 2 m+ D For r T we have n D n <, and in r = we have D n D n. D n D n D n D n =. Proof. By expanding the determinants on the first lines until the marked line, we get the following recurrence relation: [ ] [ ] D n = D n r N+ 2 (m r m + r2 N D n r N+ (m r m + r N.

16 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS 5 (6.5 D n = D n r N+ 2 (m + (r Analogously D n = D n D n [ [ ( r2 N+ (m (r m + [ r N+ (6.6 Dn = D n r N+ 2 (m + (r ( r r 2 N+ m r 2 m + D n D n m + r 2 m + + r2 N m + r N ( (m (r m + m + r 2 m + m [ m r 2 m + m 2r 2 m + ( r r 2 ]. ] ]. ] N+ D n. D n Observe that D n and D n have the same factor before the large bracket, and this factor cancels when we divide them, obtaining (6.4. We prove the other statement by induction on n. For n = we have a single vertex and m infinite edges and D = (m (r + r 2 D m(r so the statement is true in this case. Suppose that r, r 2 T and denote z = z < for proving that ( r r 2 N+ = m r 2 m D n D n m r 2 (m 2r 2 z < (m + (m r 2 z. so that we can assume that By squaring this inequality, simplifing and collecting the terms, it is equivalent to (r 2 + r [ (m z 2 + z 2] > 0 which is true and completes the induction step for this case. Suppose now that r = r 2 =. The ratios are rational functions and we write D n D n D n = f(x D n g(x = f (x g (x, f(x = (x + f (x = mx x 2N+2 D n [(m x 2] (x D n g(x = (x + g (x = (m + x x 2N+2 D n (mx (x. D n Then, for x T, f(x g(x = (x + ( x 2N+2 D n (x D n f (x g (x = x 2N+2 D n (x. D n

17 6 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS Lemma 6.2. We have D n D n ( = f ( g ( = 0 (6.7 D n = (r n (r + n h(r for h a rational function which does not vanish on T. D n D n ( =. Proof. The (r n factor is clear from (6.5 and, denoting the bracket from (6.5 by 2N+ mx D n ϕ(x = x (x m + D n we see from Lemma 6. that ϕ( = 0, so that D n = (r n (r + n h(r, for a rational function h. To prove that h does not vanish on T, by Lemma 6., we only need to prove that ϕ ( 0. ϕ ( = m(2n N + m + D n (. D n D We prove by induction that n : n D n ( (0,, which clearly implies ϕ ( 0. For n =, we have D D (x = m for which D m mx D ( (0,. For n > we use the functions f, g, f, g from Lemma 6. for which D n D n D n = f (x D n g (x ( = f g f g g 2 and f (x g (x = x 2N+2 D n D n (x. ( = f g g ( = D 2N + 2 n D n (. D (2N + 2(m + + m (m + n D n ( The last ratio is always in the interval (0, using the induction hypothesis. denominator we have used that g ( = g (. For the We write the coefficients from Lemma 4. in the vector x = [..., a, b, u 0, u 2 0,..., u m 0, u m 0 ] = [x i ]. Replacing the column i from M n by column K, like in the particular case, we get a determinant which we denote by Dn. i Then, the elements in the vector x are written x i = Di n D n. Lemma 6.. We have (6.8 D i n = (r n (r + n 2 [ α E S α q α,i (r, r + ( S α q α,i r, ] r α finite

18 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS 7 where q α,i si q α,i (6.9 x i = α E are polynomials. And S α q α,i (r, r (r (r + h(r + α finite S α q α,i (r, r (r (r + h(r, Proof. The column K contains sums of elements S α and S α, like in the particular case in Lemma 5. and the same reasoning applies here. So the formula (6.8 is true, provided that we prove that the following derivatives are zero: D i n( = 0, D i n( = 0,..., D i n( (n 2 = 0 D i n( = 0, D i n( = 0,..., D i n( (n = 0 (We write the number of derivatives in brackets placed up. We use the following known rule for differentiating a determinant with columns c, c 2,..., c r : if D(x = c (x,..., c r (x then: D (x = c (x, c 2 (x,..., c r (x + + c (x, c 2 (x,..., c r(x. So, by iterating this rule and denoting c, c 2,..., c r the columns of Dn, i it is sufficient to prove: c (i, c (i 2 2,..., c (ir r ( = 0 i + i i r n 2 c (j, c (j 2 2,..., c (jr ( = 0 j + j j r n. r Note that, in the same way as in the particular case Γ 2 we had (in M the columns and 4 equal in r = and r =, now, for M n, we have n pairs of columns and in each pair, the two columns are equal for r = r 2 = ±. This is true because, if M n has n 2 such pairs, M n will have the additional pair (a, b. Now, for r =, in D i n we also insert the column K: f K does not enter in one of these pairs of columns, as we only differentiate at most n 2 columns, for at least one of these pairs the two columns will be identical in r =. f K enters in one of the columns from the n pairs, as we differentiate at most n 2 of the others, considering the same reasoning as above, the only chance for the determinant to be non-zero is if we differentiate one column from each of the remaining pairs. n this instance: we make a linear combination of columns (different from K to be zero by induction: assuming we have such a combination from M n \ K n, we use it such that (in addition to it we only have to make a linear combination of the columns a, b, u 0, u 2 0,..., u m 0, u m 0 below the marked line, which is trivial. For r = r 2 = it is simpler because we have at most n derivatives, so we can always find a pair of columns which are equal. Equality (6.9 follows from (6.8 and (6.7.

19 8 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS Proof of Lemma 4.. We recall that we have to prove the following estimate e itλ x i r n dλ C ϕ l (n Z + t where x i are now written as in Lemma 6.: x i = α E S α q α,i (r, r (r (r + h(r + α finite Using (.2 we have that S α and S α have the form Hence (6.0 S α = k α r k ϕ α k, S α = r Nα e itλ x i r n dλ ϕ α k e itλ α E k α + ϕ α k α finite k α S α q α,i (r, r (r (r + h(r. r k r r ϕ α k. k α rk r k q α,i (r, r (r (r + h(r rn dλ r k r k rnα itλ r r e t is then sufficient to prove the following estimates q α,i (6. e itλ (r, r (r (r + h(r rm dλ and for finite edges α (6.2 e itλ r k r k r r q α,i (r, r (r (r + h(r rm dλ q α,i (r, r (r (r + h(r rn dλ C + t C. + t We claim that it is sufficient to prove the following one: (6. e itλ r m (r (r + h(r dλ C. + t ndeed, q α,i (x, y are a finite number of polynomials, since α E- a finite set and we get the same constant for all of them. Hence inequality (6. clearly follows from (6.. n the case of (6.2, since k α, α is a finite edge and the graph has a finite number of edges, it follows that α K, where the constant K depends only on the graph, and so k K. By expanding r k r k r r = r k + r k + + r (k and using the fact that q α,i are polynomials, inequality (6.2 also follows from (6...

20 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS 9 Putting now the estimates (6.0, (6. and (6.2 together we obtain that [ ] e itλ x i r n dλ C ϕ α k + ϕ α C k ϕ + t l., α E k α α finite k α + t which proves the desired inequality. Let us now prove (6.. Since h is a rational function, it has an absolutely and uniformly convergent Laurent series on an annulus and this series becomes a fourier series if restricted to T. The function h does not vanish on the unit circle so, using Wiener s theorem (see [0], /h has an absolutely and uniformly convergent Fourier series h(r = a p r, p a p < A. p Z p Z This reduces the proof of (6. to the following estimate e itλ r m (r (r + dλ C. + t (m Z. We now write r = e iθ and hence λ = 2(cos θ + and (r (r + = i sin θ exp(iθ. By making a change of variable such that we integrate over θ [0, π], the last integral becomes π e 2it(cos θ e imθ dθ C (m Z. 0 + t This last estimate has been proved in [6] by using Van der Corput s Lemma. The proof if now complete. Acknowledgements. Both C.G. and the supervisor have been supported by the grant PN--TE 4/200 of the Romanian National Authority for Scientific Research, CNCS UEFSCD. L.. has been also partially supported from ERC Advanced Grant FP NUMERWAVES. Parts of this work have been done when both authors have visited the Basque Center for Applied Mathematics. The authors also acknowledge this institution and for the hospitality and support. References [] V. Banica. Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients. SAM J. Math. Anal., 5(4: (electronic, 200. [2] Valeria Banica and Liviu. gnat. Dispersion for the schrödinger equation on networks. J. Math. Phys., 52:0870, 20. [] T. Cazenave. Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics 0. Providence, R: American Mathematical Society (AMS; New York, NY: Courant nstitute of Mathematical Sciences. xiii, 200.

21 20 DSPERSON PROPERTY FOR DSCRETE SCHRÖDNGER EQUATONS ON NETWORKS [4] Nelson Dunford and Jacob T. Schwartz. Linear operators. Part : Spectral theory, self adjoint operators in Hilbert space. With the assistance of William G. Bade and Robert G. Bartle. Repr. of the orig., publ. 96 by John Wiley & Sons Ltd., Paperback ed. Wiley Classics Library. New York etc.: John Wiley & Sons Ltd./nterscience Publishers, nc. ix, $25.95, 988. [5] Liviu. gnat. Strichartz estimates for the Schrödinger equation on a tree and applications. SAM J. Math. Anal., 42(5: , 200. [6] Liviu. gnat and Diana Stan. Dispersive properties for discrete schrödinger equations. Journal of Fourier Analysis and Applications, 7(5:05 065, 20. [7] M. Keel and T. Tao. Endpoint Strichartz estimates. Am. J. Math., 20(5: , 998. [8] P. Kuchment. Quantum graphs: an introduction and a brief survey. n Analysis on graphs and its applications, volume 77 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Providence, R, [9] Alexander Mielke and Carsten Patz. Dispersive stability of infinite-dimensional Hamiltonian systems on lattices. Appl. Anal., 89(9:49 52, 200. [0] Walter Rudin. Functional analysis. 2nd ed. nternational Series in Pure and Applied Mathematics. New York, NY: McGraw-Hill. xviii, 424 p., 99. [] A. Stefanov and P.G. Kevrekidis. Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations. Nonlinearity, 8(4:84 857, [2] T. Tao. Nonlinear dispersive equations, volume 06 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, Local and global analysis. C. Gavrus Scoala Normala Superioara Bucuresti - at the nstitute of Mathematics Simion Stoilow of the Romanian Academy, 2 Calea Grivitei Street, Bucharest, Romania. address: gcd2006@gmail.com L.. gnat nstitute of Mathematics Simion Stoilow of the Romanian Academy, 2 Calea Grivitei Street, Bucharest, Romania and BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo, 4 E Bilbao, Basque Country - Spain. address: liviu.ignat@gmail.com Web page: lignat