Introduction and basic concepts BVP on weighted paths Bibliography Boundary value problems on a weighted path Angeles Carmona, Andr´esM. Encinas and Silvia Gago Depart. Matem`aticaAplicada 3, UPC, Barcelona, SPAIN Midsummer Combinatorial Workshop XIX Prague, July 29th - August 3rd, 2013 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts BVP on weighted paths Bibliography Outline of the talk Notations and definitions Weighted graphs and matrices Schr¨odingerequations Boundary value problems on weighted graphs Green matrix of the BVP Boundary Value Problems on paths Paths with constant potential Orthogonal polynomials Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Two-side Boundary Value Problems in weighted paths MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Weighted graphs A weighted graphΓ=( V ; E; c) is composed by: V is a set of elements called vertices. E is a set of elements called edges. c : V × V −! [0; 1) is an application named conductance associated to the edges. u, v are adjacent, u ∼ v iff c(u; v) = cuv 6= 0. X The degree of a vertex u is du = cuv . v2V c34 u4 u1 c12 u2 c23 u3 c45 c35 c27 u5 c56 u7 c67 u6 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Matrices associated with graphs Definition The weighted Laplacian matrix of a weighted graph Γ is defined as di if i = j; (L)ij = −cij if i 6= j: c34 u4 u c u c u 1 12 2 23 3 0 d1 −c12 0 0 0 0 0 1 c B −c12 d2 −c23 0 0 0 −c27 C 45 B C c B 0 −c23 d3 −c34 −c35 0 0 C 35 B C c27 L = B 0 0 −c34 d4 −c45 0 0 C B C u5 B 0 0 −c35 −c45 d5 −c56 0 C c56 @ 0 0 0 0 −c56 d6 −c67 A 0 −c27 0 0 0 −c67 d7 u7 c67 u6 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Matrices associated with graphs Now consider a weighted graph with weighted vertices q4 c34 u4 u1 c12 u2 c23 u3 c c45 q q 35 q1 2 3 c27 c56 u5 u7 c67 u6 q5 q7 q6 Definition A Schr¨odingermatrix LQ on Γ with potential Q is defined as the generalization of the weighted Laplacian matrix, that is: Lq = L + Q; where Q = diag[q0;:::; qn+1] is called the potential matrix. MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Schr¨odingerequations X We call the Homogeneous Schr¨odingerequation on F to [HSE] Lqu = 0 n+2 X The Wronskian of x and y 2 R is: (w[x; y])k = xk yk+1 − xk+1yk ; for k = 0;:::; n (w[x; y])n+1 = (w[x; y])n: X Two solutions x and y of the HSE are linearly independent iff (w[x; y])k 6= 0; k = 0;:::; n + 1 X It is well-known that the product ckk+1(w[x; y])k = c for any k = 1;:::; n + 1 is constant iff Lq is a symmetric matrix. MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Definition of BVP X Considering a subset of vertices F ⊂ V (Γ), its boundary is defined as δ(F ) = fu 2 V (Γ) : u ∼ v; v 2 F g; and its inner boundary is defined as @(F ) = δ(F ) [ δ(F c ). Example: q4 c34 u4 u1 c12 u2 c23 u3 c45 c35 F = fu ; u ; u ; u ; u g q2 q3 2 3 5 6 7 q1 c27 u5 c56 δ(F ) = fu1; u4g u7 c67 u6 q5 q7 q6 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Definition of BVP X Considering a subset of vertices F ⊂ V (Γ), its boundary is defined as δ(F ) = fu 2 V (Γ) : u ∼ v; v 2 F g; and its inner boundary is defined as @(F ) = δ(F ) [ δ(F c ). Example: q4 c34 u4 u1 c12 u2 c23 u3 F = fu2; u3; u5; u6; u7g c45 c35 q2 q3 q1 c27 δ(F ) = fu1; u4g u5 c56 u c u q 7 67 6 5 @(F ) = fu1; u2; u4; u3; u5g q7 q6 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Definition of BVP Definition A boundary value problem on F consists in finding u 2 Rn+2 such that Lqu = f on F ; B1u = g1;:::; Bpu = gp; n+2 P for a given f 2 R and Bi u = j2@(F ) bij uj , gi 2 R, i = 1;:::; p. Examples of application: Chip-firing games [Chung, Lov´asz]: Lf = ci − ce on F . Hitting-time [Markov chains]: LHk = δj on V − fkg. MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Open questions: Which kind of Schr¨odingerequations can we solve? Which kind of boundary value problems can we solve? In which kind of graphs can we solve them? MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Paths with constant potential Introduction and basic concepts Orthogonal Polynomials BVP on weighted paths Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Bibliography Two-side boundary value problems in weighted paths Paths with constant potential In a preliminar work E. Bendito, A. Carmona, A.M. Encinas, Eigenvalues, Eigenfunctions and Green's Functions on a Path via Chebyshev Polynomials, Appl. Anal. Discrete Math., 3, (2009), 182-302, study BVP in a path Pn+2 with constant potential 2q − 2 u0 1 u1 1 u2 1 u3 un 1 un+1 2q − 2 2q − 2 2q − 2 2q − 2 2q − 2 2q − 2 0 2q −1 0 ::: 0 1 F = fu ;:::; u g 1 n B −1 2q −1 ::: 0 C B C δ(F ) = fu0; un+1g B 0 −1 2q ::: 0 C Lq = B C B . .. C @(F ) = fu0; u1; un; un+1g @ . 0 A 0 0 ::: 2q A linear boundary condition of F in the path is given by n+2 Bi u = ci;1u0 + ci;1u1 + ci;nun + ci;n+1un+1; for any u 2 R : MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Paths with constant potential Introduction and basic concepts Orthogonal Polynomials BVP on weighted paths Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Bibliography Two-side boundary value problems in weighted paths Paths with constant potential So the BVP with two-side conditions is given by: 8 < Lqu = f ; c10u0 + c11u1 + c1nun + c1n+1un+1 = g1; : c20u0 + c21u1 + c2nun + c2n+1un+1 = g2: A base of independent solutions of the HSE is fu; vg, where xk = Uk−1(q), yk = Uk−2(q), 1 ≤ k ≤ n. They obtain the solution of the two-side BVP problem in terms of linear combinations of the second-order Chebyshev polynomials fUk g. MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Paths with constant potential Introduction and basic concepts Orthogonal Polynomials BVP on weighted paths Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Bibliography Two-side boundary value problems in weighted paths BVP in weighted paths GOAL: To generalize this result for a path with non-constant potential MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Paths with constant potential Introduction and basic concepts Orthogonal Polynomials BVP on weighted paths Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Bibliography Two-side boundary value problems in weighted paths Families of Orthogonal Polynomials 1 1 Given fAngn=0 a real positive sequence and fBngn=0 a real sequence 1 of numbers, consider fRn(x)gn=0 a sequence of real orthogonal polynomials satisfying the recurrence relation Rn(x) = (Anx + Bn)Rn−1(x) − CnRn−2(x); n ≥ 2. (1) with Cn = An=An−1. Choosing a pair of initial polynomials R0(x) and R1(x) we obtain a family of orthogonal polynomials satisfying the recurrence relation. If we consider the two families such that: 1 First kind OP: fPngn=0 with P0(x) = 1, P1(x) = ax + b, 1 A0+A1 Second kind OP: fQng , with Q0(x) = 1, Q1(x) = P1(x). n=0 A0 they verify that P−1(x) = P1(x) and Q−1(x) = 0. MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Paths with constant potential Introduction and basic concepts Orthogonal Polynomials BVP on weighted paths Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Bibliography Two-side boundary value problems in weighted paths Schr¨odingerequations on weighted Paths Now consider the weighted path A0 A0 A0 A0 u0 A1 u1 A2 u2 A3 u3 un An+1 un+1 q0 q1 q2 q3 qn qn+1 A0(Ak+1x+Bk+1) A0 With qk = − , k = 1;:::; n and Schr¨odingermatrix: Ak+1 Ak 0 1 A0(A1x+B1) − 1 − A0 0 ::: 0 A1 A1 B − A0 A0(A2x+B2) − A0 ::: 0 C B A1 A2 A2 C B A A0(A3x+B3) A C L = B 0 − 0 ::: − 0 C q B A2 A3 An+1 C B .