1 / 40

“Real problems”

4. 1. 9. 7. 5. 6. 8. 13. 10. 2. 3. 12. 11. “Real problems”. Nodal patterns on graphs. 1.The Laplace operator on metric graphs ( “ Quantum Graphs ” ) 2.The Laplace operqtor on discrete graphs. “ Quantum graphs ”. “Discrete graphs”. The Schroedinger operator on graphs:

jenette-mclaughlin
Download Presentation

“Real problems”

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4 1 9 7 5 6 8 13 10 2 3 12 11 “Real problems” Nodal patterns on graphs 1.The Laplace operator on metric graphs (“Quantum Graphs”) 2.The Laplace operqtor on discrete graphs. “Quantum graphs” “Discrete graphs”

  2. The Schroedinger operator on graphs: 1. The wave function 2. The wave equation On each bond: 3. The boundary conditions at the vertices i = 1,…V Neumann : . Dirichlet : The Schroedinger operator is self – adjoint with real and non-negative discrete spectrum

  3. The 1stsurprising numerical evidence (Tsampikos Kottos, U.S., (1997)) = cumulative level-spacing distribution

  4. A spectral trace formula (J. P. Roth, T. Kottos and U.S)

  5. 1.

  6. Proof (idea) : 1. Traceformulae and the length spectrum (Roth, Kottos & US) Schroedinger spectrum: {kn2} 2. Identify all “1 bond” orbits: Q : (Q is constructed as the minimal basis for the length spectrum) A finite search! {Lb}, b=1,B Non commensurability of the bond lengths! 3. Construct all metric stars: Thus, the lengths and the connectivity matrix are uniquely recovered

  7. Conclusion: isospectral but not isometric graphs must have commensurate bond lengths! Example: (Michail Solomyak)

  8. Sunada isospectral quantum graphs Principle: building blocks, transpositions 73 b c Central vertex not an isometry Boundary vertex

  9. Counting nodal domains on graphs But how do we count ? Metric count ( = 8) Schapotschnikow (2006): On tree graph, the nth eigenfunction has n nodal domains Berkolaiko (2007): For a graph of rank r, the number of nodal domains of the n’th eigenfunction is bounded in the interval Discrete count ( = 1)

  10. Resolving isospectrality Isospectral tree graphs – Because of the tree nature of the isospectral graphs, metric count does not resolve isospectrality. Discrete count does resolveisospectrality for some isospectralpairs. Numerical evidence.(Shapira & Smilansky 2005) The length spectrum from the nodal sequence (numerical)

  11. a 2c 2b N D 2b b b D N a a 2a c c 2c N D Constructing isospectral pair using the Dihedral D4 symmetry Rami Band {D. Jakobson, M. Levitin, N. Nadirashvili and I.Pelterovich}, Martin Sieber

  12. Discrete nodal count N D 2b b b D N a a 2a c c 2c N D • Theorem 1 – Let GI and GII the graphs below. Denote with {in} the sequence of discrete nodal count of the graph Gi. Then {In} is different from {IIn} for half of the spectrum. GII GI

  13. Discrete nodal count GI N N N N D D plus minus N D N N N D D D D N D N • Obsevation – The transplantation that takes an eigenfunction of GI with eigenvalue k and transforms it to eigenfunction of GII with eigenvalue k is: GI GII

  14. Discrete nodal count I=1 I=1 I=2 I=2 II=1 II=2 N D N D I=2 I=2 I=1 I=1 II=2 II=1 GI D N GII • Geometrical representation of the transplantation of a certain eigenfunction

  15. Discrete nodal count • Calculate h(x) - the distribution function of

  16. Discrete nodal count • Calculate h(x) - the distribution function of Ergodic principle {In} is different from {IIn} for half of the spectrum.

  17. Metric nodal count N D 2b b b D N a a 2a c c 2c N D • Theorem 2 – Let GI and GII the graphs below. Denote with {in} the sequence of metric nodal count of the graph Gi. Then { In} is different from { IIn} for half of the spectrum. GII GI

  18. Metric nodal count N D N D GI D N GII • Presenting the formula for the number of metric nodal domains (Gnutzmann, Smilansky & Weber) • Number of nodal points on the bond (i,j) is given by ( or if ϕi=0 or ϕj=0 then) • Number of metric nodal domains:

  19. Metric nodal count N D N D GI D N GII • Recall that on tree graph, the nth eigenfunction has n nodal domains (Schapotschnikow 2006). So that: • Nodal count difference is: • Again, we use the ergodic principle to calculate the distribution function h(x) of  In- IIn : { In} is different from { IIn} for half of the spectrum.

  20. Discrete Laplacians Graphs: G: Connected, finite, simple graph with V vertices and B bonds Connectivity : Ci,j vi= valency = jCi,j Ci,j 2 {0,1} Discrete Laplacian Li,j = - Ci,j + vii,j Secular equation: ZV() = det (IV – L) = 0 The spectrum is non-negative, 0 in the spectrum Task: Generate random ensembles which have a given connectivity and study - Spectral statistics - Eigenvectors statistics - Conditions for the existence of limit distributions when V !1 Definition : A regular graph G(V,v) is a connected graph with V vertices all having the same valency vi = v.

  21. Numerics: Spectral density: (McKay) Nearest level distribution

  22. Nodal domains: Let  =(1, 2, … , V) be an eignevector of the graph Laplacian with eigenvalue . The number of nodal domains  () : # of maximally connected subgraphs on which the i have the same sign Courant (1922) : Arrange the eigenvectors by increasing values of the corresponding eigenvalues. Let n be the number of nodal domains of the n’th eigenvector. Then: n n Nodal sequence:1,2, … , V = 3

  23. Interior vertex 2. Morphology of nodal domains

  24. Example: rectangular grid ¸ = v ¸ < v ¸ > v

  25. Eigenvectors correlations G(4000,3) v = 3-regular graph The “tree approximation” is expected to be exact for k <1/2 log V TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA

  26. Counting nodal domains Courant bound Assuming the tree approximation and Guassian distribution V=3 V=6

  27. 3. Nodal domains vs Percolation on random v-regular graphs We consider random regular G(V,v) graphs in the limit

  28. Percolation on Graphs (Erdos and Renyi, Alon et. al., Nachmias and Peres) Start with a random v-regular graph on V vertices. Perform an independent p-bond percolation : Retain a bond with a probability p Delete a bond with a probability 1- p Phase transition at pc = 1/(v-1): Denote by L the number of vertices in the largest connected component pc p < pc p > pc 1/V1/3 E(L) ~o(V2/3) E(L) ~ V2/3 E(L)~o(V2/3) V1/3 Bond percolation ~ Vertex percolation

  29. 4 1 9 7 5 6 8 13 10 3 2 12 11 Phase transition in the eigenstates of the discrete Laplacian ? For a given eigenvalue, and we apply the following procedure: We enumerate the vertices according to the decreasing value of the eigenstate at the vertex.1st vertex : Maximum value Vth vertex : Minimum value

  30. Next, we delete vertices which have an index larger than (and all the bonds which are attached to them). We denote this graph by 4 1 5 3 2

  31. Compute the ratio between the cardinality of the largest connected component of the sub-graph and the cardinality of . ( ) 4 1 5 3 2

  32. phase transition ?

  33. scaling

  34. Critical percolation vs eigenvalue for 3 d-regular graphs. Numerical and Mean field results (Yehonatan Elon)

  35. nodal counting and isospectrality Given an arbitrary function on the vertices (not necessarily and eigenfunction of a Laplacian) How does the connectivity of the graph limit the number of nodal domains? How to count? Idan Oren: Theorem: Denote by max the maximum number of nodal domains on a graph, and by  the chromatic number of the graph. Then the following (optimal) bound holds: Optimal bound: for a tree graph  = 2 and max = V for a complete graph  = V and max= 2

  36. Explicit formulae for counting nodal domains (Idan Oren and Rami band) Given a graph G and a function fi on the vertices. Denote si= Sign [ fi ]  (f) = the number of sign flips on a graph = # { ( i, j ) | si sj < 0 }. (1):  (f) = ¼ (s,Ls) where L is the graph Laplacian. Proof: (s,Ls) =  (si - sj)2 Ci,j A flip contributes 4 to the sum (2): (f) = ¼ (s,Ls) – (r - c) +1; where r = B - V+1 is the number of independent cycles in G (The rank of the fundamental group of G) c = the number of independent cycles in G with the same si Proof: For a tree (f) =  (f) +1 and r = c =0. Continue by induction on r. (3): Construct an auxiliary graph with connectivity (f) = the degeneracy of the lowest eigenvalue for the corresponding Laplacian Efficient for simulations (using SVD).

  37. Corollary : Let G and H be an isospectral pair of graphs, G bipartite and H not, Then, the nodal counts for the eigenfunctions with the largest eigenvalue are different, G > H.

  38. Conclusions Nodal domains – counting and morphology A surprising junction of many fields in Math. Phys. Open Problems: For non-separable systems (Graphs incl): How to count nodal domains? Trace formula for nodal domains. Counting nodal intersections. Resolution of isospectral domains Nodal domains in d (>2) dimensions Thank you for your attention

More Related