A lower bound for nodal count on discrete and metric graphs

According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturm's result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.Comment: 15 pages, 4 figures; Minor corrections: added 2 important reference

On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $\nu_n$ of the $n$-th eigenfunction satisfies $n\ge \nu_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-\nu_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure

Dynamics of nodal points and the nodal count on a family of quantum graphs

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr\"odinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.Comment: 34 pages, 12 figure

Динамика насаждений сосны крымской (Pinus pallasiana L. ) в горном Крыму

За период 1938 – 2000 гг. произошло "смещение" мест произрастания сосновых древостоев в более богатые и влажные условия. Увеличилась площадь насаждений сосны крымской. Площадь сосновых культур больше чем в 3 раза превысила площадь естественных лесов. Средний запас сосновых лесов составляет 136 м³/га. Древостои ІІ и высших классов бонитета занимают лишь 12 % сосновых лесов.За період 1938 – 2000 рр. відбувся "зсув" місць виростання соснових деревостанів у багатші й вологіші умови. Збільшилася площа насаджень сосни кримської. Площа соснових культур у понад 3 рази перевершила площу природних лісів. Середній запас соснових лісів становить 136 м³/га. Деревостани ІІ і вищих класів бонітету займають лише 12 % соснових лісів.For 1938 – 2000 "displacement" of pine stands to more rich and moist sites has occurred. Area of P. pallasiana has increased. Pine plantation area has exceeded area of natural pine forests more than 3 times. Mean stock of pine forests is 136 m³/ha. Stands of the ІІ and higher growth classes take only 12 % of pine forests

Understanding interactions between capped nanocrystals: Three-body and chain packing effects

Self-assembly of capped nanocrystals (NC) attracted a lot of attention over the past decade. Despite progress in manufacturing of NC superstructures, the current understanding of their mechanical and thermodynamic stability is still limited. For further applications, it is crucial to find the origin and the magnitude of the interactions that keep self-assembled NCs together, and it is desirable to find a way to rationally manipulate these interactions. We report on molecular simulations of interacting gold NCs protected by capping molecules. We computed the potential of mean force for pairs and triplets of NCs of different size (1.8–3.7 nm) with varying ligand length (ethanethiol-dodecanethiol) in vacuum. Pair interactions are strongly attractive due to attractive van der Waals interactions between ligand molecules. Three-body interaction results in an energy penalty when the capping layers overlap pairwise. This effect contributes up to 20% to the total energy for short ligands. For longer ligands, the three-body effects are so large that formation of NC chains becomes energetically more favorable than close packing of capped NCs at low concentrations, in line with experimental observations. To explain the equilibrium distance for two or more NCs, the overlap cone model is introduced. This model is based on relatively simple ligand packing arguments. In particular, it can correctly explain why the equilibrium distance for a pair of capped NCs is always ?1.25 times the core diameter independently on the ligand length, as found in our previous work [Schapotschnikow, R. Pool, and T. J. H. Vlugt, Nano Lett. 8, 2930 (2008)]. We make predictions for which ligands capped NCs self-assemble into highly stable three-dimensional structures, and for which they form high-quality monolayers.Process and EnergyMechanical, Maritime and Materials Engineerin