Oscillation properties of the spectrum of a boundary value problem on a graph

Abstract

KEY WOADS: Sturm-Liouville problem, oscillation properties of the spectrum, boundary value problems on graphs.Consider the problem -(pu')' + qu = $ru (1) on an interval. In the present paper we show that for u(a) = 0 = u(b) the oscillation spectral Sturm properties can be generalized to the case in which the problem is essentially not one-dimensional. Specifically, Eq.(1) is no longer defined on the interval (a, b), but on a geometric graph (spatial network) r and the function u satisfies the conditions lar = 0,where aF is the set of boundary (dead-end) vertices (nodes) of F.w The boundary value problems on graphs (spatial networks) have been studied systematically only since the beginning of the eighties. In the pioneering papers [1-3] an equation of the form (1) given on a graph F was interpreted as a set (system) of ordinary scalar equations, each of which was defined on one of the edges Vi of I ~ . In this case, the conditions relating the solutions of the equations on adjacent edges were treated as boundary conditions. Usually, such matching conditions at each interior node a included the continuity conditions ui(a) = uj(a) (i # j)(here ui(" ) is the restriction of the solution to the edge 7/) and balance conditions of the formi where u~(a + O) denotes the derivative of ui(. ) along the edge 7i in the direction "from a." Quite a deep analogy with classical scalar problems was obtained by another approach [4,5], in which Eq. (1) was considered in the class of qealar-valued functions defined on the entire graph F and satisfying the natural condition of continuity at all interior nodes (that is, on the entire F); thus, conditions (3) are included in the definition of the solution. Then Eq. (1) considered on F becomes an intermediate object between a scalar equation on an interval and a second-order equation on a multidimensional domain; this allows us to treat the problem of the form (1), (2) defined on an interval as an analog of the Dirichlet problem.In what follows, we apply this scalar approach to problem (1), (2) on a graph (for more precise definitions,

The nodal count {0,1,2,3,…} implies the graph is a tree

“…This result for the interval is the famous Sturm oscillation theorem [1,2] and its generalization for trees was done in [43,44]. The fact that discrete tree graphs also have this nodal count is proved in [45][46][47].…”

“…Observe that Γ has infinitely many generic eigenvalues as a direct conclusion of lemma 4.1. If Γ is a tree graph then it was proved in [43,44] (see also appendix A in [45]) that the nodal counts of all generic eigenfunctions are φ n = n − 1 and ν n = n. Otherwise, if Γ has β > 0 cycles, assume by contradiction that there are only finitely many generic eigenfunctions with φ n = n − 1. In particular, this means that there is at least one generic eigenfunction for which φ n = n − 1, and thus σ n = 0.…”

Nodal Statistics on Quantum Graphs

“…since ∂F ∂α 2 = 0 at α 2 = 0 by (50). Simple examples of the block-diagonal structure of Hessian can be found in Appendices D.1 and D.2.…”

Indefinite boundary value problems on graphs

“…In parallel to the variational aspects of boundary-value problems on graphs studied here and on trees in [21], the work of Pokornyi and Pryadiev, and Pokornyi, Pryadiev and Al-Obeid, in [17] and [18], should be noted for the extension of Sturmian oscillation theory to second order operators on graphs. The idea of approximating the behaviour of eigenfunctions and eigenvalues for a boundary-value problem on a graph by the behaviour of associated problems on the individual edges, used here, was studied in the definite case in [2], [11] and [22].…”