We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms geometric parameter dependent upon number short closed geodesics surface. In particular, we show an L2 normalised eigenfunction restricted to measurable subset has squared L2-norm ? > 0, only if set relatively large size—exponential parameter. For random surfaces with respect Weil—Petersson probabi...

We consider a perturbed graph consisting of two infinite edges, loop, and glued arbitrary finite γε with small where is obtained by ε−1 times contraction some fixed ε parameter. On the graph, we Schrödinger operator whose potential on edges can singularly depend Kirchhoff condition at internal vertices Dirichlet or Neumann boundary vertices. For eigenvalue corresponding eigenfunction prove holo...

and γ = [ IN + ρH H ]−1 11 , where H is a M × N complex Gaussian matrix with independent entries and M ≥ N . These diagonal entries are related to the “signal to interference and noise ratio” (SINR) in multi-antenna communications. They depend not only on the eigenvalues but also on the corresponding eigenfunction weights, which we are able to evaluate on average constrained on the value of the...

We consider linear differential equations of the form (p(t)x′)′ + λq(t)x = 0 (p(t) > 0, q(t) > 0) (A) on an infinite interval [a,∞) and study the problem of finding those values of λ for which (A) has principal solutions x0(t;λ) vanishing at t = a. This problem may well be called a singular eigenvalue problem, since requiring x0(t;λ) to be a principal solution can be considered as a boundary co...

This work presents a new methodology for computing ground states of Bose–Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a preprocessing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties....

The equation ∆u + λu + g(λ, u)u = 0 is considered in a bounded domain in R2 with a Signorini condition on a straight part of the boundary and with mixed boundary conditions on the rest of the boundary. It is assumed that g(λ, 0) = 0 for λ ∈ R, λ is a bifurcation parameter. A given eigenvalue of the linearized equation with the same boundary conditions is considered. A smooth local bifurcation b...

If the adjacency matrix contains degenerate eigenvalues, we must modify the approach using non-degenerate eigenvalues. We denote the eigenvalues as λki, where the index k runs over different eigenvalues and the index i runs over M associated eigenvectors of the same eigenvalue. Note that there is no unique way of choosing a basis for the eigenvectors of the unperturbed network since any linear ...

We consider the following eigenvalue problem: −Δu f u λu, u u x , x ∈ B {x ∈ R3 : |x| < 1}, u 0 p > 0, u||x| 1 0, where p is an arbitrary fixed parameter and f is an odd smooth function. First, we prove that for each integer n ≥ 0 there exists a radially symmetric eigenfunction un which possesses precisely n zeros being regarded as a function of r |x| ∈ 0, 1 . For p > 0 sufficiently small, such...

and refer to [2] and [3] for more background information. Here, M is the wanted eigenvalue of the invariant mass squared operator, with associated eigenfunction ψ ≡ Ψqq̄. It is the probability amplitude for finding in the qq̄–space a quark with the effective (constituent) mass m1, longitudinal momentum fraction x, transversal momentum ~k⊥ and helicity λ1, and correspondingly for the anti– quark w...