A bstract . We study in one dimension the semi-classical limit of the exact eigenfunction Ψh E(N,h) of the Hamiltonian H = − 1 2 h2∆ + V (x), for a potential V being analytic, bounded below and lim|x|→∞ V (x) = +∞. The main result of this paper is that for any given E > minx∈R1 V (x) with two turning points, the exact L2 normalized eigenfunction |Ψh E(N,h) (q)|2 converges to the classical proba...

We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue lambda1 depends on system size N as lambda1 approximately N1/4 for l...

Abstract Let $Z$ be a unimodular real spherical space. We develop theory of constant terms for tempered functions on $Z$, which parallels the work Harish-Chandra. The $f_I$ an eigenfunction $f$ are parametrized by subsets $I$ set $S$ roots that determine fine geometry at infinity. Constant transitive i.e., $(f_J)_I=f_I$ $I\subset J$, and our main result is quantitative bound difference $f-f_I$,...

We study the covariant derivatives of an eigenfunction for Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. show that along every parallel tensor, derivative is scalar multiple eigenfunction. also polynomial depending eigenvalue and prove some properties. A conjecture motivated by vertex algebraic structure space forms announced, ...

We investigate an initial-boundary value problem for a time-fractional subdiffusion equation with the Caputo derivatives on $N$-dimensional torus by classical Fourier method. Since our solution is established eigenfunction expansion of elliptic operator, method proposed in this article can be used to arbitrary domain and operator variable coefficients. It should noted that conditions existence ...

We present an improved version of Berry’s ansatz able to incorporate exactly the existence of boundaries and the correct normalization of the eigenfunction into an ensemble of random waves. We then reformulate the Random Wave conjecture showing that in its new version it is a statement about the universal nature of eigenfunction fluctuations in systems with chaotic classical dynamics. The emerg...