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 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...

We investigate the relationship between complex symmetry of composition operators C ϕ f = ∘ induced on classical Hardy space H 2 ( D ) by an analytic self-map open unit disk and its Koenigs eigenfunction. A generalization orthogonality known as conjugate-orthogonality will play a key role in this work. show that if is Schröder map (fixes point ∈ with 0 < | ′ 1 σ eigenfunction, then symmetric on...

We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain.

For bounded smooth domains the number λ1 is the so-called principal eigenvalue. It has a unique eigenfunction, which is positive, and this eigenfunction is the only positive one (up to normalization). Two main references for this type of results, which are usually called maximum principles, are the books by Walter (1964) and by Protter and Weinberger (1967). Extensions to general bounded non-sm...

The classical eigenfunction method for the solution of contact problems involving wear is formulated in the context of the finite element method. Static reduction is used to reduce the full stiffness matrix to the N contact nodes, after which the assumption of a separated variable solution leads to a linear eigenvalue problem with N eigenvalues and eigenfunctions. A general solution to the tran...

We study an eigenvalue problem associated with a reaction-diffusionadvection equation of the KPP type in a cellular flow. We obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude A 1. It follows that the minimal pulsating traveling front speed c∗(A) satisfies the upper and lower bounds C1A ≤ c∗(A) ≤ C2A. Physically, the speed enhancement is related to the boun...

We are interested in a nonlinear boundary value problem for (|u′′|p−2u′′)′′ = λ|u|p−2u in [0,1], p > 1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to the nth eigenvalue, has precisely n− 1 zero points in (0,1). Eigenvalues of the ...

For most `nice' elliptic boundary value problems there is a general expectation that the rst eigenfunction is unique and of xed sign. And indeed, for second order elliptic di erential equations with Dirichlet boundary conditions such a result holds as a consequence of the maximum principle. It is well known that such a maximum principle does not have a direct generalization to higher order elli...