In this paper we study a number of conjectures on the behavior value distribution eigenfunctions. On two dimensional torus, observe that symmetry conjecture holds in strongest possible sense. other hand, provide counterexample for higher tori, which relies computer-assisted argument. Moreover prove theorem certain class trigonometric polynomials might be independent interest.

Abstract Every continuous-time flow on a topological space has associated to it Koopman operator, which operates by time-shifts various spaces of functions, such as C r , L 2 or functions bounded variation. An eigenfunction the vector field (and thus for operator) can be viewed an S 1 -valued function, also plays role semiconjugacy rigid rotation . This notion eigenfunctions will generalized Li...

We derive the biorthogonality condition for axisymmetric Stokes flow in a region between two concentric spheres. This biorthogonality condition is a property satisfied by the eigenfunctions and adjoint eigenfunctions, which is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding creeping flow problem.

Using gauge transformations for the corresponding generating pseudo-differential operators L in terms of eigenfunctions and adjoint eigenfunctions, we construct several types of auto-Bäcklund transformations for the KP hierarchy with self-consistent sources (KPHSCS) and mKP hierarchy with self-consistent sources (mKPHSCS) respectively. The Bäcklund transformations from the KPHSCS to mKPHSCS are...

We describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application we find, under some additional hypotheses on the spectrum, subsequences of eigenfunctions of Šeba billiards that localise around a pair of unperturbed eigenfunctions.

We estimate the behavior of the generalized eigenfunctions of critical Dirac operators (which are Dirac operators with eigenfunctions and/or resonances for E = m) under small perturbations in the potential. The results also apply for other differential operators (for example Schrödinger operators).

T he knowledge of the eigenfunctions of a linear time-varying (LTV) system is a fundamental issue from both theoretical and applications points of view. Nonetheless, no analytic expressions are available for the eigenfunctions of a general LTV system. However, approximate expressions have been proposed for slowly-varying operators. In particular, in [1] it was shown that an underspread system c...

An atomic decomposition is considered in Banach space.  A method for constructing an atomic decomposition of Banach  space, starting with atomic decomposition of  subspaces  is presented. Some relations between them are established. The proposed method is used in the  study  of the  frame  properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

We consider an eigenvalue problem for a divergence form elliptic operator Aε with locally perturbed high contrast periodic coefficients. Periodicity size ε is a small parameter. Local perturbation of coefficients for such operator could result in emergence of localized waves eigenfunctions with corresponding eigenvalues lying in the gaps of the Floquet-Bloch spectrum. We prove that, for a doubl...

The vertical structure of the low-level atmospheric response to an elevated large-scale, low-frequency heat source in the Tropics is explored using linear tidal theory on an equatorial beta plane. Through the calculation of the projection of a large-scale, low-frequency thermal source onto the meridional eigenfunctions, the contributions from a set of discrete meridional eigenfunctions with pos...