Koenigs problem and extreme fixed points

This note continues some previous studies by the authors. We consider a linear-fractional mapping $ F_A :K \to K $ F_A :K \to K generated by a triangular operator, where $ K $ K is the unit operator ball and the fixed point C of the extension of $ F_A $ F_A to $ \overline K $ \overline K is either an isometry or a coisometry. Under some natural restrictions on one of the diagonal entries of the operator matrix A, the structure of the other diagonal entry is investigated completely. It is shown that generally C cannot be replaced in all these considerations by an arbitrary point of the unit sphere. Some special cases are studied in which this is nevertheless possible.     

Essential Angular Derivatives and Maximum Growth of Koenigs Eigenfunctions

We introduce the notion ofessential angular derivativefor functionsφmapping the open unit diskUholomorphically into itself. After exploring some of its basic properties, we show how the essential angular derivative ofφdetermines the maximum growth rate of the Koenigs eigenfunctionσforφwhenφhas an attractive fixed point inU. Our work answers some questions about growth of Koenigs functions recently posed by Pietro Poggi-Corradini.     

Global structure and geodesics for Koenigs superintegrable systems

We present a new derivation of the local structure of Koenigs metrics using a framework laid down by Matveev and Shevchishin. All of these dynamical systems allow for a potential preserving their superintegrability (SI) and most of them are shown to be globally defined on either ?² or ?². Their geodesic flows are easily determined thanks to their quadratic integrals. Using Carter (or minimal) quantization, we show that the formal SI is preserved at the quantum level and for two metrics, for which all of the geodesics are closed, it is even possible to compute the classical action variables and the point spectrum of the quantum Hamiltonian.     

Concentration of symmetric eigenfunctions

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.     

Holomorphic extension of eigenfunctions

Let X = G/K be a Riemannian symmetric space of non-compact type. We prove a theorem of holomorphic extension for eigenfunctions of the Laplace–Beltrami operator on X, by techniques from the theory of partial differential equations.     

Embedding of eigenfunctions of the Johnson graph into eigenfunctions of the Hamming graph

Under study is the relationship between the eigenfunctions of the Johnson and Hamming graphs. An eigenfunction of a graph is an eigenvector of its adjacency matrix with some eigenvalue; moreover, an eigenfunction can be identically zero. We find a criterion for the embeddability of an eigenfunction of the Johnson graph J(n, w) with a given eigenvalue into a certain eigenfunction of the Hamming graph with a given eigenvalue.     

Topology of the zeros of eigenfunctions

Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 3, pp. 59–60, July–September, 1989.     

Fourier coefficients of restrictions of eigenfunctions

Let {e_j} be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g). Let H ? M be a submanifold and {ψk} be an orthonormal basis of Laplace eigenfunctions of H with the induced metric. We obtain joint asymptotics for the Fourier coefficients ■ of restrictions γHe_j of e_j to H. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum {(μ_k, λ_j)}_j,...     

Approximation of eigenfunctions in kernel-based spaces

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the “native” Hilbert space \(\mathcal {H}\) in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L ₂-orthonormal and orthogonal in \(\mathcal {H}\) (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of \(\mathcal {H}\). These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.     

Complex zeros of real ergodic eigenfunctions

We determine the limit distribution (as λ→∞) of complex zeros for holomorphic continuations φ_λ ^? to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold (M,g) with ergodic geodesic flow. If \(\{\phi_{j_{k}}\}\) is an ergodic sequence of eigenfunctions, we prove the weak limit formula \(\frac{1}{\lambda_j}[Z_{\phi_{j_k}}^{\mathbb{C}}]\ \to\ \frac{i}{\pi} \partial\bar{\partial} |\xi|_g\), where \([Z_{\phi_{j_k}^{\mathbb{C}}}]\) is the current of integration over the complex zeros and where \(\overline{\partial}\) is with respect to the adapted complex structure of Lempert-Szöke and Guillemin-Stenzel.     

Exponential decay of eigenfunctions and generalized eigenfunctions of a non-self-adjoint matrix Schrodinger operator related to NLS

We study the decay of eigenfunctions of the non-self-adjoint, for µ > 0, corresponding to eigenvalues in the strip -µ < Re E < µ.     

On eigenfunctions of nonlinear heat generation

Summary We consider the equation u+ expu=0, >0,u(boundary)0 in the formv= exp (K,v), whereK ^–1=–. We give bounds on for the latter equation to be solvable by the contraction mapping principle, and estimate theL ² norm of the solution so obtained. We also give a bound on for the topological index of the solution to be non-zero and apply Krasnoselskii's results to the least squares method of approximating the solution.

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Sommario Consideriamo l'equazione u+ expu=0, >0,u(frontiera)=0 nella formav= exp (Kv), doveK ^–1=–. In questo lavoro diamo limitazioni per per cui la seconda equazione e risolubile col metodo delle contrazioni, e diamo una stima della norma inL ² della soluzione cosi ottenuta. Diamo anche una limitazione per per cui l'indice topologico della soluzione diventa non zero, e applichiamo i risultati di Krasnoselskii al metodo dei minimi quadrati per approssimare la soluzione.

     

Gaussian decay of the magnetic eigenfunctions

We investigate whether the eigenfunctions of the two-dimensional magnetic Schrödinger operator have a Gaussian decay of type exp(–Cx ²) at infinity (the magnetic field is rotationally symmetric). We establish this decay if the energy (E) of the eigenfunction is below the bottom of the essential spectrum (B), and if the angular Fourier components of the external potential decay exponentially (real analyticity in the angle variable). We also demonstrate that almost the same decay is necessary. The behavior ofC in the strong field limit and in the small (B–E) limit is also studied.Partial support from the Hungarian National Foundation for Scientific Research, grant no. 1902.