Analysis and Mathematical Physics最新文献

We study the magnetic Laplacian in a two-dimensional exterior domain with Neumann boundary condition and uniform magnetic field. For the exterior of the disk we establish accurate asymptotics of the low-lying eigenvalues in the weak magnetic field limit. For the exterior of a star-shaped domain, we obtain an asymptotic upper bound on the lowest eigenvalue in the weak field limit, involving the (4)-moment, and optimal for the case of the disk. Moreover, we prove that, for moderate magnetic fields, the exterior of the disk is a local maximizer for the lowest eigenvalue under a (p)-moment constraint.

Properties of Riesz capacity are developed with respect to the kernel exponent (p in (-infty ,n)), namely that capacity is strictly monotonic as a function of p, that its endpoint limits recover the diameter and volume of the set, and that capacity is left-continuous with respect to p and is right-continuous provided (when (p ge 0)) that an additional hypothesis holds. Left and right continuity properties of the equilibrium measure are obtained too.

In the theory of (g_alpha )-potentials on a domain (Dsubset mathbb R^n), (ngeqslant 2), (g_alpha ) being the (alpha )-Green kernel associated with the (alpha )-Riesz kernel (|x-y|^{alpha -n}) of order (alpha in (0,n)), (alpha leqslant 2), we establish the existence and uniqueness of the (g_alpha )-balayage (mu ^F) of a positive Radon measure (mu ) onto a relatively closed set (Fsubset D), we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for (mu ^F(D)=mu (D)) to hold, given in terms of the (alpha )-harmonic measure of suitable Borel subsets of (overline{mathbb R^n}), the one-point compactification of (mathbb R^n). As a by-product, we find necessary and/or sufficient conditions for the existence of the (g_alpha )-equilibrium measure (gamma _F), (gamma _F) being understood in an extended sense where (gamma _F(D)) might be infinite. We also discover quite a surprising version of Deny’s principle of positivity of mass for (g_alpha )-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann Acad Sci Fenn Math 43:121–145, 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.

The Peetre K-functional is a key object in the development of the real method of interpolation. In this paper we point out a less known relation to wavelet theory and its applications to approximation theory and engineering applications. As a new basis for further development of these studies we present some known properties in the form appropriate for further applications and then derive new information and prove some new results concerning the K-functional and its close relation to (almost) quasi-monotone functions, various indices and interpolation theory. In particular, we extend and unify some known function parameter generalizations of the standard real interpolation spaces ((A_0, A_1)_{theta ,q}).

We study, for a continuous linear operator T acting on an F-space X, when the direct sum operator (Toplus T) is recurrent on the direct sum space (Xoplus X). In particular: we establish the analogous notion for recurrence to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each separable infinite-dimensional Banach space, solving the (Toplus T)-recurrence problem in the negative way.

We define the family of truncated Laguerre polynomials (P_n(x;z)), orthogonal with respect to the linear functional (varvec{ell }) defined by

The connection between (P_n(x;z)) and the polynomials (S_n(x;z)) (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials (P_n(x;z)) and (S_n(x;z)) are studied taking into account the relation between the parameters of the three-term recurrence relations that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear in a natural way. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameter z are given.

We show that the analytic content (lambda (cdot )) is neither subadditive nor semiadditive. To be precise, for compact sets K in the complex plane, (lambda (K)) is the K-uniform distance from the complex conjugation to the algebra of all rational functions with poles outside K. Thus, given any integer (nge 1), it is proven that each compactum K can be decomposed as the union of two new compact sets (E_1) and (E_2) with (lambda (E_j)le 1/n) for (j=1,2). Moreover, we also show that no compactum K with positive analytic content can be decomposed as the countable union of compact sets of zero analytic content.

Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, whose effective descriptions have involved generalised resolvents, we construct the functional model for a typical example of the latter. This provides a spectral representation for the generalised resolvent, which can be utilised for further analysis, in particular the construction of the scattering operator in related wave propagation setups.