Complex Analysis and Operator Theory最新文献

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in Zalar (Linear Algebra Appl 649:186–239, 2022. https://doi.org/10.1016/j.laa.2022.05.008), while the degree 6 cases in Yoo (Integral Equ Oper Theory 88:45–63, 2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle (y(ay+x^2+y^2)=0, ain {mathbb {R}}{setminus } {0}), and a union of a line and a parabola (y(x-y^2)=0). In both cases we also determine the number of atoms in a minimal representing measure.

A new variant of Newton’s method - named Backtracking New Q-Newton’s method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions, with BNQN. In general, they look more smooth than that of Newton’s method. In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton’s flow and Voronoi’s diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton’s method and Random Relaxed Newton’s method.

Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The evolution of superoscillatory initial datum under the time dependent Schrödinger equation is stable in free space, but in general it is unclear whether it can be preserved in the presence of an external potential. In this paper, we consider the two-dimensional problem of superoscillations interacting with a half-plane barrier, where homogeneous Dirichlet or Neumann boundary conditions are imposed on the negative (x_2)-semiaxis. We use the Fresnel integral technique to write the wave function as an absolute convergent Green’s function integral. Moreover, we introduce the propagator of the Schrödinger equation in form of an infinite order differential operator, acting continuously on the function space of exponentially bounded entire functions. In particular, this operator allows to prove that the property of superoscillations is preserved in the form of a similar phenomenon called supershift, which is stable over time.

The purpose of this paper, is to study and investigate a larger class of operators than the Fredholm ones, is the so-called, generalized Fredholm operators, in a right quaternionic separable Hilbert space with a left multiplication defined on it. More explicitly, we first prove some auxiliary results , in the quaternionic setting, that are needed in the development of this paper. After that, we prove that some special classes of operators are generalized Fredholm ones, and we establish a characterization of the generalized Fredholm operators in a specific quotient algebra. The obtained results leads to the study of the invariance of the generalized Fredholm S-spectrum of a bounded right linear operator defined on a right quaternionic separable Hilbert space under a finite-rank commuting operator.

For (nu in mathbb {R}), we consider the invariant Laplacians (Delta _{nu }) in the unit complex ball ({mathcal {B}}^{n}=(SU(n,1)/S(U(n)times U(1)))

and the spectral projectors ({mathcal {Q}}_{lambda ,nu }) associated to (Delta _{nu }) defined by

where (varphi _{lambda ,nu }) is the (S(U(n)times U(1)))-invariant eigenfunction of (Delta _{nu }) and ({textbf{c}}_{nu }(lambda )) the Harish-Chandra function. The goal of this paper is to give an image characterization of ({mathcal {Q}}_{lambda ,nu }) of ({mathcal {C}}_{c}^{infty }({mathcal {B}}^{n})) and (L^{2}({mathcal {B}}^{n},(1-|z|^2)^{-n-1}dm(z))).

We study L-system realizations generated by the original Weyl–Titchmarsh functions (m_alpha (z)). In the case when the minimal symmetric Schrödinger operator is non-negative, we describe Schrödinger L-systems that realize inverse Stieltjes functions ((-m_alpha (z))). This approach allows to derive a necessary and sufficient conditions for the functions ((-m_alpha (z))) to be inverse Stieltjes. In particular, the criteria when ((-m_infty (z))) is an inverse Stieltjes function is provided. Moreover, it is shown that the knowledge of the value (m_infty (-0)) and parameter (alpha ) allows us to describe the geometric structure of the L-system realizing ((-m_alpha (z))). Additionally, we present the conditions in terms of the parameter (alpha ) when the main and associated operators of a realizing ((-m_alpha (z))) L-system have the same or different angle of sectoriality which sets connections with the Kato problem on sectorial extensions of sectorial forms. An example that illustrates the obtained results is presented in the end of the paper.

We study the inverse spectral problems of recovering Dirac-type functional-differential operator with two constant delays (a_1) and (a_2) not less than one-third of the length the interval. It has been proved that the operator can be recovered uniquely from four spectra when (2a_1+frac{a_2}{2}) is not less than the length of the interval, while it is not possible otherwise.