Recent zbMATH articles in MSC 32Ahttps://www.zbmath.org/atom/cc/32A2021-04-16T16:22:00+00:00WerkzeugBasic properties of non-stationary Ruijsenaars functions.https://www.zbmath.org/1456.812102021-04-16T16:22:00+00:00"Langmann, Edwin"https://www.zbmath.org/authors/?q=ai:langmann.edwin"Noumi, Masatoshi"https://www.zbmath.org/authors/?q=ai:noumi.masatoshi"Shiraishi, Junichi"https://www.zbmath.org/authors/?q=ai:shiraishi.junichiSummary: For any variable number, a non-stationary Ruijsenaars function was recently introduced as a natural generalization of an explicitly known asymptotically free solution of the trigonometric Ruijsenaars model, and it was conjectured that this non-stationary Ruijsenaars function provides an explicit solution of the elliptic Ruijsenaars model. We present alternative series representations of the non-stationary Ruijsenaars functions, and we prove that these series converge. We also introduce novel difference operators called \({\mathcal T}\) which, as we prove in the trigonometric limit and conjecture in the general case, act diagonally on the non-stationary Ruijsenaars functions.Simple matrix models for random Bergman metrics.https://www.zbmath.org/1456.824992021-04-16T16:22:00+00:00"Ferrari, Frank"https://www.zbmath.org/authors/?q=ai:ferrari.frank"Klevtsov, Semyon"https://www.zbmath.org/authors/?q=ai:klevtsov.semyon"Zelditch, Steve"https://www.zbmath.org/authors/?q=ai:zelditch.steveMeromorphic solutions of generalized inviscid Burgers' equations and related PDEs.https://www.zbmath.org/1456.350092021-04-16T16:22:00+00:00"Lü, Feng"https://www.zbmath.org/authors/?q=ai:lu.fengSummary: The purposes of this paper are twofold. The first one is to describe entire solutions of certain type of PDEs in \(\mathbb{C}^n\) with the modified KdV-Burgers equation and modified Zakharov-Kuznetsov equation as the prototypes. The second one is to characterize entire and meromorphic solutions of generalized inviscid Burgers' equations in \(\mathbb{C}^2\).Toeplitz and asymptotic Toeplitz operators on \(H^2(\mathbb{D}^n)\).https://www.zbmath.org/1456.470082021-04-16T16:22:00+00:00"Maji, Amit"https://www.zbmath.org/authors/?q=ai:maji.amit"Sarkar, Jaydeb"https://www.zbmath.org/authors/?q=ai:sarkar.jaydeb"Sarkar, Srijan"https://www.zbmath.org/authors/?q=ai:sarkar.srijanSummary: We initiate a study of Toeplitz operators and asymptotic Toeplitz operators on the Hardy space \(H^2(\mathbb{D}^n)\) (over the unit polydisc \(\mathbb{D}^n\) in \(\mathbb{C}^n\)). Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let \(T_{z_i}\) denote the multiplication operator on \(H^2(\mathbb{D}^n)\) by the \(i\)-th coordinate function \(z_i\), \(i = 1, \dots, n\), and let \(T\) be a bounded linear operator on \(H^2(\mathbb{D}^n)\). Then the following hold: \begin{itemize}\item[(i)] \(T\) is a Toeplitz operator (that is, \(T = P_{H^2(\mathbb{D}^n)} M_\varphi |_{H^2(\mathbb{D}^n)}\), where \(M_\varphi\) is the Laurent operator on \(L^2(\mathbb{T}^n)\) for some \(\varphi \in L^\infty(\mathbb{T}^n)\)) if and only if \(T_{z_i}^\ast T T_{z_i} = T\) for all \(i = 1, \dots, n\).\item[(ii)] \(T\) is an asymptotic Toeplitz operator if and only if \(T = \text{Toeplitz} + \text{compact}\).
\end{itemize}
The case \(n = 1\) gives the well-known results of \textit{A. Brown} and \textit{P. R. Halmos} [J. Reine Angew. Math. 213, 89--102 (1963; Zbl 0116.32501)] and \textit{A. Feintuch} [Oper. Theory, Adv. Appl. 41, 241--254 (1989; Zbl 0676.47014)], respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.Multi-logarithmic differential forms on complete intersections.https://www.zbmath.org/1456.320052021-04-16T16:22:00+00:00"Aleksandrov, Alexandr G."https://www.zbmath.org/authors/?q=ai:aleksandrov.alexandr-g"Tsikh, Avgust K."https://www.zbmath.org/authors/?q=ai:tsikh.avgust-kSummary: We construct a complex \(\Omega_S^\bullet(\log C)\) of sheaves of multi-logarithmic differential forms on a complex analytic manifold \(S\) with respect to a reduced complete intersection \(C\subset S\), and define the residue map as a natural morphism from this complex onto the Barlet complex \(\omega_C^\bullet\) of regular meromorphic differential forms on \(C\). It follows then that sections of the Barlet complex can be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the residue map can be described explicitly in terms of certain integration current.A new approach towards estimating the \(n\)-dimensional Bohr radius.https://www.zbmath.org/1456.320012021-04-16T16:22:00+00:00"Bernal-González, Luis"https://www.zbmath.org/authors/?q=ai:bernal-gonzalez.luis"Cabana, Hernán J."https://www.zbmath.org/authors/?q=ai:cabana.hernan-j"García, Domingo"https://www.zbmath.org/authors/?q=ai:garcia.domingo"Maestre, Manuel"https://www.zbmath.org/authors/?q=ai:maestre.manuel"Muñoz-Fernández, Gustavo A."https://www.zbmath.org/authors/?q=ai:munoz-fernandez.gustavo-a"Seoane-Sepúlveda, Juan B."https://www.zbmath.org/authors/?q=ai:seoane-sepulveda.juan-benignoSummary: A new estimate for the Bohr radius of the family of holomorphic functions in the \(n\)-dimensional polydisk is provided. This estimate, obtained via a new approach, is sharper than those that are known up to date.Biduality in weighted spaces of analytic functions.https://www.zbmath.org/1456.460142021-04-16T16:22:00+00:00"Boyd, Christopher"https://www.zbmath.org/authors/?q=ai:boyd.christopher"Rueda, Pilar"https://www.zbmath.org/authors/?q=ai:rueda.pilarSummary: We study new conditions for non necessarily radial weights implying that the weighted Banach space \(\mathcal{H}_v(U)\) of analytic functions \(f\) such that \(vf\) is bounded on \(U\), is canonically isometrically isomorphic to the bidual of \(\mathcal{H}_{v_o}(U)\), its closed subspace formed by those functions \(f\) such that \(vf\) converges to \(0\) on the boundary of \(U\). We provide several examples of weights that satisfy these conditions. As an application, we show that whenever \(\mathcal{H}_v(U)=\mathcal{H}_{v_o}(U)''\) the norm-attaining functions are dense in \(\mathcal{H}_v(U)\).
For the entire collection see [Zbl 1444.15003].Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations.https://www.zbmath.org/1456.811652021-04-16T16:22:00+00:00"Aharonov, Yakir"https://www.zbmath.org/authors/?q=ai:aharonov.yakir"Behrndt, Jussi"https://www.zbmath.org/authors/?q=ai:behrndt.jussi"Colombo, Fabrizio"https://www.zbmath.org/authors/?q=ai:colombo.fabrizio"Schlosser, Peter"https://www.zbmath.org/authors/?q=ai:schlosser.peterSummary: In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the \(\delta\) and \(\delta^\prime\)-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.Spectral properties of Toeplitz operators on the unit ball and on the unit sphere.https://www.zbmath.org/1456.470062021-04-16T16:22:00+00:00"Akkar, Zineb"https://www.zbmath.org/authors/?q=ai:akkar.zineb"Albrecht, Ernst"https://www.zbmath.org/authors/?q=ai:albrecht.ernstSummary: In this article, we consider Toeplitz operators on the Hardy and weighted Bergman Hilbert spaces of the unit sphere, respectively on the unit ball in $\mathbb{C}^N$. Various aspects of the interplay between local and global properties of the symbols and local and global spectral properties of the corresponding Toeplitz operators are investigated. A local version of the spectral inclusion theorem of \textit{A. M. Davie} and \textit{N. P. Jewell} [J. Funct. Anal. 26, 356--368 (1977; Zbl 0374.47011)] is proved. Using some recent results of \textit{R. Quiroga-Barranco} and \textit{N. Vasilevski} [Integral Equations Oper. Theory 59, No. 3, 379--419 (2007; Zbl 1144.47024); ibid. 60, No. 1, 89--132 (2008; Zbl 1144.47025)], we describe some commutative $C^*$-subalgebras of the Toeplitz algebra for $N \geq2$. The method of \textit{G. McDonald} [Ill. J. Math. 23, 286--293 (1979; Zbl 0438.47031)] to compute the essential spectrum of Toeplitz operators with certain piecewise continuous symbols is extended to a larger class of symbols including examples where the surface measure of set of discontinuity points has strictly positive measure.
For the entire collection see [Zbl 1300.47008].A uniqueness theorem for meromorphic functions concerning total derivatives in several complex variables.https://www.zbmath.org/1456.320032021-04-16T16:22:00+00:00"Xu, Ling"https://www.zbmath.org/authors/?q=ai:xu.ling"Cao, Tingbin"https://www.zbmath.org/authors/?q=ai:cao.tingbinThe authors prove the following theorem: Let \(f\) and \(g\) be two non-constant meromorphic functions on \(\mathbb{C}^{m}\) and \(k\) be a positive integer such that \(f\) and \(g\)
share \(0\) CM, \(D^{k}f\) and \(D^{k}g\) share \(\infty\) and \(1\) CM. If \(2\delta(0; f) + (k + 4)\Theta (\infty; f) > k + 5\), then \(D^{k}f - 1= c(D^{k}g - 1)\), where \(c (\neq 0, \infty)\) is a
constant and \(D^{k}f\) represents the \(k^{\text th}\) order total derivative of \(f\).
Reviewer: Indrajit Lahiri (Kalyani)Cauchy's integral formula in the theory of analytic functions of several complex variables.https://www.zbmath.org/1456.320042021-04-16T16:22:00+00:00"Fuks, B. A."https://www.zbmath.org/authors/?q=ai:fuks.boris-abramovichThis article is a slightly modified survey lecture read in November 1946 at a meeting of the Moscow Mathematical Society.Effectiveness of Cannon and composite sets of polynomials of two complex variables in Faber regions.https://www.zbmath.org/1456.320022021-04-16T16:22:00+00:00"Adepoju, Jerome Ajayi"https://www.zbmath.org/authors/?q=ai:adepoju.jerome-ajayi"Mogbademu, Adesanmi Alao"https://www.zbmath.org/authors/?q=ai:mogbademu.adesanmi-alaoSummary: Conditions are obtained for effectiveness of Cannon and Composite sets of polynomials of two complex variables in Faber regions. It generalizes to these regions the results of Nassif on composite sets in balls of centre origin whose constituents are also cannon sets.