Lower Half-plane Research Articles

THE BEREZIN TRANSFORMATION ON L 2 (U+)

Let L 2 a (U+) be the Bergman space of the upper half plane U+. In this paper, we consider the integral operator H from L 2 (U+) into L 2 (U+) defined by (Hf)(w) = fe(w) = Z U+ f(s)|dw(s)| 2 dAe(s), w ∈ U+, where dw(s) = 1 √ π w + i w − i (−2i)Im w (s + w) 2 and dAe is the area measure on U+. We refer the map H as the Berezin transformation defined on L 2 (U+). We have derived various algebraic properties of the operator and showed that ||H|| ≤ 3π 4 considered as an operator on L 2 a (U+).

Simple vertex algebras arising from congruence subgroups

For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.

Duality, BMO and Hankel operators on Bernstein spaces

In this paper we deal with the problem of describing the dual space (Bκ1)⁎ of the Bernstein space Bκ1, that is the space of entire functions of exponential type (at most) κ>0 whose restriction to the real line is Lebesgue integrable. We provide several characterizations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type κ whose restriction to the real line are in a suitable BMO-type space, or as the space of symbols b for which the Hankel operator Hb is bounded on the Paley–Wiener space Bκ/22. We also provide a characterization of (Bκ1)⁎ as the BMO space w.r.t. the Clark measures of the inner function ei2κz on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant 1-spaces on the torus. Furthermore, we show that the orthogonal projection Pκ:L2(R)→Bκ2 induces a bounded operator from L∞(R) onto (Bκ1)⁎.Finally, we show that Bκ1 is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator Hb on the Paley–Wiener space Bκ/22 is compact.

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front.

Generalisation of the Asai-Kaneko-Ninomiya identity to higher level

Let j be the classical modular invariant, which is a modular function of weight zero for SL2(Z). Then for every z,τ in the upper half-plane, the Asai-Kaneko-Ninomiya (AKN) identity relates j(z)−j(τ) to the generating function of the Hecke system jm(τ) of modular functions for SL2(Z). In [7] the modular functions jm(z) were generalised to modular functions JN,m for higher level using harmonic weak Maass forms. In this paper, we show that under certain conditions on JN,m, the AKN identity can be generalised to higher level Γ0(N) using Poincaré series of weight 2 and level N. We also prove an infinite product formula for JN,1 which generalises the denominator formula for the Monster Lie algebra.

Brown-Halmos type theorems for Toeplitz operators on the Bergman space of the upper half-plane

We investigate Brown-Halmos type theorems which consist in characterizing when the product of two Bergman space Toeplitz operators is a Toeplitz operator whose symbol is the product of their symbols. The main involved technique is the characterization of functions that are in the range of the Berezin transform, and those which have finite rank Berezin symbols. We establish the upper half-plane analogues of the results of P. Ahern and Ž. Čučković (2001), P. Ahern (2004), and N.V. Rao (2018). In particular, we consider two classes of Toeplitz operators, namely those with bounded harmonic symbols and those with bi-analytic or tri-analytic symbols. Due to the unbounded character of the domain, many challenging difficulties occur, requiring more careful analysis.

Quasinormal modes and ringdown waveforms of a Frolov black hole

In this paper we investigate scalar perturbation over a Frolov black hole (BH), which is a regular BH induced by the quantum gravity effect. The quasinormal frequencies of a scalar field always consistently reside in the lower half-plane, and the time-domain evolution of the field demonstrates a decaying behavior, with the late-time tail exhibiting a power-law pattern. These observations collectively suggest the stability of a Frolov BH against scalar perturbation. Additionally, our study reveals that the quantum gravity effect leads to slower decay modes. For the case of the angular quantum number l = 0, the oscillation exhibits non-monotonic behavior with the quantum gravity parameter α 0. However, once l ≥ 1, the angular quantum number surpasses the influence of the quantum gravity effect.

Decomposition of the Unitary Representation of $SL_{2}(\mathbb{R})$ on the Upper Half Plane into Irreducible Components

The main purpose of the paper is to find the inversion formula for the covariant transform. This formula is equivalent to the decomposition of the unitary representation of $SL_{2}(\mathbb{R})$ on the upper half plane into irreducible components. We consider an eigenvalue $1+s^{2}$ of the Casimir operator: $$d\rho_{k}(C)=-4v^{2}\left(\partial_{u}^{2}+\partial_{v}^{2}\right),\quad \text{where}\, k=0.$$ To find the inversion formula, first we study the representation of $SL_{2}(\mathbb{R})$, $\rho_{k}$ and $\rho_{\tau}$, induced from the complex characters of $K$ and $N$ respectively. Then, we find the induced covariant transform $\mathcal{W}_{\varphi_{0}}^{\rho_{k}}$ with $N$-eigenvector to obtain a transform in the space $\FSpace{L}{2}(SL_{2}(\mathbb{R}/N)$. Thereafter, we compute the contravariant transform with $K$-eigenvector $$\mathcal{M}_{\phi_{0}}^{\rho_{\tau}}:\FSpace{L}{2}(SL_{2}(\mathbb{R})/N)\rightarrow \FSpace{L}{2}(SL_{2}(\mathbb{R})/K).$$

Zero dissipation limit of the anisotropic Boussinesq equations with Navier-slip and Neumann boundary conditions

In this paper, we study the vanishing dissipation limit of the 2D anisotropic Boussinesq equations with the Navier-slip boundary condition for velocity field and the fixed flux boundary condition for temperature in the upper half plane. By constructing boundary layer correctors to compensate for the discrepancies between dissipative equations and non-dissipative equations at the boundary, we prove that the solutions of the anisotropic Boussinesq equations converge to the solutions of the non-dissipative Boussinesq equations in L2-norm. Particularly, we find that the anisotropic dissipation coefficients only affect the rate of convergence, which is different from the phenomenon of the Dirichlet problem of the anisotropic Boussinesq equations in Wang & Xu (2021).

Renormalization of translated cone exchange transformations

In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a geometric construction for the first return map to a neighbourhood of the vertex of the middle cone for a large class of parameters, then we show a recurrence in the first return map tied to Diophantine properties of the parameters, and subsequently prove the infinite renormalizability of the first return map for these parameters.

Dirichlet Problem for Nonlinear Higher-Order Equations in Upper Half Plane

In this article, we consider the Dirichlet problem for nonlinear higher-order equations in upper half plane. Firstly we introduce the solutions of inhomogeneous polyanalytic equation in upper half plane. Then we investigate the properties of relevant integral operators. Lastly we transform the Dirichlet problem for nonlinear higher-order equations in upper half plane into the system of integro-differential equations and we obtain the existence of unique solution using Banach fixed point theorem.

A hyperbolic Kac-Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.

On approximation of functions from Hölder classes by biharmonic Poisson integrals defined in the upper half-plane

On approximation of functions from Hölder classes by biharmonic Poisson integrals defined in the upper half-plane

On pole-skipping with gauge-invariant variables in holographic axion theories

We study the pole-skipping phenomenon within holographic axion theories, a common framework for studying strongly coupled systems with chemical potential (μ) and momentum relaxation (β). Considering the backreaction characterized by μ and β, we encounter coupled equations of motion for the metric, gauge, and axion field, which are classified into spin-0, spin-1, and spin-2 channels. Employing gauge-invariant variables, we systematically address these equations and explore pole-skipping points within each sector using the near-horizon method. Our analysis reveals two classes of pole-skipping points: regular and singular pole-skipping points in which the latter is identified when standard linear differential equations exhibit singularity. Notably, pole-skipping points in the lower-half plane are regular, while those elsewhere are singular. This suggests that the pole-skipping point in the spin-0 channel, associated with quantum chaos, corresponds to a singular pole-skipping point. Additionally, we observe that the pole-skipping momentum, if purely real or imaginary for μ = β = 0, retains this characteristic for μ ≠ 0 and β ≠ 0.

Duality of the Nonreflexive Bergman Space of the Upper Half Plane and Composition Groups

We identify the predual of the nonreflexive Bergman space of the upper half plane with the little Bloch space of the upper half plane consisting of those functions vanishing at point i. Using the duality pairing as well as the composition groups on the nonreflexive Bergman space, we obtain the groups of composition operators defined on the identified predual. We identify the infinitesimal generator of each group and prove the strong continuity property. We then obtain the spectra of the generator Γ, determine the resolvents and further obtain the spectra and the norms of the resulting resolvents.

The pseudospectra of black holes in AdS

We study the stability of quasinormal modes (QNMs) in electrically charged black brane spacetimes that asymptote to AdS by means of the pseudospectrum. Methodologically, we adopt ingoing Eddington-Finkelstein coordinates to cast QNMs in terms of a generalised eigenvalue problem involving a non-selfadjoint operator; this simplifies the computation significantly in comparison with previous results in the literature. Our analysis reveals spectral instability for (neutral) scalar as well as gravitoelectric perturbations. This indicates that the equilibration process of perturbed black branes is sensitive to external perturbations. Particular attention is given on the hydrodynamic modes, which are found to be the least unstable. In contrast with computations in hyperboloidal coordinates, we find that the pseudospectral contour lines cross to the upper half plane. This indicates the existence of pseudo-resonances as well as the possibility of transient instabilities. We also investigate the asymptotic structure of pseudospectral contour levels and we find remarkable universality across all sectors, persistent in the extremal limit.

The Heisenberg covering of the Fermat curve

Abstract For N integer $\ge 1$ , K. Murty and D. Ramakrishnan defined the Nth Heisenberg curve, as the compactified quotient $X^{\prime }_N$ of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over $\mathbf {Z}[\mu _N,1/N]$ of the Nth Heisenberg curve as covering of the Nth Fermat curve. We show that the Manin–Drinfeld principle holds for $N=3$ , but not for $N=5$ . We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves $X_N$ and the classical modular curves $X(n)$ , for n even integer, both dominate $X(2)$ , which produces a morphism between Jacobians $J_N\rightarrow J(n)$ . We prove that the latter has image $0$ or an elliptic curve of j-invariant $0$ . In passing, we give a description of the homology of $X^{\prime }_{N}$ .

An inverse problem for Hankel operators and turbulent solutions of the cubic Szegő equation on the line

We construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of this theory, we prove the genericity of turbulent solutions of the cubic Szegő equation on the real line.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

Classical solution of the Cauchy problem for a semilinear hyperbolic equation in the case of two independent variables

In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution of some integral equation. The local solvability of this equation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray-Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.