arXiv subjects
Publications and source records attributed to Haakan Hedenmalm.
We combine McShane's identity for a once-punctured torus with the length-spectrum identity of Wang and Xue. An exact identity in each primitive homology class yields a finite limit for the difference of two divergent length-spectrum sums over self-intersecting hyperbolic geodesics. The limit is $\frac18$ and is equivalent to McShane's identity. We also derive higher-iteration identities governed by integer divisors.
We establish a discrete Hadamard variational formula for Green operators on weighted graphs along an increasing family of subgraphs. The increment of the Green operator factors through harmonic extension from the newly added layer and the positive square root of its boundary Green operator. This gives a dimension-free decomposition of the Green kernel and an associated isometric Hadamard operator. As an application, the discrete Gaussian free field is constructed by successively adding independent harmonically extended Gaussian contributions. The formulation uses only the graph structure and its conductances, without smoothness assumptions. For nested square-lattice approximations of smooth planar domains, we also prove convergence of the growing fields in finite-dimensional distributions and convergence of the integrated Hadamard increments as Green-operator forms.
We develop the geometry of conformal mappings through Dirichlet operator symbols and the nonlinear wave equation. These symbols, introduced in the work of Hedenmalm and Shimorin (2020), are holomorphic functions on the bidisk associated with contractions on $L^2(\mathbb{D})$. For Grunsky operators, the nonlinear wave equation and the related Riccati equation induce a hierarchy of of differential relations along the diagonal. We use this jet hierachy to obtain alternative coordinates and a characterization of the manifold $Σ/\mathbb{C}$ of normalized exterior univalent functions modulo translations, without an explicit univalence test. This manifold may be viewed as an extension of the universal Teichmüller space beyond the quasicircle boundary setting. The diagonal jets also give rise to a hierarchy of \emph{diagonal series Schwarzian derivatives}. Already the fourth Schwarzian $S_4(ψ)$ leads to differential equations connected with the Weierstrass elliptic $\wp$ function and Painlevé I, revealing a comparison between the geometry of conformal mappings and that of complex tori. As a quantitative application of the framework, we interpret asymptotic variance through the Littlewood-Paley formula as an asymptotic density of quadratic nonlinearity energy under logarithmic averaging. We introduce Schwarzian asymptotic variance and obtain the upper bound $12.566563\ldots$ for its averaged squared hyperbolically normalized amplitude, in comparison with the classical pointwise bound $36$.
We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary hyperbolic partial differential equation, after the wave equation itself. Relative to the origin, the spacetime splits according to the light cones, and we find four quarter-planes, two of which are timelike while the remaining two are spacelike. Not unexpectedly, the solutions behave quite differently in the two types of quarter-planes. It turns out that the spacelike quarter-planes exhibit a Liouville phenomenon, where insufficient growth forces the solutions to display a certain kind of symmetry, where the values on the two linear edges are in a one-to-one relation. This phenomenon shares features with the classical Liouville theorem as well as the Phragmen-Lindelof principle for harmonic functions. We obtain sharp transition from uniqueness to nonuniqueness.
We extend the packing results of Etayo, Hedenmalm, and Ortega-Cerda for complex tori to arbitrary compact Riemann surfaces. The construction uses two different approaches, one based on complex geometry of line bundles, and the other on Laplace designs. Both approaches give the same basic existence of close-to-optimal point configurations on the given surface. This version was written with the help of ChatGPT.
We study the exponential Green energies associated with a finite point configuration on a Riemann surface. We focus on studying optimal or close-to-optimal configurations, and show that they must be well-spread out on the surface. We study in particular the round sphere and obtain values for the minimal Green energy of a configuration. The first version of this paper was written with the help of ChatGPT.
It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schrödinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+αLu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $α$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-Pólya idea to the spectral problem at hand.
In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass $σ$ function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.
We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density $B(z,\cdot)$ for the polynomial Bergman space, where the point $z\in\mathbb{C}$ is fixed. When $z=\infty$, the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial $P$. We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the $\bar\partial$-operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of $|P|^2$ in place of $\overline{P}$. Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities $B(z,w)$ in the off-spectral regime, that is, when $z$ is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.
In this article we explain the essence of the interrelation described in [PNAS 118, 15 (2021)] on how to write explicit interpolation formula for solutions of the Klein-Gordon equation by using the recent Fourier pair interpolation formula of Viazovska and Radchenko from [Publ Math-Paris 129, 1 (2019)]. We construct explicitly the sequence in $L^1 (\mathbb{R} )$ which is biorthogonal to the system $1$, $\exp ( i πn x)$, $\exp ( i πn/ x)$, $n \in \mathbb{Z} \setminus \{0\}$, and show that it is complete in $L^1 (\mathbb{R})$. We associate with each $f \in L^1 (\mathbb{R}, (1+x^2)^{-1} d x)$ its hyperbolic Fourier series $h_{0}(f) + \sum_{n \in \mathbb{Z}\setminus \{0\}}(h_{n}(f) e^{ i πn x} + m_{n}(f) e^{-i πn / x} )$ and prove that it converges to $f$ in the space of tempered distributions on the real line. Applied to the above mentioned biorthogonal system, the integral transform given by $U_φ (x, y):= \int_{\mathbb{R}} φ(t) \exp \left( i x t + i y / t \right) d t $, for $φ\in L^{1} (\mathbb{R})$ and $(x, y) \in \mathbb{R}^{2}$, supplies interpolating functions for the Klein-Gordon equation.
We look at the work of Oleg Ivrii connected with the dimension of quasicircles for asymptotically small quasiconformality parameter $k$. We intend to make this work more easily accessible. Our main focus is the integral means spectrum associated with normalized conformal mappings of the exterior disk which have quasiconformal extensions to the whole plane with small dilatation parameter $k$. Moreover, we address the estimates from above only, not the sharpness from below.
We introduce a novel collection of uniqueness problems, with related sampling and interpolation issues. We call them deep zero problems, as they are concerned with local properties at a small number of given points.
Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, e.g. in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to planar exponentially varying weights, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), using a technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a matrix dbar-problem (Its, Takhtajan), which we call a soft Riemann-Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann-Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann-Hilbert problems along the interface, which appears as the free boundary for an obstacle problem. Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real-analytic weights, the asymptotics holds in the $L^2$ sense with error $O(e^{-δ\sqrt{m}})$ in a fixed neighborhood of the closed exterior of the interface, for some constant $δ>0$, where $m\to+\infty$. Here, $m$ is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface a distance proportional to $m^{-\frac14}$.
We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies - sequences of scalar Riemann-Hilbert problems - which allows us to express all higher order correction terms in closed form. In fact, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.
We show that the planar normalized orthogonal polynomials $P_{m,n}(z)$ of degree $n$ with respect to an exponentially varying planar measure $\mathrm{e}^{-2mQ}\mathrm{dA}$ enjoy an asymptotic expansion \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{ϕ_τ'(z)}[ϕ_τ(z)]^n \mathrm{e}^{m\mathcal{Q}_τ(z)}\left(\mathcal{B}_{τ, 0}(z) +m^{-1}\mathcal{B}_{τ, 1}(z)+m^{-2} \mathcal{B}_{τ,2}(z)+\ldots\right), \] as $n,m\to\infty$ while the ratio $τ=\frac{n}{m}$ is fixed. Here $\mathcal{S}_τ$ denotes the droplet, the boundary of which is assumed to be a smooth simple closed curve, and $ϕ_τ$ is a conformal mapping from the complement $\mathcal{S}_τ^c$ to the exterior disk $\Bbb{D}_\mathrm{e}$. The functions $\mathcal{Q}_τ$ and $\mathcal{B}_{τ, j}$ are bounded holomorphic functions which may be expressed in terms of $Q$ and $\mathcal{S}_τ$. We apply these results to obtain boundary universality in the random normal matrix model for smooth droplets, i.e., that the limiting rescaled process is the random process with correlation kernel \[ \mathrm{k}(ξ,η)= \mathrm{e}^{ξ\barη\,-\frac12(\lvertξ\rvert^2+\lvert η\rvert^2)} \,\mathrm{erf}\,(ξ+\barη). \] A key ingredient in the proof of the asymptotic expansion of the orthogonal polynomials is the construction of an orthogonal foliation -- a smooth flow of closed curves near $\partial\mathcal{S}_τ$, on each of which $P_{m,n}$ is appropriately orthogonal to lower order polynomials. To compute the coefficient functions, we develop an algorithm which determines the coefficients $\mathcal{B}_{τ, j}$ successively in terms of inhomogeneous Toeplitz kernel conditions. These inhomogeneous Toeplitz kernel conditions may be understood in terms of scalar Riemann-Hilbert problems.
We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, i.e., finite dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type) we establish an analogue of de Branges' Ordering Theorem. We then construct examples which show that the result fails for general Fock-type spaces of larger growth.
We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.
We study a new type of extremal problem in complex analysis, referred to as "geometric zero packing", which is the hyperbolic analogue of a problem considered by Abrikosov in the 1950s concerning Bose-Einstein condensates. We relate the corresponding minimal discrepancy density with the asymptotic variance for Bloch functions of the form "Bergman projection of bounded functions" and obtain a corresponding identity. Together with related work of Ivrii, this gives the asymptotic behavior of the universal quasiconformal integral means spectrum for small values of quasiconformality k and small exponents t. In particular, the conjectured behavior is shown to be smaller than conjectured by Prause and Smirnov, which also shows that there are no quasidisks with dimension 1+k^2, at least for small k.