More Like this

1. Painlevé-III Monodromy Maps Under the $$D_6\to D_8$$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

   https://doi.org/10.3842/SIGMA.2024.019  

   Barhoumi, Ahmad; Lisovyy, Oleg; Miller, Peter D; Prokhorov, Andrei (March 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   The third Painlevé equation in its generic form, often referred to as Painlevé-III($$D_6$$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $$u_0(x)$$ corresponding to parameters $$\alpha$$, $$\beta$$, denoted as the triple $$(u_0(x),\alpha,\beta)$$, we apply an explicit Bäcklund transformation to generate a family of solutions $$(u_n(x),\alpha + 4n,\beta + 4n)$$ indexed by $$n \in \mathbb N$$. We study the large $$n$$ behavior of the solutions $$(u_n(x), \alpha + 4n, \beta + 4n)$$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $$u_n(z/n)$$. Our main result is a proof that the limit of solutions $$u_n(z/n)$$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($$D_8$$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($$D_6$$), which are constructed using the seed solution $(1,4m,-4m)$ where $$m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $$D_6$$ and $$D_8$$ at $z = 0$. We also deduce the large $$n$$ behavior of the Umemura polynomials in a neighborhood of $z = 0$. 

   more » « less
2. Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

   https://doi.org/10.1093/imrn/rnz262  

   Liu, Wencai (November 2019, International Mathematics Research Notices)

   Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $$H$$ as a perturbation of the free operator $$H_0$$, where $$(H_0u)(n)= u({n+1})+u({n-1})$$. For $$H_0$$ (no perturbation), $$\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$$ and $$H_0$$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $$H_0+V$$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $$\limsup _{n\to \infty } n|V(n)|=a<\infty .$$ We obtain a lower/upper bound of $$a$$ such that $$H_0+V$$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $$\{ E_j\}_{j=1}^N$$ in $(-2,2)$ with $$0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$$, we construct the explicit potential $$V(n)=\frac{O(1)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}_{j=1}^N$$.5: Given any countable set of points $$\{ E_j\}$$ in $(-2,2)$ with $$0\notin \{ E_j\}+\{ E_j\}$$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $$|V(n)|\leq \frac{h(n)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}$$. 

   more » « less
3. Stochastic Ricci Flow on Compact Surfaces

   https://doi.org/10.1093/imrn/rnab015  

   Dubédat, Julien; Shen, Hao (April 2021, International Mathematics Research Notices)

   Abstract In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville conformal field theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full “$L^1$ regime” $$\sigma < \sigma _{L^1}=2 \sqrt \pi $$ where $$\sigma $$ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces and list some open questions. 

   more » « less
4. Negative Moments of L –Functions With Small Shifts Over Function Fields

   https://doi.org/10.1093/imrn/rnad118  

   Florea, Alexandra (June 2023, International Mathematics Research Notices)

   Abstract We consider negative moments of quadratic Dirichlet $$L$$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $$\mathbb{F}_{q}[x]$$, we obtain an asymptotic formula for the $$k^{\textrm{th}}$$ shifted negative moment of $$L(1/2+\beta ,\chi _{D})$$, in certain ranges of $$\beta $$ (e.g., when roughly $$\beta \gg \log g/g $$ and $k<1$). We also obtain non-trivial upper bounds for the $$k^{\textrm{th}}$$ shifted negative moment when $$\log (1/\beta ) \ll \log g$$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$. 

   more » « less
5. The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

   https://doi.org/10.1007/s00208-021-02219-1  

   Hofmann, Steve; Li, Linhan; Mayboroda, Svitlana; Pipher, Jill (July 2021, Mathematische Annalen)

   Grafakos, Loukas (Ed.)

   We establish the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti- symmetric part. In particular, the coefficients are not necessarily bounded. We prove that for some $$p\in (1,\infty)$$, the Dirichlet problem for the elliptic equation $$Lu= \dv A\nabla u=0$$ in the upper half-space $$\mathbb{R}^{n+1},\, n\geq 2,$$ is uniquely solvable when the boundary data is in $$L^p(\mathbb{R}^n,dx)$$, provided that the coefficients are independent of the vertical variable. This result is equivalent to saying that the elliptic measure associated to $$L$$ belongs to the $$A_\infty$$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity. 

   more » « less