Related papers: Full semiclassical asymptotics near transition poi…
Let $A$ be an elliptic operator on a compact manifold with boundary $M$, and let $\wp : \partial\M \to Y$ be a covering map, where $Y$ is a closed manifold. Let $A_C$ be a realization of $A$ subject to a coupling condition $C$ that is…
Using variational methods combined with perturbation arguments, we study the existence of nontrivial classical solution for the quasilinear Schr\"{o}dinger equation \begin{equation*}\label{1.1} -\Delta u+ V(x)u+ \frac{\kappa}{2}[\Delta…
We study the asymptotic behavior as |x| \to \infty of Schr\"odinger operators with homogeneous potentials. For this purpose, we use methods from semiclassical analysis and investigate semiclassical defect mesures. We prove their…
In this manuscript, we study modified scattering for the nonlinear defocusing Schr\"odinger equation with a critical gauge-invariant nonlinearity of order 1+2/n. We address the following question: Given initial data in an appropriate…
In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian…
We study the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its…
We investigate the connection between semilinear elliptic PDEs with isolated singularities and stationary nonlinear Schr\"odinger equations with point interactions. In dimensions $d=2,3$, we provide a detailed equivalence result between the…
Sharp $L^\infty$ estimates are obtained for general classes of fully non-linear PDE's on non-K\"ahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the K\"ahler case. The key idea is…
In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the…
By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in…
For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for…
Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical…
Our paper investigates one-dimensional Schr\"odinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We…
This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This…
The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques…
We consider the one-dimensional nonlinear Schr\"odinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schr\"odinger flow and we…
We consider the limiting process that arises at the hard edge of Muttalib--Borodin ensembles. This point process depends on $\theta > 0$ and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti…
We prove an explicit weighted estimate for the semiclassical Schr\"odinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is…
In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schr\"odinger equation with $\delta$- and $\delta'$-potentials. It is shown that the sequence of solutions converges uniformly on…
Schr\"odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to…