arXiv subjects
Publications and source records attributed to Haoren Xiong.
We construct a Banach algebra of Gevrey asymptotic symbols that is stable under the semiclassical quantization product and elliptic inversion (microlocal parametrix), and derive sharp quantitative estimates for the coefficients of the full asymptotic inverse. As an application, we obtain exponential estimates for adiabatic projectors in the Gevrey setting.
We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey-$s$ regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is of size ${\mathcal O}(h^{1-\frac{1}{s}})$ where the resolvent is at most fractional exponentially large in $h$, as the semiclassical parameter $h\to 0^+$. This is a natural Gevrey analogue of a result by N. Dencker, J. Sj{\"o}strand, and M. Zworski in the $C^{\infty}$ and analytic cases.
We establish precise asymptotic expansions for solutions to semilinear wave equations with power-type nonlinearities on asymptotically flat spacetimes. Our analysis focuses on two key cases: cubic nonlinearities and higher-order power nonlinearities. For cubic nonlinearities of the form $a(t,x) \, \phi^3$, we prove asymptotic expansions for the solution globally in the spacetime. In the special case of compact spatial regions, solutions exhibit the asymptotic behavior $\phi(t, x) = c \, t^{-2} + \mathcal{O}(t^{-3+})$. For higher-order nonlinearities $a(t,x) \, \phi^p$ with $p \geq 4$, we prove the solution satisfies $\phi(t, x) = d \, t^{-3} + \mathcal{O}(t^{-4+})$, thereby extending the classical Price's law (a late-time tail postulated in 1972) to nonlinear settings in a precise fashion. These results sharpen previous decay estimates for nonlinear waves. We develop a radiation field expansion and a low-energy resolvent expansion adapted to conormal asymptotic inputs, extending Hintz's approach for linear waves to the semilinear setting. Our methods connect geometric microlocal analysis (b-calculus) with classical physical-space techniques, providing a convenient tool for analyzing asymptotic behavior of nonlinear waves.
We extend the direct approach to the semiclassical asymptotics for Bergman projections, developed by Deleporte--Hitrik--Sjöstrand for real analytic exponential weights and Hitrik--Stone for smooth exponential weights, to the case of Gevrey weights. We prove that the amplitude of the asymptotic Bergman projection forms a Gevrey symbol whose asymptotic coefficients obey certain Gevrey-type growth rate, and it is constructed by an asymptotic inversion of an explicit Fourier integral operator up to a Gevrey-type small remainder.
We study Toeplitz operators on the Bargmann space, whose Toeplitz symbols are exponentials of complex inhomogeneous quadratic polynomials. Extending a result by Coburn--Hitrik--Sjöstrand, we show that the boundedness of such Toeplitz operators implies the boundedness of the corresponding Weyl symbols, thus completing the proof of the Berger--Coburn conjecture in this case. We also show that a Toeplitz operator is compact precisely when its Weyl symbol vanishes at infinity in this case.
We show that all resonances in Dirichlet obstacle scattering (in $\mathbb{C}$ in odd dimensions and in the logarithmic cover of $\mathbb{C}\setminus\{0\}$ in even dimensions) are generically simple in the class of obstacles with $C^k$ (and $C^\infty$) boundaries, $k \geq 2$.
In this note we discuss the complex version of the Higgs oscillator on the hyperbolic space. The eigenvalues and resonances of the complex Higgs oscillator are computed in different examples in the hyperbolic setting. We also propose open problems like whether the complex absorbing potential (CAP) method works for asymptotically hyperbolic manifolds and finding hyperbolic analogues of the complex harmonic oscillator.
We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to an abstractly defined class of black box perturbations of the Laplacian in $\mathbb{R}^n$ which can be analytically extended from $\mathbb{R}^n$ to a conic neighborhood in $\mathbb{C}^n$ near infinity. The black box setting allows a unifying treatment of diverse problems ranging from obstacle scattering to scattering on finite volume surfaces.
We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Δ+ V - iεx^2$, $|V(x)|\leq C e^{-2γ|x|}$ converge, as $ ε\to 0+ $, to the poles of the meromorphic continuation of $ ( -Δ+ V -λ^2 )^{-1} $ uniformly on compact subsets of $\textrm{Re}\,λ>0$, $\textrm{Im}\,λ>-γ$, $\argλ> -π/8$.
For exterior dilation analytic potential, $V$, we use the method of complex scaling to show that the resonances of $ - Δ+ V $, in a conic neighbourhood of the real axis, are limits of eigenvalues of $ - Δ+ V - i εx^2 $ as $ ε\to 0+ $, if $V$ can be analytically extended from $\mathbb{R}^n$ to a truncated cone in $\mathbb{C}^n$.