Full semiclassical asymptotics near transition points

TL;DR

This work develops a complete semiclassical framework for 1D Schrödinger-type ODEs with transition points, allowing higher-order vanishing ($\kappa\ge 2$) and controlled singularities in the lower-order term. By embedding the problem in a manifold-with-corners $M$ obtained via a quasihomogeneous blowup, the authors obtain exponential-polyhomogeneous expansions that are differentiable in the semiclassical parameter $h$ and valid along multiple $h\to 0^+$ scales. The main technical advance is a two-edge patching scheme (ze and fe) with two matched $h\to 0^+$ expansions, enabling Airy/Weber-type data to appear naturally as boundary data; this improves upon classical Langer–Olver results by allowing logarithms and preserving full asymptotic expansions. The results apply to a broad class of potentials, including Coulomb-like and anharmonic cases, and yield rigorous quasimode constructions, with $O(h^\infty)$ error control, offering new asymptotic descriptions in regimes previously inaccessible. Overall, the paper provides a robust geometric-analytic pipeline for uniform semiclassical asymptotics across transition points, with potential PDE analogues and broad applicability in quantum-mechanical and gravitational settings.

Abstract

We construct complete asymptotic expansions of solutions of the 1D semiclassical Schrödinger equation near transition points. There are three main novelties: (1) transition points of order $κ\geq 2$ (i.e.\ trapped points -- the simple turning point is $κ=1$, the simple pole is $κ=-1$) are handled, (2) various terms in the operator are allowed to have controlled singularities of a form compatible with the geometric structure of the problem (some applications are given in the text), and (3) the term-by-term differentiability of the expansions with respect to the semiclassical parameter is included. We prove that any solution to the semiclassical ODE with initial data of exponential type is of exponential-polyhomogeneous type on a suitable manifold-with-corners compactifying the $h\to 0^+$ regime. Consequently, such a solution has an atlas of full asymptotic expansions in terms of elementary functions, and these expansions are well-behaved. The Airy and Bessel functions show up in the expected way, as the asymptotic data at one boundary edge. We are able to handle cases that Langer--Olver could not because the framework of polyhomogeneous functions on manifolds-with-corners provides more flexibility (two matched $h\to 0^+$ expansions, possibly with logarithms, in this case) than that employed by Langer--Olver (one uniform $h\to 0^+$ expansion without logarithms). We work entirely in the $C^\infty$ category. No analyticity is ever assumed, nor proven.

Figures (23)

• Figure 1: Left: The manifold-with-corners $M$ (the portion with $h<h_0$, for $h_0>0$ arbitrary). Some local coordinate charts are depicted. Here, as elsewhere in the paper, ${\lambda=z/h^{2/(\kappa+2)}}$. The quasihomogeneous blowup used to create $M$ is the one which resolves curves $\Gamma_{\lambda_0}=\{\lambda=\lambda_0\}$ of constant $\lambda$, defined in \ref{['eq:Gamma_curve']}. Right: some curves in $M$, including (the lift of) $\Gamma_1$, which hits the front face of the blowup. Also shown is a level set of $z$, a curve $\Gamma_{\mathrm{H}}$ probing the "intermediate" regime $\mathrm{ze}\cap \mathrm{fe}$ (the high corner of $\mathrm{fe}$), and one, ${\Gamma_{\mathrm{L}}}$, probing the other such regime $\mathrm{fe}\cap \mathrm{be}$ (the low corner of $\mathrm{fe}$). One of the upshots of this paper is that we can understand the asymptotics of solutions of the ODE along each of these curves, even $\Gamma_{\mathrm{H}},\Gamma_{\mathrm{L}}$.
• Figure 2: Left: the rectangle $[0,Z]_z\times [0,\infty)_{h^2}$ whose lower-left corner is blown up (quasihomogeneously) to create $M$. Three families of curves are shown in $M$. These are the level sets $\Gamma^{(z_0)}=\{z=z_0\}$ of $z$ (horizontal lines), the level sets of $h$ (vertical lines), and the level sets $\Gamma_\lambda$ of $\lambda$. We depict the $\kappa\geq 1$ case. Middle: some of the curves $\Gamma_\lambda$ in the $\kappa=-1,0$ cases, where their convexity differs from the $\kappa\geq 1$ case. In the $\kappa=0$ case, they are lines. In the $\kappa=-1$ case, they are parabolas. Right: the same curves in $M$. Note that, in this compactification, the $\Gamma_\lambda$ limit to distinct points of $\mathrm{fe}^\circ$. This is what the quasihomogeneous blowup of the corner $\{z=0,h=0\}$ accomplishes. Note that the level sets of $h,z$ (dotted lines) appear distorted in this rendering.
• Figure 3: Left:$\Omega\subset \mathbb{R}^{2}_{x,y}$, where $\Omega$ is as in \ref{['eq:Omega']}. Level curves of $h(x,y)$, $z(x,y)$ are shown. This example was computed using \ref{['eq:fur_Nick']}. Middle: The function $f$ defined in \ref{['eq:f_ex']} plotted against $h,z$. Note how many contour lines converge at the bottom-left corner. This shows that $f$ is not continuous there. Right: the pullback $f\circ T$ of $f$ to $\Omega$, via $T:(x,y)\mapsto (z,h)$. As is evident from the behavior of the contour lines, $f\circ T$ is continuous on $\Omega$.
• Figure 4: (a) The rectangle $[-Z,Z]_z\times [0,\infty)_{h^2}$ and (b) the mwc $X$ that results from blowing up the points $(z_n,0)$, as described in §\ref{['subsec:JWKB']}. Here, $\Gamma_{n,\lambda}=\{z= z_n+\lambda h^{2/3}\}$.
• Figure 6: The mwc $M_2$, with the subset $M\cong M'\subset M_2$ labeled, with the edges labeled by the corresponding edges in $M$. A smooth atlas is depicted.

Theorems & Definitions (52)

• Theorem A
• Remark 1.1: Reduction from the general second-order case to that above
• Remark 1.2: $\kappa\leq -2$
• Remark 1.3: Fractional $\kappa$
• Example
• Example
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Example : SussmanACL