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arXiv 2104.10028math.CAmath-phmath.MP
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Laplace-Fourier型积分的渐近展开

Asymptotic Expansion of Laplace-Fourier-Type Integrals

Sara Konrad, Matthias Bartelmann

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13 pages, to be submitted to SciPost Physics
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摘要：

本文研究了Laplace-Fourier型$P(k) = \int_Ω\mathrm{e}^{-|k|^sf(x)}\mathrm{e}^{\mathrm{i} kx}\mathrm{d} x\;, $与$k\in\mathbb{R}^d$在$d\ge1$维数上的积分的渐近行为，以及与$Ω\subset\mathbb{R}^d$和充分良函数$f:Ω\to\mathbb{R}$的积分的渐近行为.我们的主要结果是$ P(k) \sim \frac{\mathrm{e}^{-|k|^sf(0)}}{|k|^{sd/2}}\sqrt{\frac{(2π)^d}{\det A}} \exp\left(-\frac{k^\top A^{-1}k}{2|k|^s}\right) $为$|k|\to\infty$，其中$A$是Hessian矩阵的功能$f$在其临界点，假设是在$x_0 = 0$。在一维中，Hessian由二阶导数$A = f''(0)$代替。我们还表明，集成域$Ω$可以扩展到$\mathbb{R}^d$不改变渐近行为。

英文摘要：

We study the asymptotic behaviour of integrals of the Laplace-Fourier type $P(k) = \int_Ω\mathrm{e}^{-|k|^sf(x)}\mathrm{e}^{\mathrm{i} kx}\mathrm{d} x\;, $ with $k\in\mathbb{R}^d$ in $d\ge1$ dimensions, with $Ω\subset\mathbb{R}^d$ and sufficiently well-behaved functions $f:Ω\to\mathbb{R}$. Our main result is $ P(k) \sim \frac{\mathrm{e}^{-|k|^sf(0)}}{|k|^{sd/2}}\sqrt{\frac{(2π)^d}{\det A}} \exp\left(-\frac{k^\top A^{-1}k}{2|k|^s}\right) $ for $|k|\to\infty$, where $A$ is the Hessian matrix of the function $f$ at its critical point, assumed to be at $x_0 = 0$. In one dimension, the Hessian is replaced by the second derivative, $A = f''(0)$. We also show that the integration domain $Ω$ can be extended to $\mathbb{R}^d$ without changing the asymptotic behaviour.

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