Expansion of Integrals. 4

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\Hess}{\operatorname{Hess}}$ $\newcommand{\supp}{\operatorname{supp}}$ $\newcommand{\sgn}{\operatorname{sgn}}$

### Oscillatory integrals. II. Multidimensional theory

We consider $$I(k)= \int_X e^{k \phi(x)}f(x)\,dx \label{eq-2.4.1}$$ where now $X= \bR^d$ and $\phi \in C^\infty(X)$, $\phi$ is a real-valued function, $f\in C_0^\infty(X)$ which means that $f=0$ as $|x|\ge R$.

We are interested in the asymptotics of $I(k)$ as $k\to +\infty$.

#### No stationary points

First of all we need

Theorem 1. If $\phi$ has no stationary points on $\supp f$ then $$I(k)= O(k^{-\infty})\qquad\text{as }\ k\to +\infty \label{eq-2.4.2}$$ which means that $I(k)= O(k^{-N})$ for any $N$.

Proof. We can decompose $f=f_1+\ldots+f_d$ such that $\partial_{x_j}\phi \ne 0$ on $\supp f_j$. Then applying we can apply $1$-dimensional result.

#### Single stationary point

Assume now that $\phi$ has a single stationary point $\bar{x}$ on $\supp f$ and this stationary point is non-degenerate. Then $f=f_0+f_1$ where $f_0$ is supported in the small vicinity of $\bar{x}$ and $\phi$ has no stationary points on $\supp f_1$. So without any loss of the generality one can assume that $f=f_0$.

Applying Morse theory (Theorem 2.2.2) we can assume without any loss of the generality that $$\phi (y)=\phi (\bar{x})+ \sum_{1\le j\le d} \lambda_j z_j^2 \label{eq-2.4.3}$$ and applying $1$-dimensional theory (Theorem 2.3.5) we arrive to

Theorem 2. Let $\phi$ have a single stationary point $\bar{x}$ on $\supp f$ and $\phi''(\bar{x})0$ non-degenerate. Then $$I(k) \sim e^{ik\phi (\bar{x})}\sum _{n=0}^\infty \kappa_{2n} k^{-\frac{d}{2}-n} \label{eq-2.4.4}$$ in the sense that $$|I(k)- e^{ik\phi (c)}\sum _{n=0}^{N-1} \kappa_{2n}k^{-\frac{d}{2}-n}|\le C_N k^{-N-\frac{d}{2}}. \label{eq-2.4.5}$$ Here the main coefficient is $$\kappa_0=(2\pi)^{\frac{d}{2}}|\det \phi''(\bar{x} )|^{-\frac{1}{2}} e^{i\frac{\pi}{4} \sgn \phi''(\bar{x})} f(\bar{x}) \label{eq-2.4.6}$$ and $\sgn A$ is a signature of non-degenerate Hermitean matrix $A$: $\sgn A= d_+ - d_-$ where $d_\pm$ is a number of positive and negative eigenvalues of $A$ (or equivalently the dimension of positive or negative space of the quadratic form $\langle Ax,x\rangle$).

#### Several stationary points

Let now $\phi$ has several stationary points (or stationary points in which $\Im \phi=0$ in the framework of Theorem 3 below) on $\supp f$: $x_1,\ldots ,x_K$ each of the type considered above. Then asymptotics of $I(k)$ is given by the sum of the contributions of all these points.

#### Degenerate stationary points

The degenerate stationary points of the function of several variables can be of the very different types. What is more: these functions usually depend on extra parameters $\phi=\phi(x; y)$ where we integrate with respect to $x$ only and want remainder estimate uniform with respect to $y$. In this case we using the above approach eliminate integration with respect to "some of $x$".

#### Complex phase

Theorem 3. Assume now that $\phi$ is a complex-valued function, $\Im\phi \ge 0$ and there exists a single point $\bar{x}$ such that $\Im \phi (\bar{x})=0$, $\nabla \phi (\bar{x})=0$ and $\phi''(\bar{x})$ is non-degenerate.

Then decomposition (\ref{eq-2.4.4}) holds.

Proof. Observe, that $\Re i\phi = -\Im \phi \le 0$. Also observe that if $\Im \phi>0$ on $\supp f$ then $I(k) =O(e^{-\epsilon k})$. Further, if $\nabla \phi \ne 0$ on $\supp f$ then $I(k) =O(k^{-\infty})$.

Thus we need to consider a small vicinity of $\bar{x}$. Without any loss of the generality one can assume that $\bar{x}=0$ and $\phi(0)=0$.

Let $Q(x)=\frac{1}{2}\langle \phi''(0)x,x\rangle$ be a quadratic part of $\phi(x)$, and $S(x)=\phi(x)-Q(x)$. Then \begin{equation*} e^{ikS (x)}\sim \sum_{n\ge 0} \frac{1}{n!} (iS(x))^n \sim \sum _{m\ge 0} k^m \sum_{\alpha:|\alpha|\ge 3m} c_{m,\alpha} x^\alpha \end{equation*} in the sense that the remainder is $O(|x|^{3N}k^N)$ and since $f(x)$ could be decomposed into asymptotic Taylor series similarly \begin{equation*} e^{ikS (x)}f(x)\sim \sum _{m\ge 0} k^m \sum_{\alpha:|\alpha|\ge 3m} c'_{m,\alpha} x^\alpha \end{equation*} and then \begin{multline} I(k)\sim \sum _{m\ge 0} k^m \sum_{\alpha:|\alpha|\ge 3m} c'_{m,\alpha} \int e^{ikQ(x)}x^\alpha\,dx \sim\\ \sum _{m\ge 0} \sum_{\alpha:|\alpha|\ge 3m} c'_{m,\alpha} k^{m- \frac{d}{2}-\frac{|\alpha|}{2}} \int e^{iQ(x)}x^\alpha\,dx \label{eq-2.4.7} \end{multline} and terms with odd $|\alpha|$ vanish. One can prove (\ref{eq-2.4.7}) by integration by parts using that $|\nabla \phi|\asymp |x|$ since $\phi''$ is non-degenerate.

This proves decomposition (\ref{eq-2.4.4}).

Remark 1. One can prove that $$\kappa_0 = (2\pi)^{\frac{d}{2}} \bigl(\det (-i\phi ''(\bar{x})^{\frac{1}{2}}\bigr)^{-1}= \pm (2\pi)^{\frac{d}{2}} \bigl(\det (-i\phi ''(\bar{x})\bigr)^{-\frac{1}{2}}. \label{eq-2.4.8}$$ Since $\Im \phi''(\bar{x})$ is non-negative definite matrix, $-i\phi''(\bar{x})$ is a sectorial matrix in the sense that its spectrum belongs to ${z\in \bC:\, \Re z\ge 0, \, z\ne 0}$ and since it cannot belong $(-\infty,0]$ the square root of it $(-i\phi ''(\bar{x})^{\frac{1}{2}}$ is properly defined. However $(\det (-i\phi ''(\bar{x})\bigr)^{\frac{1}{2}}$ is defined up to a factor $\pm 1$.

Remark 2. If $f= O(|x-\bar{x}|^l)$ then $\kappa_n=0$ as $n=\lceil l/2\rceil$.