Expansion of Integrals. 2

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

### Laplace integrals. II. Multidimensional theory

We consider $$I(k)= \int_X e^{k \phi(x)}f(x)\,dx \label{eq-2.2.1}$$ 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 $k\to +\infty$. Naturally we expect that the main contribution to $I(k)$ is delivered by the vicinities of the points $x\in X$ in which $\phi(x)$ reaches its maximum. In such points $\nabla\phi=0$. We assume that $\phi$ has only non-degenerate maxima, i.e. $\phi'':=\Hess \phi=\Bigl(\frac{\partial^2\phi}{\partial x_j\partial x_k}\Bigr)_{j,k=1,\ldots, d}$ is a non-degenerate matrix at such points (then it is strictly negative matrix, since we talk about maxima).

#### Morse theory

Morse theory contains several important theorems concerning stationary points of smooth functions.

Definition 1. Function $\phi$ is Morse function if all its stationary points are non-degenerate.

Theorem 1.

1. Let $\phi \in C^\infty (X)$. Then for any $R$, $m$ and $\epsilon>0$ there exists function Morse function $\varphi^* \in C^\infty (X)$ such that all derivatives of order $\le m$ of $(\phi-\varphi)$ do not exceed (by absolute value) $\epsilon$ (in the ball $B(0,R)=\{x:\, |x|\le R\}$).
2. Let $\phi$ be Morse function. Then there exists $\epsilon >0$ such that if all the derivatives of order $\le 2$ of $(\phi - \varphi)$ do not exceed (by absolute value) $\epsilon$ then $\varphi$ is also Morse function (in the ball $B(0,R)$).

We say that Morse functions are generic and all functions with degenerate stationary points are exceptional.

Theorem 2. Let $\phi$ be Morse function. Then

1. All stationary points of $\phi$ are isolated and thus there is only a finite number of them (in the ball $B(0,R)$).
2. Near each stationary point $\bar{x}$ there exists a change of variables $y=y(x)$ such that $$\phi (y)=\phi (\bar{x})+ \sum_{1\le j\le d} \lambda_j z_j^2, \qquad \lambda_j=\pm 1. \label{eq-2.2.2}$$
3. Further, $\#\{j: \lambda_j=1\}$ (number of coefficients equal to $1$) equals to number of positive eigenvalues of $\phi''(\bar{x})$ and $\#\{j: \lambda_j=-1\}$ (number of coefficients equal to $-1$) equals to number of negative eigenvalues of $\phi''(\bar{x})$.
4. Finally, absolute value of Jacobian $|\det J(\bar{x})|= |\det \frac{1}{2}\phi''(\bar{x})^{\frac{1}{2}}$ where $J=\Bigl(\frac{\partial z_j}{\partial x_k}\Bigr)_{j,k=1,\ldots, d}$ is a Jacobi matrix.

#### Single non-degenerate maximum

Theorem 3. Let $\phi$ reach its single maximum at $\bar{x}$ and $\nabla\phi(\bar{x})=0$, $\phi''(\bar{x})<0$. Then $$I(k) \sim e^{k\phi (\bar{x})}\sum _{n=0}^\infty \kappa_{2n} k^{-\frac{d}{2}-n} \label{eq-2.2.3}$$ in the sense that $$|I(k)- e^{k\phi (c)}\sum _{n=0}^{N-1} \kappa_{2n}k^{-\frac{d}{2}-n}|\le C_N k^{-N-\frac{d}{2}} e^{k\phi(\bar{x})}. \label{eq-2.2.4}$$ Here the main coefficient is $$\kappa_0=(2\pi)^{\frac{d}{2}}|\det \phi''(\bar{x} )|^{-\frac{1}{2}} f(\bar{x}). \label{eq-2.2.5}$$

Proof. Clearly, without any loss of the generality we can assume that $\bar{x}=0$ and $\phi(\bar{x})=0$. Also in virtue of Theorem 2 without any loss of the generality we can assume that $f(x)$ is supported in $B(0,\epsilon)$ and $\phi(x)=-x_1^2-\ldots x_d^2$. Then \begin{equation*} I(k)= \int e^{-k|z|^2}g(z)\,dz. \end{equation*} In such integral we can assume that $f$ and its derivatives have no more than a polynomial growth and take integral over $\bR^d$. Decomposing $g(z)$ into Taylor series we get after change of variables $y=k^{\frac{1}{2}}z$ that \begin{equation*} I(k)\sim\sum_{\alpha} \frac{g^{(\alpha)}(\bar{x})}{\alpha!} k^{-\frac{1}{2}(|\alpha|+1)} \int e^{-|y|^2} y^\alpha \,dy \end{equation*} where $\alpha=(\alpha_1,\ldots,\alpha_d)\in \bZ^{+\,d}$ is multiindex, $\bZ^+$ is the set of non-negative integers, $|\alpha|:=\alpha_1+\ldots+\alpha_d$, $\alpha!=\alpha_1!\cdots \alpha_d!$, $y^\alpha:= y_1^{\alpha_1}\cdots y_d^{\alpha_d}$.

Then we arrive to decomposition (\ref{eq-2.2.4}) with \begin{equation*} \kappa_0= g(0)\int e^{-|z|^2} \,dz. \end{equation*} Observe that since we integrate over $\bR$ then $\kappa_n=0$ for odd $n$. Also observe that $\kappa_0= g(0)\pi^{d/2}$ and we use Statement (d) of Theorem 2.

#### Multiple maxima

Let now $\phi$ has several maxima on $X$: $x_1,\ldots ,x_K$ each of the type considered above; $\phi(x_1)=\ldots =\phi(x_K)$ (because we are looking only for absolute maxima). Then asymptotics of $I(k)$ is given by the sum of the contributions of all these points.