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

2.2. Laplace integrals. II. Multidimensional theory

1. Morse theory
2. Single non-degenerate maximum inside
3. Multiple maxima

We consider \begin{equation} I(k)= \int_X e^{k \phi(x)}f(x)\,dx \label{eq-2.2.1} \end{equation} 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 Morse function $\phi^* \in C^\infty (X)$ 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 \begin{equation} \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} \end{equation}
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 $\nabla\phi(\bar{x})=0$, $\Hess \phi (\bar{x})<0$. Then \begin{equation} I(k) \sim e^{k\phi (\bar{x})}\sum _{n=0}^\infty \kappa_{2n} k^{-\frac{d}{2}-n} \label{eq-2.2.3} \end{equation} in the sense that \begin{equation} |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} \end{equation} Here the main coefficient is \begin{equation} \kappa_0=(2\pi)^{\frac{d}{2}}|\det \Hess \phi (\bar{x} )|^{-\frac{1}{2}} f(\bar{x}). \label{eq-2.2.5} \end{equation}

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.

Degenerate maxima

The structure of the degenerate maxima could be rather complicated in dimensions $m >1$, albeit more simple than the structure of the general stationary point. If we consider non-zero coefficients in the Taylor decomposition and mark in $\bZ_+^m$ corresponding powers, then Newton polyhedra are very handy. See Newton polytop.

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