I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\sum_{i=1}^{k}y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_{k},$$ where this notation means that I want to integrate over $\mathbb{R}^k$ restricted to the plain where $\sum_{i=1}^{k}y_i=x$ (a convolution of gaussians) and $N$ and $r$ are positive real constants. I have tried two different methods for computing this integral, but they are yielding different results. I would appreciate it very much if someone could take a look and tell me what I'm doing wrong.

**Method 1**

In method 1 I just wrote it as $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\sum_{i=1}^{k}y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_{k}=\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty} e^{-N^2r((x-y_1)^2+\sum_{i=1}^{k-2}(y_i-y_{i+1})^2+y_{k-1}^2-\frac{1}{k}x^2)} dy_1\dots dy_{k-1}=\sqrt{\frac{1} {\pi r^{k-1}k}}\frac{\pi^{k}}{N^{k-1}}$$

I deduced this formula by induction, first integrating in $y_{k-1}$, then $y_{k-2}$ and so on.

**Method 2**

In method 2 I tried writting the function in a matrix form $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\sum_{i=1}^{k}y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_{k}=\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\vec{y},Q\vec{y})} dy_1\dots dy_{k}$$ where \begin{equation} Q:=\left(\begin{array}{cccccccc} (1-\frac{1}{k})& -\frac{1}{k} & -\frac{1}{k} & \cdots & -\frac{1}{k} \\ -\frac{1}{k} & (1-\frac{1}{k}) & -\frac{1}{k} & \cdots & -\frac{1}{k} \\ \vdots & \ddots & & &\vdots \\ -\frac{1}{k} & \dots & &-\frac{1}{k} &(1-\frac{1}{k}) \end{array}\right) \end{equation}.

This matrix $Q$ has eigenvalues $\lambda_0=0$, $\lambda_l=1$ and corresponding normalized eigenvetors \begin{equation} \vec{\lambda}_l=\frac{1}{\sqrt{k}}\left(\begin{array}{c} 1 \\ e^{\frac{2\pi i}{k}1l} \\ \vdots \\ e^{\frac{2\pi i}{k}(k-1)l} \\ \end{array}\right) \end{equation} for $0\le l\le k-1$.

As I understand it, the restriction in the integral means that I shouldn't integrate in the $\lambda_0$ direction, since in this direction I must have all components equal, and the only place where the components are equal and the bound is satisfied is $(\frac{x}{k},\dots,\frac{x}{k})$. So my integration should occour in the orthogonal complement of this vector, which is a hyperplane of dimension $k-1$. Everything seems to check to this point, so I diagonalized the matrix $Q=U\Lambda U^{-1}$ and so

$$(\vec{y},Q\vec{y})=(\vec{\xi},\Lambda\vec{\xi})=\sum_{i=1}^{k-1}\xi_i^2.$$

The change of variables $\vec{\xi}=U^{-1}\vec{y}$ has a Jacobian $\frac{1}{\sqrt{k^{k-1}}}$, since $U^{-1}$ is the DFT matrix times $\frac{1}{\sqrt{k^{k-1}}}$ and the DFT matrix is known to be unitary. So

$$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\vec{y},Q\vec{y})} dy_1\dots dy_{k}=\idotsint\limits_{\mathbb{R}^k} e^{-N^2r\sum_{i=1}^{k-1}\xi_i^2} \frac{1}{\sqrt{k^{k-1}}}d\xi_1\dots d\xi_{k-1}= \sqrt{\frac{\pi^{k-1}}{k^{k-1}r^{k-1}}}\frac{1}{N^{k-1}}$$.

These two results are different and I cannot figure out why.

Thank you all in advance for your help!