Let $M$ be a manifold with smooth boundary. We can consider the Dirichlet or the Neumann problem on $M$. Let $(\phi\_k)$ be an orthonormal basis of eigenfunctions to the Dirichlet problem and let $(\psi\_k)$ and orthonormal basis of eigenfunctions to the Neumann problem.


What can be said about the functions $\partial\_{\mathbf{n}}\phi\_k|\_{\partial M}$ and $\psi\_k|\_{\partial M}$? For example, does a subsequence of them constitute an orthonormal basis of $L^2(\partial M)$? Are they solutions to some PDE?



Boundary values of boundary value problems
Consider the Dirichlet eigenfunctions. Let $\Delta \phi\_k + \lambda\_k^2\phi\_k=0$, where $0<\lambda\_1\leq \lambda\_2\leq \lambda\_3\dots$, and $\displaystyle u\_k = \frac{\partial \phi\_k}{\partial n}\bigr|\_{\,\partial M}$. Here is a quotation from [Bäcker, Fürstberger, Schubert, and Steiner](http://arxiv.org/abs/nlin/0207034) when $\dim M=2$:



> 
> Our results show that the mean behaviour of the normalized boundary functions is very similar to the mean behaviour of eigenfunctions. The crucial difference between the two sequences of functions $\{\phi\_k\}\_{k\in\mathbb N}$, $\{u\_k\}\_{k\in\mathbb N}$ is that the eigenfunctions live on a two–dimensional space whereas the boundary functions live on a one–dimensional space. Since both $u\_k$ and $\phi\_k$ oscillate roughly with the same de Broglie wave length $2π/\lambda\_k$, this leads to an overcompleteness of the set $\{u\_k\}\_{k\in\mathbb N}$. This statement can be made more explicit $\ldots$
> 
> 
> More precisely, this means that for every $w \in C^\infty(\partial M)$ 
>  $$ 
>  w(s) = \sum\_{k\in\mathbb N} \rho(\lambda-\lambda\_k) w\_k u\_k(s) +
>  O\left(1/\lambda\right) \tag{54}
> $$
>  holds with coefficients 
>  $$
>  w\_k := \frac{\pi}{2\lambda\_k^2}\int\_{\partial M} w(s)u\_k(s)\,ds. \tag{55}
> $$ 
>  This follows from ($\ldots$) by the method of stationary phase (see $\ldots$). Since $\rho$ is a rapidly decreasing function, this means that the boundary functions with spectral parameter $\lambda\_k$ in an interval of fixed width around $\lambda$ form a complete set in the limit $\lambda\to\infty$. The number of these states grows like
>  $\lambda$, in contrast to the number of all states up to energy $\lambda^2$, which, according to the Weyl formula, grows like $\lambda^2$. Therefore this result gives a quantitative measure of the overcompleteness of the set $\{u\_k\}$.
> 
> 
> 


In formulas (54), (55), $s$ is the generic point in the boundary $\partial M$,
while $\rho\in S(\mathbb R)$ has Fourier transform which is even and compactly supported in a small neighborhood of $0$. Also the notation has been slightly adjusted (e.g., $\lambda\_k$ in place of $k\_n$ which is used by BFSS)


Similar results hold for the Neumann eigenfunctions and in higher dimensions, see e.g. [Hassell and Barnett](http://arxiv.org/abs/1107.2172).


