Suppose we have two eigenfunctions $f_n(x,y)$ and $f_m(x,y)$ and corresponding eigenvalues $\lambda_n<\lambda_m$ of a differential operator $L$. How can we determine whether there exists another eigenvalue $\lambda_i$ between these given, i.e. $\lambda_n<\lambda_i<\lambda_m$, if the problem is 2D and higher?

I know that one could just count zeros of the eigenfunctions in 1D case, but according to Courant, Hilbert "Methods of Mathematical Physics Vol. 1" $\S\mathrm{VI}.6$, this is not generally true for higher dimensions:

In the case of eigenvalue problems of partial differential equations, arbitrary values of $n$ may exist for which the nodes of the eigenfunctions $u_n$ subdivide the entire fundamental domain into only two subdomains.



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