Maybe the simplest non-trivial settings in which the spectrum of the Laplacian be can be computed is on the round sphere $\mathbf{S}^n$, and for products of manifolds. I want to use the two as examples to situate my question, which goes as follows.
Question. Can the first few eigenvalues of the manifold \begin{equation} \mathbf{S}^n \quad \text{or} \quad (M \times N,g \oplus h) \end{equation} say $\lambda_1 \leq \cdots \leq \lambda_D$ be computed 'directly', using an argument that does not involve the other, higher eigenspaces?
I would be happy with an argument that produces the first $D = 5$ eigenvalues for example in either setting. To be clear, in the latter case one would assume knowledge of the spectra of $M$ and $N$. The question arose in a more complicated setting, in which the larger eigenvalues seem inaccessible even a posteriori.
To explain potential difficulties, let me point out a step that the two computations have in common. (I am following the book of Berger, Gauduchon and Mazet [1].) Both rely on the Stone-Weierstrass theorem, combined with the following lemma [Lemma A.II.1 on p.143,1].
Lemma. Let $(M,g)$ be a compact manifold and $(V_i \mid i \in \mathbf{N})$ be a collection of vector subspaces of $C^\infty(M)$ so that for all $i \in \mathbf{N}$ there is $\lambda_i \in \mathbf{R}$ for which $\Delta \varphi = \lambda_i \varphi$ for all $\varphi \in V_i$. If $\sum_i V_i$ is $L^2$-dense in $C^\infty(M)$ then the spectrum of $(M,g)$ is $(\lambda_i \mid i \in \mathbf{N})$ and the $V_i$ are its eigenspaces.
I want to avoid going into too much detail, but just quickly illustrate how this is used in the latter example, to compute the spectrum of the product manifold $(M \times N,g \oplus h)$ [Prop. A.II.3 on p. 144,1]. Basically this states that its eigenspaces all factor into \begin{equation} \mathcal{E}_\nu(M \times N) = \mathcal{E}_\lambda(M) \otimes \mathcal{E}_\mu(N) \quad \text{with $\nu = \lambda + \mu$.} \end{equation} The first inclusion $\mathcal{E}_\nu(M \times N) \supset \mathcal{E}_\lambda(M) \otimes \mathcal{E}_\mu(N)$ is easy. It is the reverse inclusion that uses Stone-Weierstrass and the lemma cited above. A by-product of this is that the relation above is simultaneously established for all eigenvalues.
[1] Berger, M., Gauduchon, P., Mazet, E. Le spectre d'une variete riemannienne. Lect. Notes Math. 194 (1971).
