If we know Spec($M_1$) and Spec($M_2$), what could we say about Spec($M_1 \cup M_2$)?

Question

Let two domains $M_1$ and $M_2$ (Dirichlet conditions). If we know the spectrum of the Laplacian on $M_1$ and $M_2$, what could we say about Spect($M_1 \cup M_2$)? Is there a theorem that might give us some information about it?

Answer 1

Let $\{\lambda_k\}$ be the eigenvalues of $M_1\cup M_2$; similarly, let $\{\lambda_k^{(j)}\}$ be the eigenvalues of $M_j$, $j=1,2$; in both cases arrange them in non-decreasing order. The domain monotonicity of Dirichlet eigenvalues implies that $$\lambda_k\ge \max(\lambda_k^{(1)}, \lambda_k^{(2)})\tag1$$ for every $k$. Equality holds in $(1)$ when $M_1=M_2$.

There is no upper bound for $\lambda_k$ in terms of $\lambda_k^{(j)}$. Indeed, let $M_1$ and $M_2$ be very thin spiral-like domains such that their union is a disk of radius $1$. Then the eigenvalues $\lambda_1^{(j)}$ can be made arbitrarily large while $\lambda_1$ remains the same.