Let $D\subset \mathbb R^2$ be a region and let $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ be the set of eigenvalues of the Laplacian $-\Delta$ (with boundary condition $\psi=0$ on $\partial D$).

Mark Kac, in his famous article *Can One Hear the Shape of a Drum?*, only considers asymptotics of the function $\sum_{\lambda\in\Lambda}e^{-\lambda t}$ as $t\to0^+$, hence cannot detect if finitely many $\lambda_i$'s are modified. This lead me to wonder:

Can there be two planar regions $D_1$ and $D_2$ with Laplace eigenvalues $\Lambda_1$ and $\Lambda_2$, respectively, such that there exists a $\lambda>0$ such that $\Lambda_1\cap\mathbb R_{>\lambda}=\Lambda_2\cap\mathbb R_{>\lambda}$ but $\Lambda_1\ne\Lambda_2$? In other words, can two regions be "almost" isospectral without actually being isospectral?

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