As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what minimal information on the eigenfunctions one would need to carry out the reconstruction. One answer to this problem was given in Can one count the shape of a drum? (2006).

The "counting" refers to the number $\nu_n$ of nodal domains of the $n$-th eigenvalue $E_n$. (A nodal domain is a connected region in which the eigenfunction has a fixed sign. In one dimension $\nu_n=n$, in higher dimensions $\nu_n\leq n$.) The conjecture in the paper is that the sequence $\{\nu_n, E_n\}$, $n=1,2,\ldots$, is sufficient to recover the shape of the domain. The conjecture has been proven for certain classes of isospectral domains, but there are counter-examples (here is a recent overview).

As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what minimal information on the eigenfunctions one would need to carry out the reconstruction. One answer to this problem was given in Can one count the shape of a drum? (2006).

The "counting" refers to the number $\nu_n$ of nodal domains of the $n$-th eigenvalue $E_n$. (A nodal domain is a connected region in which the eigenfunction has a fixed sign. In one dimension $\nu_n=n$, in higher dimensions $\nu_n\leq n$.) Gnutzmann, Karageorge, and Smilansky conjecture in the cited paper that the sequence $\{\nu_n, E_n\}$, $n=1,2,\ldots$, is sufficient to recover the shape of the domain. The conjecture has been proven for certain classes of isospectral domains (flat tori in three and four dimensions). It does not hold for the discrete Laplacian on a graph (see Isospectral graphs with identical nodal counts).

As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what minimal information on the eigenfunctions one might need to carry out the reconstruction. One answer to this problem was given in Can one count the shape of a drum? (2006).

The "counting" refers to the number $\nu_n$ of nodal domains of the $n$-th eigenvalue $E_n$. (A nodal domain is a connected region in which the eigenfunction has a fixed sign. In one dimension $\nu_n=n$, in higher dimensions $\nu_n\leq n$.) The conjecture in the paper is that the sequence $\{\nu_n, E_n\}$, $n=1,2,\ldots$, is sufficient to recover the shape of the domain.

As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what minimal information on the eigenfunctions one would need to carry out the reconstruction. One answer to this problem was given in Can one count the shape of a drum? (2006).

The "counting" refers to the number $\nu_n$ of nodal domains of the $n$-th eigenvalue $E_n$. (A nodal domain is a connected region in which the eigenfunction has a fixed sign. In one dimension $\nu_n=n$, in higher dimensions $\nu_n\leq n$.) The conjecture in the paper is that the sequence $\{\nu_n, E_n\}$, $n=1,2,\ldots$, is sufficient to recover the shape of the domain. The conjecture has been proven for certain classes of isospectral domains, but there are counter-examples (here is a recent overview).