Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically decreasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous functions on $M$.

Does there exist/can one construct:

• A smooth Riemannian metric $g$ on $M$,
• A self-adjoint (wrt. a positive density $\mu$ on $(M,g)$) second-order elliptic operator $T$ on $(M,g)$ with smooth coefficients.

Such that the eigendecomposition of $T$ on $L^2(M,g)$ is $\{(f_k,\lambda_k)\}_{k=1}^{\infty}$?

If not, are there known sufficient conditions on $\{f_k,\lambda_k)\}_{k=1}^{\infty}$ for this to be possible?

Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous functions on $M$.

Does there exist/can one construct:

• A smooth Riemannian metric $g$ on $M$,
• A self-adjoint (wrt. a positive density $\mu$ on $(M,g)$) second-order elliptic operator $T$ on $(M,g)$ with smooth coefficients.

Such that the eigendecomposition of $T$ on $L^2(M,g)$ is $\{(f_k,\lambda_k)\}_{k=1}^{\infty}$?

If not, are there known sufficient conditions on $\{f_k,\lambda_k)\}_{k=1}^{\infty}$ for this to be possible?

Let $(M,g)$ be a closed Riemannian manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically decreasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be an orthonormal basis in $L^2(M)$ (defined wrt. the Riemannian volume measure on $(M,g)$).

Does there exist/can we construct a second-order elliptic operator $T$ on $M$ with smooth coefficients?

What if we are allowed to perturb $g$?

Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically decreasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous functions on $M$.

Does there exist/can one construct:

• A smooth Riemannian metric $g$ on $M$,
• A self-adjoint (wrt. a positive density $\mu$ on $(M,g)$) second-order elliptic operator $T$ on $(M,g)$ with smooth coefficients.

Such that the eigendecomposition of $T$ on $L^2(M,g)$ is $\{(f_k,\lambda_k)\}_{k=1}^{\infty}$?

If not, are there known sufficient conditions on $\{f_k,\lambda_k)\}_{k=1}^{\infty}$ for this to be possible?