I am familiar with the notion of infinite-dimensional linear operators from quantum mechanics, such as the Hamiltonian $\hat H = -\frac{1}{2m} \frac{\partial^2}{\partial x^2}$ which has eigenstates $e^{ipx}$ and eigenvalues $\frac{p^2}{2m}$ , for $-\infty < p < \infty$ .
Now for Hermitian matrices knowing the eigenvalues determines the matrix up to a unitary similarity transformation, or in other words two finite-dimensional Hamiltonians with exactly the same eigenvalues will describe the same physical system, since we can map the eigenstates of one to the other with a unitary transformation, and the time evolution will be the same because the eigenvalues are the same.
How we can determine if two infinite-dimensional Hermitian operators with a continuous spectrum are similar ?
e.g. if we know that some other Hamiltonian has eigenvalues $0 < E < \infty$ , how can we tell if it describes the same physical system as the one above ?
More precisely is there a way to describe the spectrum such that we can tell if two operators are similar just by comparing their spectra (without having to know what the eigenvectors are) ?
