"Physically distinct" Hermitian operators with the same eigenspaces

Question

I understand that Hermitian operators can be decomposed in terms of their eigenbasis: \begin{equation} H = \sum_i\lambda_i|i\rangle\langle i| \end{equation} where the $\lambda_i$ are all real. I've kept the summation index ambiguous since I am interested in both finite- and infinite-dimensional operators. My question is, does anyone know any examples of two "physically relevant" Hermitian operators (e.g meaningful observables, or Hamiltonians for some real-world systems) that have the same eigenspaces but not all the same eigenvalues? That is: \begin{equation} H_1 = \sum_i\lambda_i|i\rangle\langle i|;\;\;\;H_2 = \sum_{i}\mu_i|i\rangle\langle i| \end{equation} where $\lambda_i\neq\mu_i$ for at least one $i$. Ideally $H_1$ and $H_2$ are also "physically distinct," e.g not just two differently scaled versions of the same system. For example, one could assemble an operator $H_{HO}'$ with the same eigenstates as the harmonic oscillator Hamiltonian $H_{HO}$, but with $H_{HO}'|n\rangle = n^2|n\rangle$ or $\frac{1}{n^2}|n\rangle$ or something. $H_{HO}'$ then commutes with $H_{HO}$, but I feel that they correspond to very different physics.

I'd appreciate any examples of pairs of operators as described above, or a more general/abstract statement on such pairs of operators if anyone knows one.

Answer 1

Let $A$ be a self-adjoint operator with spectral decomposition $$A = \int_{\sigma_A} \lambda \ \mathrm dP^A(\lambda)$$ which reduces to your expression $\sum_i \lambda_i |i\rangle\langle i|$ in the case that the spectrum is purely discrete and all of the eigenspaces are 1D. Given some (Borel-measurable) function $f(A)$, we can define $$f(A) := \int_{\sigma_A} f(\lambda) \ \mathrm dP^A(\lambda) \rightsquigarrow \sum_i f(\lambda_i) |i\rangle\langle i|$$

$A$ and $f(A)$ satisfy the requirements you're looking for. Certainly one ubiquitous example would be the position operator $X$ and any arbitrary potential operator $V(X)$, with the caveat that these operators are a bit subtle because of their continuous spectra. The Hamiltonian operator $H$ and the time-evolution operator $U=\exp[-iHt/\hbar]$ are another example, in both finite and infinite dimensions.

Answer 2

This is intended as an extended comment: let $A$ and $B$ two hermitian matrices sharing a common eigenbasis $(\phi_i)_i$, such that each of the multiplicities of the eigenvalues ($(\lambda_i)_i$ for $A$ and $(\mu_i)_i$ for $B$) is one. Then not only do $A$ and $B$ commute: each of them is a polynomial in the other one.

Indeed, there are, from Lagrange interpolation, two polynomials $P$ and $Q$ such that for all $i$, $P(\lambda_i) = \mu_i$ and $Q(\mu_i) = \lambda_i$. Therefore, $P(A) = B$ and $Q(B) = A$.

So, both $A$ and $B$ can be seen as relabellings of each other, and in some loose sense, they can be thought as equivalent.

Of course, if the eigenvalues have a dimension, it may not be meaningful to feed them to nonmonomial polynomials, so at some point physics may consider $A$ and $B$ as inequivalent.