Extend scalar Lanczos to off‑diagonal correlators for overlaps and matrix elements

Develop a rigorous extension of the scalar Lanczos analysis to treat overlap factors and external‑current matrix elements extracted from off‑diagonal correlation‑function entries C_{ab}(t) with a ≠ b, ensuring physically consistent handling of non‑positive Ritz values and sign conventions when analyzing off‑diagonal correlators.

Background

In the paper’s noiseless demonstrations, the authors limit scalar Lanczos applications to diagonal correlator entries because off‑diagonal entries can produce non‑positive Ritz values that complicate interpretation. Although scalar Lanczos can, in principle, be applied to off‑diagonals for spectroscopy, the formal machinery to consistently extract overlaps and matrix elements from off‑diagonal entries is not yet available.

The authors explicitly indicate that proper treatment of overlaps and matrix elements for off‑diagonal Lanczos requires nontrivial extensions to their previous scalar formalism, and they defer this development.