Nonlocality and Infinite LDOS in Lossy Media

While I have written many posts on this blog about various topics in physics or math unrelated to my graduate work as well as posts promoting papers from my graduate work, it is rare that I've written direct technical posts about my graduate work. It is even more unusual that I should be doing so 2 years after leaving physics as a career. However, I felt compelled to do so after meeting again with my PhD advisor (a day before the Princeton University 2020 Commencement, which was held in person after a delay of 2 years due to this pandemic), as we had a conversation about the problem of infinite local density of states (LDOS) in a lossy medium.

Essentially, the idea is the following. Working in the frequency domain, the electric field produced by a polarization density in any EM environment is \( E_{i}(\omega, \vec{x}) = \int G_{ij}(\omega, \vec{x}, \vec{x}')P_{j}(\omega, \vec{x}')~\mathrm{d}^{3} x' \) which can be written in bra-ket notation (dispensing with the explicit dependence on frequency) as \( |\vec{E}\rangle = \hat{G}|\vec{P}\rangle \). The LDOS is proportional to the power radiated by a point dipole and can be written as \( \mathrm{LDOS}(\omega, \vec{x}) \propto \sum_{i} \mathrm{Im}(G_{ii}(\omega, \vec{x}, \vec{x})) \). This power should be finite as long as the power put into the dipole to keep it oscillating forever at a given frequency \( \omega \) is finite. However, there appears to be a paradox in that if the position \( \vec{x} \) corresponds to a point embedded in a local lossy medium, the LDOS diverges there.

I wondered if an intuitive explanation could be that loss should properly imply the existence of energy leaving the system by traveling out of its boundaries, so the idea of a medium that is local everywhere (in the sense that the susceptibility operator takes the form \( \chi_{ij}(\omega, \vec{x}, \vec{x}') = \chi_{ij}(\omega, \vec{x})\delta^{3} (\vec{x} - \vec{x}') \) at all positions) and is lossy at every point in its domain may not be well-posed as energy is somehow disappearing "into" the system instead of leaving it. Then, I wondered if the problem may actually be with locality and whether a nonlocal description of the susceptibility could help. This is where my graduate work could come in. Follow the jump to see a very technical sketch of how this might work (as I won't work out all of the details myself).