# Bound states and continua in the spectral function

Okay, let me try hard to pose this question as clear as I can.

Let's take a quantum system where a single charge carrier interacts with a bosonic mode. Examples would be the Holstein model where a tight-binding carrier interacts with dispersionless phonons, or maybe a single carrier in a magnetic lattice where it can excite magnons.

The typical single particle spectral function then has a bunch of generic features: There might be a bound state, appearing as a sharp peak in the spectrum, and there'll also be a continuum of states that are associated with the "free" particle plus a bosonic excitation. For example, in the Holstein model there'll be a sharp peak for the Holstein polaron, at some energy $E_0$, and then a continuum of "polaron + a free phonon" states starting at energy $E_0 + \Omega$ where $\Omega$ is the energy of the bosonic mode (with $\hbar = 1$).

So far, so good, I've worked with the Holstein model and some more complicated variatons of it for quite some time. Now I'm looking at something more complicated, and I'm looking at it in the two-particle sector, and here's a simple question:

When I see a continuum in the spectrum, should there always be a simple interpretation for that continuum? As in, I look at where the continuum starts and then I can tell immediately: "Ah yes, this is the continuum of two independent single-carrier bound objects" or "Yes, this is the continuum of ground-state-plus-one-bosonic-excitation states"? Or could there be continua that are more complicated in nature and aren't amenable to a straightforward interpretation?

I'm asking because I'm trying to decide whether something my simulation spits out is a bug or a feature... I see a continuum of states well below where the "two independent single-particle-bound-states" continuum would lie, but below this continuum I'd only expect a bound state...

• If I understand the question correctly, I am afraid it is not simple to characterize continuum spectrum of operators. In general, you may have eigenvalues (corresp. to bound states) embedded in the continuum spectrum. Also, you have a distinction between the absolutely continuous spectrum and singular continuous spectrum (but maybe this is not so interesting to you). It really depends on the form of your Hamiltonian. For example, the Hamiltonian $H=-\partial^2_x +(-i\partial_y +x)^2$ has eigenvalues (of infinite multiplicity) embedded in the continuum spectrum. – yuggib Jul 9 '14 at 14:46
• I am not familiar with your model. But some time ago I also had to filter bound states out of extended states. Here is my method: journals.aps.org/pra/abstract/10.1103/PhysRevA.87.023613 – Jiang-min Zhang Jul 9 '14 at 18:16
• @Lagerbaer Could you please give some references about typical single particle spectral function with the features you mentioned? Thanks. – ZJX Nov 22 '17 at 7:16