N−VPotential in the Lee Model

An investigation is made of states in the Lee model representing a "physical" $V$-particle bound to an $N$-particle. It is found that the energy eigenvalue problem can be reduced to a single transcendental equation involving the parity, energy, renormalized coupling, and $N-V$ separation. Only states of odd parity are considered. For such states this equation leads to a single-valued real potential if the renormalized coupling constant is less than the critical value for the appearance of the so-called "ghost" states of the free physical $V$-particle. For stronger couplings the potential becomes many-valued with half its real branches corresponding to the presence of "normal" $V\text{'}\mathrm{s}$, and half to "ghost" $V\text{'}\mathrm{s}$. For still stronger coupling no real energies exist. It is shown that complex energies appear, at least in the case where the virtual $\theta$ particles are assumed nonrelativistic and the $N-V$ separation is sufficiently small. A possible reason for the appearance of these difficulties is suggested.