Resonances as poles of the S-matrix

In the present Section we recall the standard theory of the S-matrix and introduce the so-called virtual states, which may appear in the single-particle phase space for small energies, and therefore are important for the discussion of pairing correlations in weakly bound systems, see Sec. 5.

Poles of the S-matrix can be located in four different regions of the complex k-plane, corresponding to four regions of the two-sheet complex energy surface [65] (see Fig. 2). The position of a pole determines both the behavior of the respective wave function and the physical interpretation of the solution. The first region corresponds to the positive imaginary k-axis. The wave functions in this region are normalizable, negative energy solutions of the Schrödinger equation and correspond to the bound states of the system. The second region is the negative imaginary k-axis. Here solutions of the Schrödinger equation are not normalizable (they are exponentially diverging) and, hence, they are not physical. These solutions correspond to negative energies on the unphysical sheet of the energy surface and they are said to be virtual or antibound (see Fig. 2c). The third region is the sector between the positive, real k-axis and the bisection of the fourth quadrant. Asymptotically, the solutions in this region, which correspond to the resonant states with the complex energy, are oscillating and exponentially decreasing functions. The imaginary part of the energy which is negative in this case, is interpreted as a width of the state. Finally, poles located in the remaining region of the complex k-plane, are also said to be virtual.

Figure 2a presents different regions of the complex k-plane where the poles of the S-matrix are located, and the corresponding regions on the two sheets of the energy surface (Figs. 2b and c). The two sheets are connected along the real positive semiaxis. The arrows in Fig. 2 represent the movement of poles, which results from decreasing the depth of the potential well. In the general case, the poles corresponding to a bound state and to an antibound state move pairwise and cross at a given point (denoted $\kappa$ in Fig. 2a). For $L\neq 0$, this point is situated at the origin. For L=0, $\kappa$ can be found lower on the imaginary axis and the determination of its position is in general not trivial (see [65]). After the crossing, the two poles move pairwise on the lower part of the complex k-plane. One considers that they are associated with the resonance phenomenon, once the pole on the right half-plane has crossed the dashed line: $\Re(k)=-\Im(k)$ (see Fig. 2c)), so that one can interpret its complex energy ( $\Re(\epsilon)>0$, $\Im(\epsilon)<0$) as the energy and the width of a resonance, respectively. Such poles are situated on the unphysical energy sheet but the lower one can influence the positive-energy solution on the real positive energy axis.

Let us now discuss how the poles of the S-matrix are related to the widths of resonances. For that, let us consider the short ranged potential V(R) which tends to zero sufficiently fast when $R \rightarrow \infty$. The asymptotic ( $R \rightarrow \infty$) solution of the Schrödinger equation can be written as:

 

$${\Psi}(R \rightarrow \infty) A(k)\exp (-ikR) + B(k) \exp (+ikR)$$ =

$$\simeq \exp (-ikR) + S(k) \exp (+ikR) .$$ (10)

We will be interested in the poles of S(k)when A(k) vanishes and we shall consider only those poles which are embedded in the fourth quarter ($\Re(k)>0$, $\Im(k)<0$) of the complex k-plane, i.e., those which are associated with the resonance phenomenon. Near the isolated i^th pole, S(k) can be written as:

 

$$\frac{d\left(\ln S(k)\right)}{dk} = -\frac{1}{k-k_i} ,$$ (11)

where k_i is the complex pole.

The number of poles in the quarter $\Re(k)>0$, $\Im(k)<0$, can be found following the residue theorem:

 

$$N=\frac{1}{2\pi i} {\oint}_{\cal C}^{} \frac{\partial \ln S(k)}{\partial k} dk .$$ (12)

Consequently,

 

$$\frac{\partial N}{\partial k} = (2\pi i)^{-1} \frac{\partial \ln S(k)}{\partial k} .$$ (13)

The density of states:

 

$$\rho = \left( \frac{\partial N}{\partial k} \right) \left( \frac{\partial k}{\partial \epsilon} \right)^{-1} ,$$ (14)

can be expressed then as follows:

 

$$\rho = (2\pi i)^{-1} \frac{\partial \ln S(k)}{\partial \epsilon} .$$ (15)

Inserting (11) into (15), it is then easy to see that the level density has a local maximum whenever:

 

$$k = \Re (k_i) ~~~ \ , ~~~ i = 1, \cdots, N$$ (16)

The value of density at the maximum is:

 

$$\rho (k=\Re (k_i)) = -\frac{1}{2\pi \Im (\epsilon_i)} ,$$ (17)

and corresponds to the complex energy:

 

$$\epsilon_i-\epsilon_{th}=\frac{({\hbar}k_i)^2}{2m} ,$$ (18)

where $\epsilon_{th}$ is the threshold energy. The density peaks have the Lorentzian shape and the full-width half-maximum of i^th peak is given by:

 

$${\Gamma}_i=\frac{1}{\pi \rho (k=\Re (k_i))} = -2\Im (\epsilon_i) .$$ (19)

Following this simple example, we assume that the poles of S-matrix on nonphysical energy sheets near the real axis correspond to almost all resonance states. Unfortunately, this highly plausible assertion remains only a hypothesis because the relation between resonances and the S-matrix poles is not determined so rigorously as the correspondence between the bound states and the S-matrix poles on the real axis of the first energy sheet. To affirm the correspondence between the observed resonances and the S-matrix poles on nonphysical sheets, certain conditions should be satisfied. First of all, the potential has to be sufficiently analytic and has to fall-off sufficiently rapidly at $R \rightarrow \infty$, so that the corresponding S-matrix can be safely continued to the unphysical sheets. These conditions are satisfied for the PTG potential, though the examples of potentials where the analytic continuation in the eigenvalue problem brings around the redundant solutions are known as well [66,67]. In general, it is safe to speak about the resonance phenomenon in practical problems if the width is not large, i.e., $\Gamma_{nLj}/\epsilon_{nL j}<1$ or, in other words, if the distance of the resonance pole from the physical region is small. This latter condition, as we shall show in sect. 4, is never satisfied in the PTG potential for low-lying, near threshold resonances.