How to normalize resonant states?

What is not provided in (11) is a formula for the coefficients $c_j$ (the formulæ for $a_j$'s an $b_j$'s are the same as in (6)). If $u_j(x)$ is a resonant state then the assumption of compact support on initial data in (7) shows that the integrals

$$\int_\mathbb{R}u_j(x) w_0(x) dx, \ \int_\mathbb{R}u_j(x) w_1(x) dx,$$

are well defined. However it is not at all clear what normalization for $u_j$'s should be chosen.

One possibility is to use the Lax-Phillips semigroup, $Z(t)$ - see [25],[23], and also [31] for a concise treatment in a very general setting. The semigroup $Z(t)$ not only provides the most elegant way of defining resonances, but is in fact the only purely dynamical definition: resonances are simply the eigenvalues of its generator. However, it is perhaps the beauty of this definition that makes its applicability limited (even in one dimension some modifications have to be made [22]). It is also rather cumbersome numerically.

Instead, let us go back to Green's function,

$$G_V(\lambda, x, y) = G_V(\lambda, y, x) = \overline{G_V(-\bar \lambda, y, x)}.$$ (13)

Near a simple pole, that is near a simple resonance, $\lambda_j$, we can write

$$G_V(\lambda, x, y ) = \frac{1}{2 \lambda_j} \frac{ u_j(x) v_j(y) }{ \lambda - \lambda_j } + T_{V, \lambda_j}(\lambda, x, y),$$

where $T_{V, \lambda_j}$ is holomorphic near $\lambda_j$. The symmetry recalled in (13) shows that $u_j(x) = c v_j(x)$ in (14). Replacing $u_j$ by $u_j(x) / \sqrt{c}$ we obtain

$$G_V(\lambda, x, y) = \frac{1}{2 \lambda_j} \frac{ u_j(x) u_j(y) }{ \lambda - \lambda_j } + T_{V, \lambda_j}(\lambda, x, y),$$ (14)

with the resonant state $u_j(x)$ uniquely determined up to sign.

Suppose for simplicity that $w_0(x) = 0$ in the initial value problem (7). Then in the expansion (11) we can choose the resonant functions to be given by $u_j$'s and we have

$$c_j = i \int_\mathbb{R}w_1(x) u_j(x) dx, u_j(x)$$ (15)

That provides a preferred choice of a resonant state but not a normalization.

To find a normalization of $u_j(x)$ we observe that (9) and (10) shows that we can continue $u_j(x)$ analytically to

$$\mathbb{C}\setminus ([-L, L] \cup e^{i \theta} \mathbb{R}),$$

for any $\theta$. We denote the continuation by $u_j(z)$. If $\Gamma_\theta \subset \mathbb{C}$ is a $C^1$ curve with the property that

$$\begin{array}{c} \Gamma_\theta \cap \{z \;:\; \vert z\vert \l... ...ta} \mathbb{R}\cap \{\vert z\vert > (1 + C_0) L \}, \end{array}$$ (16)

for some $C_0 > 0$, then

$$-\arg \lambda_j < \theta < \pi - \arg \lambda_j \implies \int_{\Gamma_\theta} \vert u_j(z)\vert^2 \vert dz\vert < \infty.$$ (17)

It turns out that the normalization which determines $u_j$ in (14) and (15) is

$$\theta -\arg \lambda_j < \theta < \pi - \arg \lambda_j \int_{\Gamma_\theta} u_j(z) ^2 dz = 1 .$$ (18)

Contour deformation shows that this condition is independent of the choice of $\theta$ and of the contour $\Gamma_\theta$.

We cannot fully explain the origin of the normalization given in (18). It comes from the celebrated complex scaling method discovered by Aguilar-Combes and Balslev-Combes in the early 70s, and greatly developed in mathematics, computational physics, and chemistry - see [30] and [35] for a discussion and references.

Roughly speaking, we can restrict the operator $H_V$ to the contour $\Gamma_\theta$. It then acts (as an unbounded operator) on $L^2(\Gamma_\theta)$ rather than $L^2(\mathbb{R})$. The resonances with $-\arg \lambda_j < \theta$ are the eigenvalues of this new non-self-adjoint operator, denoted by $H_{V,\theta}$. Its resolvent,

$$(H_{V,\theta} - \lambda^2)^{-1} \;:\; L^2(\Gamma_\theta) \; \longrightarrow \; L^2(\Gamma_\theta) ,$$

has poles at eigenvalues and since we assumed that they are simple,

$$\frac{1}{2 \pi i} \operatorname{tr}\oint_{\vert\lambda - \lambda_... ..., \theta} - \lambda^2)^{-1} \; 2 \lambda d\lambda = 1, \ 0 < \delta \ll 1.$$

The resolvent can be obtained by an analytic continuation of the Green's function. Evaluation of the trace for the continuation gives the normalization (18).