Multichannel scattering for the Schrödinger
equation on a line with different thresholds at both infinities

The scattering problem for a one-dimensional matrix Schrödinger equation is a classical problem of quantum theory. The exhaustive treatment of general properties of such scattering on semiaxis, in particular, the proof of unitarity of the $S$-matrix in the presence of closed scattering channels, is given in the book [1], Chap. 17. As far as multichannel scattering on the whole line is concerned, this problem was investigated in many works especially in regard to the inverse scattering problem and the construction of exact solutions to the hierarchies of integrable nonlinear partial differential equations [2]. Nevertheless, to our knowledge, the description of properties of the $S$-matrix, of the Jost solutions, and of the bound states in the general case of multichannel scattering on a line with different thresholds at both left and right infinities is absent in the literature. Our aim is to fill this gap.

The study of the analytical structure of the Jost solutions and the $S$-matrix for one-channel scattering on the whole line in relation to the inverse scattering problem was being conducted already at the end of the 50s [3, 4, 5, 6]. As for the relatively recent papers regarding one-channel scattering including scattering on potentials with asymmetric asymptotics at left and right infinities, see, e.g., [7, 2, 8, 9, 10, 11, 12]. The general properties of two-channel scattering for both identical and distinct thresholds were considered in [13, 14, 15, 16], the thresholds being the same at both infinities. In the papers [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], these results were generalized to multichannel scattering on a line for the case when all the reaction thresholds coincide at both left and right infinities. The multichannel scattering problem with different thresholds identical at both infinities was investigated in [27, 28]. In the works [2, 29, 30], the multichannel scattering problem for systems of a Hamiltonian type with matrix potentials vanishing at both infinities was studied. However, unitarity of the $S$-matrix in the presence of closed channels was not proved in these papers. In the present paper, we prove unitarity of the scattering matrix for a stationary matrix Schrödinger equation on a line in the general case where the reaction thresholds do not coincide at both infinities and some of the channels are closed. Furthermore, we obtain the other relations connecting the transmission and reflection matrices for open and closed channels.

For such a scattering problem, we describe the analytical structure of the Jost solutions and of the transition matrix relating the bases of the Jost solutions. We prove the necessary and sufficient condition specifying the positions of the bound states. The form of this condition is well-known for multichannel scattering (see, e.g., [2, 17, 19, 26]). However, we show that this condition also holds for the scattering problem with different thresholds at both left and right infinities.

The paper is organized as follows. In Sec. 2, the analytical properties of the Jost solutions are described. Sec. 3 is devoted to analytical properties of the transition matrix. In Sec. 4, the basic identities for the scattering matrix are discussed. In Sec. 5, the relations between the transmission and reflection matrices in open channels are investigated. Sec. 6 is devoted to the proof of unitarity of the $S$-matrix in open channels. In Sec. 7, we discuss the necessary and sufficient condition determining the location of bound states. In Sec. 8, we obtain the asymptotics of the Jost solutions and of the transition matrix for a large spectral parameter. In Conclusion, we summarize the results. The summation over repeated indices is always understood unless otherwise stated. Furthermore, wherever it does not lead to misunderstanding, we use the matrix notation.

Consider the matrix ordinary differential equation

$$\big{[}\partial_{z}g_{ij}(z)\partial_{z}+V_{ij}(z;\lambda)\big{]}u_{j}(z)=0,\qquad z\in\mathbb{R},\quad i,j=\overline{1,N},$$

(1)

where $\lambda\in\mathbb{C}$ is an auxiliary parameter,

$$V_{ij}(z;\lambda)=V_{ij}(z)-\lambda g_{ij}(z),$$

(2)

and the matrices $g_{ij}(z)$ and $V_{ij}(z)$ are real and symmetric. The elements of these matrices are piecewise continuous functions. The matrix $g_{ij}(z)$ is positive-definite. We also assume that there exists $L_{z}>0$ such that

$$g_{ij}(z)|_{z>L_{z}}=g^{+}_{ij},\qquad V_{ij}(z)|_{z>L_{z}}=V^{+}_{ij},\qquad g_{ij}(z)|_{z<-L_{z}}=g^{-}_{ij},\qquad V_{ij}(z)|_{z<-L_{z}}=V^{-}_{ij},$$

(3)

where $g^{\pm}_{ij}$ and $V^{\pm}_{ij}$ are constant matrices. The spectral parameter $\lambda$ is not the energy, in general. For example, the energy enters into the matrices $g_{ij}(z)$ and $V_{ij}(z)$ as a parameter for the scattering problems in electrodynamics of dispersive media and the physical value of $\lambda$ is zero in this case. The corresponding nonstationary scattering problem is not described by the nonstationary Schrödinger equation associated with (1). Notice that all the results of the present paper are applicable to the case where the matrices $g_{ij}(z)$ and $V_{ij}(z)$ are not real and symmetric but Hermitian. In that case, one just has to separate the real and imaginary parts of the initial matrix Schrödinger equation. The resulting system of equations will be of the form (1) but of twice the size of the initial system.

Let

$$g^{\pm}_{ij}f^{\pm}_{js}\Lambda^{\pm}_{s}=V^{\pm}_{ij}f^{\pm}_{js}\qquad\text{(no summation over $s$)},\qquad s=\overline{1,N},$$

(4)

where $\Lambda^{\pm}_{s}\in\mathbb{R}$ are eigenvalues and the following normalization condition is true

$$f_{\pm}^{T}g_{\pm}f_{\pm}=1.$$

(5)

We consider the general case where all of $\Lambda^{+}_{s}$ and all of $\Lambda^{-}_{s}$ are different. The degenerate case is obtained by going to the respective limit. Let us introduce the diagonal matrices

$$K^{\pm}_{ss^{\prime}}:=\sqrt{\Lambda^{\pm}_{s}-\lambda}\delta_{ss^{\prime}},$$

(6)

where the principal branch of the square root is chosen. In particular, if $\lambda>\Lambda^{\pm}_{s}$, then $\sqrt{\Lambda^{\pm}_{s}-\lambda}=i\sqrt{\lambda-\Lambda^{\pm}_{s}}$. By definition, the Jost solutions to Eq. (1) have the asymptotics

$$(F^{+}_{\pm})_{is}(z;\lambda)\underset{z\rightarrow\infty}{\rightarrow}(f_{+})_{is^{\prime}}(e^{\pm iK_{+}z})_{s^{\prime}s},\qquad(F^{-}_{\pm})_{is}(z;\lambda)\underset{z\rightarrow-\infty}{\rightarrow}(f_{-})_{is^{\prime}}(e^{\pm iK_{-}z})_{s^{\prime}s}.$$

(7)

For these solutions, we can write

$$\begin{split}F^{+}_{+}(z;\lambda)&=f_{+}e^{iK_{+}z}+\int_{z}^{\infty}dtf_{+}\frac{\sin K_{+}(z-t)}{K_{+}}f^{T}_{+}U_{+}(t;\lambda)F^{+}_{+}(t;\lambda),\\ F^{+}_{-}(z;\lambda)&=f_{+}e^{-iK_{+}z}+\int_{z}^{\infty}dtf_{+}\frac{\sin K_{+}(z-t)}{K_{+}}f^{T}_{+}U_{+}(t;\lambda)F^{+}_{-}(t;\lambda),\\ F^{-}_{+}(z;\lambda)&=f_{-}e^{iK_{-}z}-\int_{-\infty}^{z}dtf_{-}\frac{\sin K_{-}(z-t)}{K_{-}}f^{T}_{-}U_{-}(t;\lambda)F^{-}_{+}(t;\lambda),\\ F^{-}_{-}(z;\lambda)&=f_{-}e^{-iK_{-}z}-\int_{-\infty}^{z}dtf_{-}\frac{\sin K_{-}(z-t)}{K_{-}}f^{T}_{-}U_{-}(t;\lambda)F^{-}_{-}(t;\lambda),\end{split}$$

(8)

where

$$U_{\pm}(z;\lambda):=\partial_{z}g(z)\partial_{z}+V(z;\lambda)-\partial_{z}g_{\pm}\partial_{z}-V_{\pm}+\lambda g_{\pm}.$$

(9)

In virtue of the assumption (3), the integration in the integral representations of the Jost solutions is performed over a finite interval. Therefore, the Jost solutions are analytic functions of $\lambda$ on a double-sheeted Riemann surface. The solutions $(F^{+}_{\pm})_{is}$ are the different branches of the same vector-valued analytic function of $\lambda$ with the branching point $\lambda=\Lambda^{+}_{s}$, whereas the solutions $(F^{-}_{\pm})_{is}$ are the different branches of the same vector-valued analytic function of $\lambda$ with the branching point $\lambda=\Lambda^{-}_{s}$.

The Jost solutions $F^{+}_{\pm}$ and $F^{-}_{\pm}$ constitute bases in the space of solutions of Eq. (1). Consequently,

$$F^{+}_{+}=F^{-}_{+}\Phi_{+}+F^{-}_{-}\Psi_{+},\qquad F^{+}_{-}=F^{-}_{+}\Psi_{-}+F^{-}_{-}\Phi_{-},$$

(10)

where $(\Phi_{\pm})_{ss^{\prime}}(\lambda)$ and $(\Psi_{\pm})_{ss^{\prime}}(\lambda)$ are some $z$-independent matrices. It is clear, that the Wronskian,

$$w[\varphi,\psi]:=\varphi^{T}(z)g(z)\partial_{z}\psi(z)-\partial_{z}\varphi^{T}(z)g(z)\psi(z),$$

(11)

of two solutions $\varphi(z)$ and $\psi(z)$ of Eq. (1) is independent of $z$ and defines a skew-symmetric scalar product on the space of solutions of Eq. (1). From the asymptotics (7) we have

$$w[F^{\pm}_{+},F^{\pm}_{+}]=w[F^{\pm}_{-},F^{\pm}_{-}]=0,\qquad w[F^{\pm}_{+},F^{\pm}_{-}]=-2iK_{\pm}.$$

(12)

This skew-symmetric scalar product allows one to express the matrices $\Phi_{\pm}$ and $\Psi_{\pm}$ in terms of Wronskians of the Jost solutions

	$\displaystyle 2iK_{-}\Phi_{+}$	$\displaystyle=w[F^{-}_{-},F^{+}_{+}],$	$\displaystyle\qquad-2iK_{-}\Phi_{-}$	$\displaystyle=w[F^{-}_{+},F^{+}_{-}],$		(13)
	$\displaystyle-2iK_{-}\Psi_{+}$	$\displaystyle=w[F^{-}_{+},F^{+}_{+}],$	$\displaystyle\qquad 2iK_{-}\Psi_{-}$	$\displaystyle=w[F^{-}_{-},F^{+}_{-}].$

We see from these relations that $(\Phi_{\pm})_{ss^{\prime}}$ and $(\Psi_{\pm})_{ss^{\prime}}$ are analytic functions of $\lambda$ with branching points of the square root type at $\lambda=\Lambda^{-}_{s}$ and $\lambda=\Lambda^{+}_{s^{\prime}}$. The four functions $(\Phi_{\pm})_{ss^{\prime}}$, $(\Psi_{\pm})_{ss^{\prime}}$ are the four branches of the same analytic function of $\lambda$. Indeed, bypassing the branching point $\lambda=\Lambda^{-}_{s}$, we have

$$(K_{-})_{s}\rightarrow-(K_{-})_{s},\qquad(F^{-}_{\pm})_{is}\rightarrow(F^{-}_{\mp})_{is}.$$

(14)

Then, using (13), we obtain

$$(\Phi_{\pm})_{ss^{\prime}}\rightarrow(\Psi_{\pm})_{ss^{\prime}},\qquad(\Psi_{\pm})_{ss^{\prime}}\rightarrow(\Phi_{\pm})_{ss^{\prime}}.$$

(15)

Similarly, bypassing the branching point $\lambda=\Lambda^{+}_{s^{\prime}}$, we come to

$$(\Phi_{\pm})_{ss^{\prime}}\rightarrow(\Psi_{\mp})_{ss^{\prime}},\qquad(\Psi_{\pm})_{ss^{\prime}}\rightarrow(\Phi_{\mp})_{ss^{\prime}}.$$

(16)

Hence, starting from $(\Phi_{+})_{ss^{\prime}}$ and bypassing successively the branching points $\lambda=\Lambda^{-}_{s}$ and $\lambda=\Lambda^{+}_{s^{\prime}}$, we obtain all the four functions $(\Phi_{\pm})_{ss^{\prime}}$, $(\Psi_{\pm})_{ss^{\prime}}$.

Consider the action of complex conjugation on the matrices $\Phi_{\pm}(\lambda)$ and $\Psi_{\pm}(\lambda)$. If $\lambda$ does not belong to the cuts of the functions $(K_{-})_{s}$ and $(K_{+})_{s^{\prime}}$, then, using (8) and (13), we obtain the following relations

$$(\Phi_{\pm})^{*}_{ss^{\prime}}(\lambda)=(\Phi_{\mp})_{ss^{\prime}}(\lambda^{*}),\qquad(\Psi_{\pm})^{*}_{ss^{\prime}}(\lambda)=(\Psi_{\mp})_{ss^{\prime}}(\lambda^{*}).$$

(17)

If $\lambda$ lies on the cut of the function $(K_{-})_{s}$ but does not belong to the cut of the function $(K_{+})_{s^{\prime}}$, then

$$(\Phi_{\pm})^{*}_{ss^{\prime}}(\lambda)=(\Psi_{\mp})_{ss^{\prime}}(\lambda).$$

(18)

If $\lambda$ lies on the cut of the function $(K_{+})_{s^{\prime}}$ but does not belong to the cut of function $(K_{-})_{s}$, then

$$(\Phi_{\pm})^{*}_{ss^{\prime}}(\lambda)=(\Psi_{\pm})_{ss^{\prime}}(\lambda).$$

(19)

If $\lambda$ lies on the cuts of the functions $(K_{-})_{s}$ and $(K_{+})_{s^{\prime}}$, we have

$$(\Phi_{\pm})^{*}_{ss^{\prime}}(\lambda)=(\Phi_{\pm})_{ss^{\prime}}(\lambda),\qquad(\Psi_{\pm})^{*}_{ss^{\prime}}(\lambda)=(\Psi_{\pm})_{ss^{\prime}}(\lambda),$$

(20)

i.e., in this case $(\Phi_{\pm})_{ss^{\prime}}(\lambda)$ and $(\Psi_{\pm})_{ss^{\prime}}(\lambda)$ are real.

Formulas (10), (12) yield the relations

	$\displaystyle\Phi^{T}_{+}K_{-}\Psi_{+}-\Psi^{T}_{+}K_{-}\Phi_{+}$	$\displaystyle=0,$	$\displaystyle\qquad\Phi^{T}_{+}K_{-}\Phi_{-}-\Psi^{T}_{+}K_{-}\Psi_{-}$	$\displaystyle=K_{+},$	$\displaystyle\qquad\Phi^{T}_{-}K_{-}\Psi_{-}-\Psi^{T}_{-}K_{-}\Phi_{-}$	$\displaystyle=0,$		(21)
	$\displaystyle\Phi_{+}K_{+}^{-1}\Psi_{-}^{T}-\Psi_{-}K_{+}^{-1}\Phi_{+}^{T}$	$\displaystyle=0,$	$\displaystyle\qquad\Phi_{+}K_{+}^{-1}\Phi_{-}^{T}-\Psi_{-}K_{+}^{-1}\Psi_{+}^{T}$	$\displaystyle=K_{-}^{-1},$	$\displaystyle\qquad\Phi_{-}K_{+}^{-1}\Psi_{+}^{T}-\Psi_{+}K_{+}^{-1}\Phi_{-}^{T}$	$\displaystyle=0.$

Let us introduce the transmission matrices $t_{(1,2)}$ and the reflection matrices $r_{(1,2)}$:

$$F^{+}_{+}t_{(1)}=F^{-}_{+}+F^{-}_{-}r_{(1)},\qquad F^{-}_{-}t_{(2)}=F^{+}_{-}+F^{+}_{+}r_{(2)}.$$

(22)

Combining the relations (10) and comparing the result with (22), we arrive at

$$t_{(1)}=\Phi_{+}^{-1},\qquad r_{(1)}=\Psi_{+}\Phi_{+}^{-1},\qquad t_{(2)}=\Phi_{-}-\Psi_{+}\Phi_{+}^{-1}\Psi_{-},\qquad r_{(2)}=-\Phi_{+}^{-1}\Psi_{-}.$$

(23)

The relations (21) imply the symmetry properties

$$K_{-}t_{(2)}=t_{(1)}^{T}K_{+},\qquad K_{-}r_{(1)}=r_{(1)}^{T}K_{-},\qquad K_{+}r_{(2)}=r_{(2)}^{T}K_{+}.$$

(24)

Define the $S$-matrix as

$$S:=\left[\begin{array}[]{cc}t_{(1)}&r_{(2)}\\ r_{(1)}&t_{(2)}\\ \end{array}\right].$$

(25)

To shorten the notation, we also introduce the operation that acts on the products of matrices $\Phi_{\pm}$, $\Psi_{\pm}$ and their inverses by the rule

$$\bar{\Phi}_{\pm}:=\Phi_{\mp},\qquad\bar{\Psi}_{\pm}:=\Psi_{\mp},\qquad\overline{A^{-1}}=(\bar{A})^{-1},\qquad\overline{AB}:=\bar{A}\bar{B}.$$

(26)

Then the $S$-matrix possesses the symmetries

$$\begin{gathered}\left[\begin{array}[]{cc}0&K_{-}\\ K_{+}&0\\ \end{array}\right]S=S^{T}\left[\begin{array}[]{cc}0&K_{+}\\ K_{-}&0\\ \end{array}\right],\qquad\left[\begin{array}[]{cc}0&K_{-}\\ K_{+}&0\\ \end{array}\right]\bar{S}=\bar{S}^{T}\left[\begin{array}[]{cc}0&K_{+}\\ K_{-}&0\\ \end{array}\right],\\ \bar{S}^{T}\left[\begin{array}[]{cc}K_{+}&0\\ 0&K_{-}\\ \end{array}\right]S=\left[\begin{array}[]{cc}K_{-}&0\\ 0&K_{+}\\ \end{array}\right].\end{gathered}$$

(27)

It is more difficult to prove unitarity of the $S$-matrix in the case when some of the channels are closed for $z\rightarrow-\infty$ and/or $z\rightarrow\infty$, i.e., when for some fixed $\lambda\in\mathbb{R}$ some of $(K_{\pm})_{s}$ are purely imaginary. Let the number of open channels for $z\rightarrow-\infty$ be $l_{o}$, and the number of closed channels be $l_{c}$. As for the numbers of open and closed channels for $z\rightarrow\infty$, we introduce the notation $r_{o}$ and $r_{c}$, respectively. For definiteness, we assume that $l_{o}\geqslant r_{o}$. It is clear that $l_{o}+l_{c}=r_{o}+r_{c}=N$. We split the relations (22) into blocks with respect to the indices $s$, $s^{\prime}$ in accordance with the splitting into open and closed channels,

	$\displaystyle(F^{+}_{+})_{o}t_{(1)oo}+(F^{+}_{+})_{c}t_{(1)co}$	$\displaystyle=(F^{-}_{+})_{o}+(F^{-}_{-})_{o}r_{(1)oo}+(F^{-}_{-})_{c}r_{(1)co},$		(31a)
	$\displaystyle(F^{+}_{+})_{o}t_{(1)oc}+(F^{+}_{+})_{c}t_{(1)cc}$	$\displaystyle=(F^{-}_{+})_{c}+(F^{-}_{-})_{o}r_{(1)oc}+(F^{-}_{-})_{c}r_{(1)cc},$		(31b)
	$\displaystyle(F^{-}_{-})_{o}t_{(2)oo}+(F^{-}_{-})_{c}t_{(2)co}$	$\displaystyle=(F^{+}_{-})_{o}+(F^{+}_{+})_{o}r_{(2)oo}+(F^{+}_{+})_{c}r_{(2)co},$		(31c)
	$\displaystyle(F^{-}_{-})_{o}t_{(2)oc}+(F^{-}_{-})_{c}t_{(2)cc}$	$\displaystyle=(F^{+}_{-})_{c}+(F^{+}_{+})_{o}r_{(2)oc}+(F^{+}_{+})_{c}r_{(2)cc},$		(31d)

where, for example,

$$t_{(1)}=\left[\begin{array}[]{cc}t_{(1)oo}&t_{(1)oc}\\ t_{(1)co}&t_{(1)cc}\\ \end{array}\right],\qquad F^{+}_{\pm}=\left[\begin{array}[]{cc}(F^{+}_{\pm})_{o}&(F^{\pm}_{\pm})_{c}\\ \end{array}\right].$$

(32)

Taking complex conjugate of these equations, bearing in mind the above conditions on $\lambda$, and using the expressions for the Jost solutions (8), we arrive at

	$\displaystyle(F^{+}_{-})_{o}t^{*}_{(1)oo}+(F^{+}_{+})_{c}t^{*}_{(1)co}$	$\displaystyle=(F^{-}_{-})_{o}+(F^{-}_{+})_{o}r^{*}_{(1)oo}+(F^{-}_{-})_{c}r^{*}_{(1)co},$		(33a)
	$\displaystyle(F^{+}_{-})_{o}t^{*}_{(1)oc}+(F^{+}_{+})_{c}t^{*}_{(1)cc}$	$\displaystyle=(F^{-}_{+})_{c}+(F^{-}_{+})_{o}r^{*}_{(1)oc}+(F^{-}_{-})_{c}r^{*}_{(1)cc},$		(33b)
	$\displaystyle(F^{-}_{+})_{o}t^{*}_{(2)oo}+(F^{-}_{-})_{c}t^{*}_{(2)co}$	$\displaystyle=(F^{+}_{+})_{o}+(F^{+}_{-})_{o}r^{*}_{(2)oo}+(F^{+}_{+})_{c}r^{*}_{(2)co},$		(33c)
	$\displaystyle(F^{-}_{+})_{o}t^{*}_{(2)oc}+(F^{-}_{-})_{c}t^{*}_{(2)cc}$	$\displaystyle=(F^{+}_{-})_{c}+(F^{+}_{-})_{o}r^{*}_{(2)oc}+(F^{+}_{+})_{c}r^{*}_{(2)cc}.$		(33d)

Introducing the notation,

$$(K_{\pm})_{o}=:\varkappa_{\pm}^{o},\qquad(K_{\pm})_{c}=:i\varkappa_{\pm}^{c},\qquad\varkappa^{o,c}_{\pm}>0,$$

(34)

the symmetry relations (24) become

$$\begin{gathered}\varkappa^{o}_{-}t_{(2)oo}=(t_{(1)oo})^{T}\varkappa^{o}_{+},\quad\varkappa^{o}_{-}t_{(2)oc}=(t_{(1)co})^{T}i\varkappa^{c}_{+},\quad i\varkappa^{c}_{-}t_{(2)co}=(t_{(1)oc})^{T}\varkappa^{o}_{+},\quad\varkappa^{c}_{-}t_{(2)cc}=(t_{(1)cc})^{T}\varkappa^{c}_{+},\\ \varkappa^{o}_{-}r_{(1)oo}=(r_{(1)oo})^{T}\varkappa^{o}_{-},\quad\varkappa^{o}_{-}r_{(1)oc}=(r_{(1)co})^{T}i\varkappa^{c}_{-},\quad i\varkappa^{c}_{-}r_{(1)co}=(r_{(1)oc})^{T}\varkappa^{o}_{-},\quad\varkappa^{c}_{-}r_{(1)cc}=(r_{(1)cc})^{T}\varkappa^{c}_{-},\\ \varkappa^{o}_{+}r_{(2)oo}=(r_{(2)oo})^{T}\varkappa^{o}_{+},\quad\varkappa^{o}_{+}r_{(2)oc}=(r_{(2)co})^{T}i\varkappa^{c}_{+},\quad i\varkappa^{c}_{+}r_{(2)co}=(r_{(2)oc})^{T}\varkappa^{o}_{+},\quad\varkappa^{c}_{+}r_{(2)cc}=(r_{(2)cc})^{T}\varkappa^{c}_{+}.\end{gathered}$$

(35)

Without loss of generality, we can assume that the rank of the matrix $t_{(1)oo}$ is maximal and is equal to $r_{o}$. Then it follows from the first relation in (35) that the rank of the matrix $t_{(2)oo}$ is also $r_{o}$.

In this case we have

$$t_{(1)oo}t^{\vee}_{(1)oo}=t^{\vee}_{(2)oo}t_{(2)oo}=1,\qquad t^{\vee}_{(1)oo}t_{(1)oo}=:L_{(1)},\qquad t_{(2)oo}t^{\vee}_{(2)oo}=:L_{(2)},$$

(36)

where $A^{\vee}$ is a pseudo-inverse matrix to $A$, and $L_{(1)}$ and $L_{(2)}$ are Hermitian projectors of rank $r_{o}$ in the subspace of open channels at $z\rightarrow-\infty$. It follows from the definition of pseudo-inverse matrix that

$$t_{(1)oo}\bar{L}_{(1)}=\bar{L}_{(1)}t^{\vee}_{(1)oo}=0,\qquad\bar{L}_{(2)}t_{(2)oo}=t^{\vee}_{(2)oo}\bar{L}_{(2)}=0,$$

(37)

where $\bar{L}_{(1,2)}:=1-L_{(1,2)}$. Further, we express the functions $(F^{+}_{-})_{o}$ from (33a) and substitute them into (33c). This gives rise to

$$(F^{+}_{+})_{o}+(F^{+}_{+})_{c}[r^{*}_{(2)co}-t^{*}_{(1)co}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}]=\\ =(F^{-}_{+})_{o}[t^{*}_{(2)oo}-r^{*}_{(1)oo}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}]-(F^{-}_{-})_{o}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}+(F^{-}_{-})_{c}[t^{*}_{(2)co}-r^{*}_{(1)co}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}].$$

(38)

Then multiplying (31a) by $t_{(1)oo}^{\vee}$ and comparing the result with the equation above, we have

	$\displaystyle t_{(1)oo}^{\vee}$	$\displaystyle=t^{*}_{(2)oo}-r^{*}_{(1)oo}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo},$		(39a)
	$\displaystyle r_{(1)oo}L_{(1)}$	$\displaystyle=-(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}t_{(1)oo},$		(39b)
	$\displaystyle t_{(1)co}L_{(1)}$	$\displaystyle=[r^{*}_{(2)co}-t^{*}_{(1)co}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}]t_{(1)oo},$		(39c)
	$\displaystyle r_{(1)co}L_{(1)}$	$\displaystyle=[t^{*}_{(2)co}-r^{*}_{(1)co}(t^{*}_{(1)oo})^{\vee}r^{*}_{(2)oo}]t_{(1)oo}.$		(39d)

We infer from (39b) that

$$\bar{L}^{*}_{(1)}r_{(1)oo}L_{(1)}=0.$$

(40)

Then (39a) implies

$$\bar{L}_{(1)}t^{*}_{(2)oo}=0.$$

(41)

Consequently,

$$\bar{L}_{(1)}L^{*}_{(2)}=0\;\Rightarrow\;L_{(1)}L^{*}_{(2)}=L^{*}_{(2)}=L^{*}_{(2)}L_{(1)}.$$

(42)

As the ranks of $L_{(1)}$ and $L^{*}_{(2)}$ are the same, we have

$$L_{(1)}=L^{*}_{(2)}=:L.$$

(43)

The first relation in (35) implies

$$t^{\vee}_{(2)oo}=(\varkappa_{+}^{o})^{-1}(t^{T}_{(1)oo})^{\vee}\varkappa_{-}^{o}L^{*},$$

(44)

whence

$$L^{*}=t_{(2)oo}t^{\vee}_{(2)oo}=(\varkappa_{+}^{o})^{-1}L^{T}\varkappa_{+}^{o}L^{*}.$$

(45)

Therefore,

$$\varkappa_{+}^{o}L=L\varkappa_{+}^{o}L=L\varkappa_{+}^{o}.$$

(46)

Thus we see that $L$ is a diagonal matrix and hence $L^{*}=L$.

Now we are in position to prove unitarity of the $S$-matrix in open channels as well as to obtain the other relations involving the different components of $t_{(1,2)}$ and $r_{(1,2)}$.