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Could somebody provide the proof (or reference to some accessible literature) of relation $$S(E) = 1 + 2πiW^{†} (H_M − E − iπW W^{†} )^{−1} W \tag{2}$$

of arXiv:0806.4889, which relates $S$-matrix to an effective Hamiltonian?

People usually refer to the book

"Shell-model approach to nuclear reactions" by Mahaux, Claude; Weidenmüller, Hans A

which I do not have access to.

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  • $\begingroup$ Would be interesting to know a bit more what are $H_M$, and $W$. I guess $E$ is the energy. I have the feeling this is just the Dyson equation for the scattering matrix written in Fourier component (in energy, not in time) as @Meng-Cheng said below [ physics.stackexchange.com/a/168478/16689 ]. Note that your equation is wrong (the term $(\cdots)-1W$ makes no dimensional sense, and should read $(\cdots)^{-1}W$ It can be prove generically starting from the Schrödinger equation for the S. The $i$ and the $\pi$ are convention, so this comment is a loop: please define $W$ ... $\endgroup$ – FraSchelle Mar 5 '15 at 7:53
  • $\begingroup$ Posts about Dyson: physics.stackexchange.com/q/10718/16689 and physics.stackexchange.com/q/105120/16689 I do not understand the link between your question and the notion of effective Hamiltonian, if you could precise it in your edit of your question in order to make everything clear. Thanks in advance. $\endgroup$ – FraSchelle Mar 5 '15 at 8:12
  • $\begingroup$ @FraSchelle You are right! I've edited the question. What i know about this relation is that it is relation between Scattering matrix $$S$$ of a system described by $$H_{M}$$ connected to leads (scattering regions) via a matrix $$W$$. I actually don't know how to derive it and hence it's validity. I've seen many people using it to compute S-matrix on lattice. $\endgroup$ – Krishna Tripathi Mar 5 '15 at 10:50
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It's been some time since this question has been asked, but allow me to post a new answer. Its piece of my thesis, where I've put the derivation (since its so hard to find). I hope it'll help anyone, who encounters this problem. It's mostly based on the introduction of the paper "The decay of quantum systems with a small number of open channels" by F. M. Dittes (http://www.sciencedirect.com/science/article/pii/S037015730000065X).

Consider a system described by a Hamiltonian $H=H_0+V$, where $V$ describes the perturbation upon which the scattering one wishes to study -- in our case it will be the coupling between the leads and the device. The system set in the state $|i\rangle$ at the initial time $t=-\infty$ is subject to evolution governed by the Schrödinger equation in the interaction picture. At the final time $t=+\infty$ its state is given by $\mathcal{T}\mathrm{e}^{-\mathrm{i} \int_{-\infty}^{\infty} \mathrm{d} t\,V_{\rm I}(t)}|i\rangle = U|i\rangle$, where $V_{\rm I}(t) = \mathrm{e}^{\mathrm{i} H_0 t}V\mathrm{e}^{-\mathrm{i} H_0 t}$ is the perturbation in the interaction picture, and $\mathcal{T}$ denotes time-ordering. The scattering amplitude to some state $|f\rangle$ is given by the projection of the final state: \begin{multline} \langle f|U|i\rangle = \langle f | i \rangle+\\+ \sum_{j_1,j_2,\dots,j_{n-1}} \sum_{n=1}^\infty (-\mathrm{i})^n \int_{-\infty}^\infty\mathrm{d} {t_1} \int_{-\infty}^{t_1}\mathrm{d} t_2 \dots \int_{-\infty}^{t_{n-1}}\mathrm{d} t_n \langle f|V_{\rm I}(t_1)|j_1\rangle\langle j_1| V_{\rm I}(t_2)|j_2\rangle\dots \langle j_{n-1}|V_{\rm I}(t_n)|i\rangle, \end{multline} where $j_k$-s are summed over the eigenbasis of $H_0$. Performing the last integral: \begin{align} \int_{-\infty}^{t_{n-1}}\mathrm{d} t_n \langle j_{n-1}|V_{\rm I}(t_n)|i\rangle &= \int_{-\infty}^{t_{n-1}}\mathrm{d} t_n \mathrm{e}^{-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+) t_n} \langle j_{n-1}| V|i\rangle = \frac{\mathrm{e}^{-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+) t_{n-1}}} {-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+)} \langle j_{n-1}| V|i\rangle. \end{align} Performing the second last integral: \begin{multline} \int_{-\infty}^{t_{n-2}}\mathrm{d} t_{n-1} \langle j_{n-2}| V_{\rm I}(t_{n-1})|j_{n-1}\rangle \frac{\mathrm{e}^{-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+) t_{n-1}}} {-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+)} \langle j_{n-1}| V|i\rangle =\hfill\\ \begin{aligned} &=\int_{-\infty}^{t_{n-2}}\mathrm{d} t_{n-1} \mathrm{e}^{-\mathrm{i} (E_i-E_{j_{n-2}} + \mathrm{i} 0^+) t_{n-1}} \langle j_{n-2}| V|j_{n-1}\rangle \frac{1}{-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+)} \langle j_{n-1}| V|i\rangle\\ &=\frac{\mathrm{e}^{-\mathrm{i} (E_i-E_{j_{n-2}} + \mathrm{i} 0^+) t_{n-2}}} {-\mathrm{i} (E_i-E_{j_{n-2}} + \mathrm{i} 0^+)} \langle j_{n-2}| V|j_{n-1}\rangle \frac{1}{-\mathrm{i} (E_i-E_{j_{n-1}} + \mathrm{i} 0^+)} \langle j_{n-1}| V|i\rangle. \end{aligned} \end{multline} It is straightforward to see that the remaining integrals (except the first one) will follow identical pattern. Summing over $j_k$-s one can contract the notation: $\sum_{j_k}|j_k\rangle\langle j_k|(E_i-E_{j_k}+\mathrm{i} 0^+)^-1 = (E_i-H_0+\mathrm{i} 0^+)^{-1} \equiv G(E_i+\mathrm{i}0^+)$, which yields: \begin{align} \langle f|U|i\rangle &= \langle f | i \rangle + \sum_{n=1}^\infty \frac{(-\mathrm{i})^n}{(-\mathrm{i})^{n-1}} \int_{-\infty}^\infty\mathrm{d} t_1 \mathrm{e}^{-\mathrm{i}(E_i-E_f)t_1} \langle f|V(GV)^{n-1}|i\rangle \\ &= \langle f | i \rangle -2\pi\mathrm{i}\delta(E_i-E_f) \langle f|V\sum_{n=0}^\infty(GV)^n|i\rangle. \label{eq:mwe1} \end{align} Note that in the case of our interest $G$ acts separately in the Hilbert spaces of the leads ($\mathscr{H}_{\rm L}$) and the device ($\mathscr{H}_{\rm D}$), whereas $V$ mediates between the two. Since both $|i\rangle$ and $|f\rangle$ belong to the $\mathscr{H}_{\rm L}$, only terms with even number of $V$ operators (including the one in front of the sum) in the equation above will contribute to the scattering amplitude. Writing explicitly $G=G_{\rm L} + G_{\rm D}$ and $V = V_{\rm LD} + V_{\rm DL}$, where $G_{\rm L}$ ($G_{\rm D}$) acts within $\mathscr{H}_{\rm L}$ ($\mathscr{H}_{\rm D}$) space and $V_{DL}$ ($V_{LD}$) acts from $\mathscr{H}_{\rm L}$ to $\mathscr{H}_{\rm D}$ (from $\mathscr{H}_{\rm D}$ to $\mathscr{H}_{\rm L}$) one gets: \begin{align} \langle f|U|i\rangle &= \langle f | i \rangle -2\pi\mathrm{i}\delta(E_i-E_f) \langle f|V_{\rm LD}\sum_{n=0}^\infty(G_{\rm D}V_{\rm DL}G_{\rm L}V_{\rm LD})^n G_{\rm D}V_{\rm DL}|i\rangle \\ &= \langle f | i \rangle -2\pi\mathrm{i}\delta(E_i-E_f) \langle f|V_{\rm LD}\frac{1}{1-G_{\rm D}V_{\rm DL}G_{\rm L}V_{\rm LD}} G_{\rm D}V_{\rm DL}|i\rangle\\ &= \langle f | i \rangle -2\pi\mathrm{i}\delta(E_i-E_f) \langle f|V_{\rm LD}\frac{1}{G_{\rm D}^{-1}-V_{\rm DL}G_{\rm L}V_{\rm LD}} V_{\rm DL}|i\rangle. \end{align} The $S$-matrix by summing over $|f\rangle$ and $|i\rangle$. Note that $\sum_{f} \delta(E_i-E_f)|f\rangle\langle f| = \sum_{f:E_f=E_i} \rho(E_i)|f\rangle\langle f| = \rho(E_i)\Pi(E_i)$, where $\rho(E_i)$ is the density of states at energy $E_i$ and $\Pi(E_i)$ is the projector onto the eigensubspace of $\mathscr{H}_{\rm L}$ of energy $E_i$. Then: \begin{align} S(E) &= \sum_{\substack{f,i:\\E_i=E}} |f\rangle\langle f|U|i\rangle\langle i| = \Pi(E) -2\pi\mathrm{i}\rho(E)\Pi(E)V_{\rm LD}\frac{1}{E-H_{\rm D}-V_{\rm DL}G_{\rm L}V_{\rm LD}} V_{\rm DL}\Pi(E). \end{align} Furthermore, the self energy $V_{\rm DL}G_{\rm L}V_{\rm LD}$ can be expressed in terms of its Hermitian and antihermitian parts: \begin{align} V_{\rm DL}G_{\rm L}V_{\rm LD} &= \int\mathrm{d} E'\, V_{\rm DL}\frac{\rho(E')\Pi(E')}{E-E'+\mathrm{i} 0^+}V_{\rm LD} \\ &= \int\mathrm{d} E'\, V_{\rm DL}\rho(E')\Pi(E')V_{\rm LD} \left(\mathcal{P}\frac{1}{E-E'} - \mathrm{i}\pi\delta(E-E')\right) \\ &= \mathcal{P}\int\mathrm{d} E'\, \frac{W(E')W^\dagger(E')}{E-E'} -\mathrm{i}\pi W(E)W^\dagger(E), \end{align} where $W(E)\equiv \sqrt{\rho(E)} V_{\rm DL}\Pi(E)$ and $\mathcal{P}$ denotes the principal value of the integral. The $S$ matrix finally reads: \begin{equation} S(E) = 1 - 2\pi\mathrm{i} W^\dagger(E)\frac{1}{E-H_{\rm D}-\mathcal{P}\int\mathrm{d} E'\, \frac{W(E')W^\dagger(E')}{E-E'} +\mathrm{i}\pi W(E)W^\dagger(E)} W(E). \end{equation} where we replaced the first projector with identity, sice the $S$-matrix acts in the relevant eigenspace anyway. The effective Hamiltonian is given by $H_{\rm D}+\mathcal{P}\int\mathrm{d} E'\,\frac{W(E')W^\dagger(E')}{E-E'}$, but the second term is often omited if the coupling depends weakly on the energy.

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  • $\begingroup$ Hi, I can't understand one simplification: how do you bring $G_D$ in the denominator in the last step to get the matrix element of U? Does it commute with everything? $\endgroup$ – Costantino Mar 23 '18 at 14:10
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This is the relation between the scattering matrix and the Green's function (notice that $(H_M-E-i\pi WW^\dagger)^{-1}$ is basically the Green's function, where $i\pi WW^\dagger$ is the self-energy correction due to coupling to the leads). For a pedagogical account, a good reference is Datta's "Electronic transport in mesoscopic systems", which is in general a great introduction to the subject.

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  • $\begingroup$ Thanks! I've gone through supriyo dutta s, i wanted some reference which derives it explicitly with all the assumptions. I have feeling that this may not hold true in general? $\endgroup$ – Krishna Tripathi Mar 5 '15 at 3:12
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    $\begingroup$ I do remember Datta derived it quite explicitly, although his notation may be very different what you write. $\endgroup$ – Meng Cheng Mar 5 '15 at 3:17

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