Scattering channels, stationary resolvents and time delay

Assume the Axiom of Choice. Let ${\mathcal{H}}_0,\mathcal{H}$ be separable complex Hilbert spaces, $H_0$ and $H$ self-adjoint operators on them, and $J:{\mathcal{H}}_0\to \mathcal{H}$ a bounded linear map. Write $E_{H_0}$ and $E_H$ for their spectral projection valued measures. The absolutely continuous subspace ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$ consists of vectors $u$ for which the scalar measure $B\mapsto \mathrm{\parallel }{E}_{H_0}(B)u{\mathrm{\parallel }}^2$ is absolutely continuous with respect to Lebesgue measure; $P_{\mathrm{a}\mathrm{c}}(H_0)$ is its orthogonal projection. Define the two-space wave operators, when the pointwise strong limits exist on ${\mathcal{H}}_0$, by

${\mathrm{\Omega }}_{\pm }(H,H_0;J):={\mathrm{s-lim}}_{t\to \pm \mathrm{\infty }}{e}^{itH}J{e}^{-it{H}_0}{P}_{\mathrm{a}\mathrm{c}}(H_0),$

where a strong limit means that for every $u\in {\mathcal{H}}_0$ the vector-valued expression converges in $\mathcal{H}$. The two signs are separate limits. When $J=I$ and ${\mathcal{H}}_0=\mathcal{H}$, write ${\mathrm{\Omega }}_{\pm }(H,H_0)$. A wave operator is complete when its range is all of ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H)$. When both wave operators exist and are isometries on ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$, the scattering operator on that subspace is

$S(H,H_0;J):={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}\mathrm{.}$

Whenever both wave operators exist, are isometries on ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$, and are complete with range ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H)$, the scattering operator $S(H,H_0;J)$ is unitary. For a fixed integer $r\ge 1$ and an open energy interval $I\subset \mathbb{R}$, finite-channel data consists of the standard space ${\mathbb{C}}^r$ and a matrix field $S:I\to U(r)$, where

$U(r):=\{U\in M_r(\mathbb{C}):U^{\ast }U=U{U}^{\ast }=I_r\}\mathrm{.}$

Here $U^{\ast }$ is conjugate transpose, $\mathrm{tr}U={\sum }_{j=1}^r{U}_{jj}$, and $\det U$ is the usual alternating determinant. This matrix field is separate specified data; the definition does not assert that it is obtained from the abstract wave operators above.

The upper arrow is ${\mathrm{\Omega }}_{-}$ restricted to the free absolutely continuous subspace. The right arrow is the full adjoint ${\mathrm{\Omega }}_{+}^{\ast }:\mathcal{H}\to {\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$; its codomain restriction follows from the factor $P_{\mathrm{a}\mathrm{c}}(H_0)$ in the definition of ${\mathrm{\Omega }}_{+}$. Whenever both limits exist, the diagonal arrow is their composite; if both are also isometries, this is the scattering operator $S={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}$ defined above. The diagram records this composition only and asserts neither existence nor completeness.

The Axiom of Choice

A bounded linear operator between normed spaces

Symmetric, self-adjoint and essentially self-adjoint operators

A self-adjoint operator generates a strongly continuous unitary group

Canonical decomposition into pure point, absolutely continuous and singular continuous parts

Let $T$ be self-adjoint on a complex Hilbert space $\mathcal{H}$ whose inner product is linear in the first variable, use the resolvent convention $R_T(z)=(T-zI{)}^{-1}$, and let $X\subseteq \mathcal{H}$ be a specified vector subspace. For $f,h\in X$, the scalar resolvent matrix element is

$r_{f,h}(z):=⟨R_T(z)f,h⟩,\mathrm{Im}z>0.$

Given an open real interval $I$, a point $E_0\in I$, and a disk $D$ centered at $E_0$, a local analytic continuation of these matrix elements through $E_0$ is a specified family of holomorphic functions ${\tilde{r}}_{f,h}:D\to \mathbb{C}$ agreeing with $r_{f,h}$ on $D\cap \{\mathrm{Im}z>0\}$, for the stated $f,h\in X$. A meromorphic continuation on $D$ is instead a specified family ${\tilde{r}}_{f,h}$ agreeing with $r_{f,h}$ on $D\cap \{\mathrm{Im}z>0\}$, holomorphic on $D$ except for isolated poles. At every crossing point where no pole occurs it is a regular analytic continuation on a smaller neighborhood. For such a meromorphic continuation, if $z_0\in D$ is nonreal and for some $f,h\in X$ the continued function satisfies

${\tilde{r}}_{f,h}(z)=(z-z_0{)}^{-k}q(z)$

on a punctured neighborhood of $z_0$, for an integer $k\ge 1$ and a holomorphic $q$ with $q(z_0)\mathrm{\ne }0$, then $z_0$ is a pole of order $k$ of that continued matrix element. Such a pole is called a resonance matrix-element pole for the specified pair $(T,X)$ and continuation. This definition asserts neither that a continuation exists nor that a pole is an eigenvalue of $T$; the continuation and the chosen test vectors are part of the data.

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Resolvent and spectrum of an unbounded operator

Symmetric, self-adjoint and essentially self-adjoint operators

Resolvent of a self-adjoint operator: nonreal resolvents and the estimate

Assume the Axiom of Choice and Countable Choice. On $\mathcal{H}=L^2({\mathbb{R}}^3,dp;\mathbb{C})$ with $⟨f,h⟩={\int }_{{\mathbb{R}}^3}f(p)\overline{h(p)}dp$, let $H_0=M_{\mathrm{\mid }p{\mathrm{\mid }}^2}$ on its maximal domain $\{f:\mathrm{\mid }p{\mathrm{\mid }}^{2}f\in L^2\}$, let $g(p)={\pi }^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}2}$, and put

$m(z):=⟨(H_0-zI{)}^{-1}g,g⟩,\mathrm{Im}z>0.$

This is a dimensionless model; no physical units are assigned to $p$ or $z$.

For $w\in \mathbb{C}$, define the entire function

$\mathcal{E}(w):=\frac{2}{\sqrt{\pi }}{\int }_{{\mathrm{\Gamma }}_w}{e}^{-{\zeta }^2}d\zeta ,{\mathrm{\Gamma }}_w=\{w+t:t\ge 0\},$

where ${\mathrm{\Gamma }}_w$ is the horizontal ray oriented from $w$ toward $+\mathrm{\infty }+w$. For any $E_0>0$, take a disk $D$ centered at $E_0$ with $0\mathrm{\notin }D$, and the holomorphic square root $k:D\to \mathbb{C}$ with $k(E_0)=\sqrt{E_0}>0$. Then $m$ has the local analytic continuation through $E_0$

$m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}(z)=2+2i\sqrt{\pi }k(z)e^{-z}\mathcal{E}(-ik(z)),z\in D\mathrm{.}$

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Continued resolvent matrix elements and resonance poles

Dominated convergence

The Gaussian integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$

Identity theorem for holomorphic functions

Real multiplication operators and free plus bounded-potential Schrödinger operators

Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Assume the Axiom of Choice and Countable Choice. Let $V\in L^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{R})$ and suppose there is a compact set $K\subset {\mathbb{R}}^3$ such that $V=0$ almost everywhere outside $K$. Let $\mathcal{H}=L^2({\mathbb{R}}^3;\mathbb{C})$, and set $H_0=-\mathrm{\Delta }$ and $H=H_0+M_V$ on $D(H_0)=D(H)=H^2({\mathbb{R}}^3;\mathbb{C})$. Use

$\widehat{f}(p)=(2\pi {)}^{-3\mathrm{/}2}{\int }_{{\mathbb{R}}^3}{e}^{-ip\cdot x}f(x)dx,{\mathrm{\Omega }}_{\pm }={\mathrm{s-lim}}_{t\to \pm \mathrm{\infty }}{e}^{itH}{e}^{-it{H}_0},S={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}\mathrm{.}$

Here $M_{V}f=Vf$, and the strong limits act on every $f\in \mathcal{H}$. Then:

(a) The wave operators exist, are isometries, and satisfy

$\mathrm{ran}{\mathrm{\Omega }}_{+}=\mathrm{ran}{\mathrm{\Omega }}_{-}={\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H)={\mathcal{H}}_{\mathrm{c}}(H),{\mathcal{H}}_{\mathrm{s}\mathrm{c}}(H)=\{0\}\mathrm{.}$

For $u\in \mathcal{H}$, write ${\mu }_u(B)=\mathrm{\parallel }{E}_H(B)u{\mathrm{\parallel }}^2$. The subspace ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H)$ consists of vectors with ${\mu }_u\ll dE$; ${\mathcal{H}}_{\mathrm{s}\mathrm{c}}(H)$ is the subspace with atomless scalar spectral measures singular with respect to Lebesgue measure; ${\mathcal{H}}_{\mathrm{p}\mathrm{p}}(H)$ is the closed span of eigenspaces, and ${\mathcal{H}}_{\mathrm{c}}(H)={\mathcal{H}}_{\mathrm{p}\mathrm{p}}(H{)}^{\perp }$. These are the canonical spectral-type subspaces of the PVM.

(b) The scattering operator $S$ is unitary on $\mathcal{H}$ and commutes with $e^{-it{H}_0}$ for every $t\in \mathbb{R}$.

Use time in the corresponding $\mathrm{\hslash }=1$ evolution convention and the standard scaled Schrödinger normalization in which the kinetic coefficient ${\mathrm{\hslash }}^2\mathrm{/}(2m)$ is set to $1$. When length is dimensional, $E$ and $V$ then have units $L^{-2}$ and $k=\sqrt{E}$ has units $L^{-1}$; the proof is an operator deduction in this normalization.

(c) Put $\mathfrak{h}=L^2(S^2,d\sigma )$, where $d\sigma (S^2)=4\pi$, and define the normalized free energy transform

$(F_{0}f)(E,\omega )=\frac{E^{1\mathrm{/}4}}{\sqrt{2}}\widehat{f}(\sqrt{E}\omega ),F_0:\mathcal{H}\to L^2((0,\mathrm{\infty }),dE;\mathfrak{h})\mathrm{.}$

Fix a bounded ball $B$ containing $K$. For $E>0$, $k=\sqrt{E}$, define bounded maps $Z_0(E):L^2(B)\to \mathfrak{h}$ and $Z_0(E{)}^{\ast }:\mathfrak{h}\to L^2(B)$ by

$(Z_0(E)f)(\omega )=\frac{\sqrt{k}}{4{\pi }^{3\mathrm{/}2}}{\int }_B{e}^{-ik\omega \cdot x}f(x)dx,(Z_0(E{)}^{\ast }a)(x)=\frac{\sqrt{k}}{4{\pi }^{3\mathrm{/}2}}{\int }_{S^2}{e}^{ik\omega \cdot x}a(\omega )d\sigma (\omega )\mathrm{.}$

Let $R_H(E+i0):L^2(B)\to L^2(B)$ be the operator-norm limit ${\lim }_{ϵ\downarrow 0}{\mathbf{1}}_B(H-(E+iϵ)I{)}^{-1}{\mathbf{1}}_B$; existence and continuity for $E>0$ are part of the stated supplier input. The products below are these bounded maps on $B$. Then $F_0$ is unitary and there is a measurable unitary field $S_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}(E)\in \mathcal{B}(\mathfrak{h})$ such that

$F_{0}S{F}_0^{\ast }=M_{S_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}(E)},S_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}(E)=I_{\mathfrak{h}}-2\pi i{Z}_0(E)[V-V{R}_H(E+i0)V]Z_0(E{)}^{\ast },E>0.$

The formula uses only positive energies; it makes no claim at the threshold $E=0$, and no long-range Coulomb potential is included.

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Wave, scattering and channel operators

Real multiplication operators and free plus bounded-potential Schrödinger operators

Compact-support asymptotic completeness

Stationary and dynamical scattering in the full energy-angular representation

Canonical decomposition into pure point, absolutely continuous and singular continuous parts

Let $I\subset \mathbb{R}$ be an open energy interval with fixed finite channel number $r\ge 1$, let $S:I\to U(r)$ be $C^1$ in the finite-channel convention of Wave, scattering and channel operators, and let $\mathrm{\hslash }>0$. Define

$Q(E):=-i\mathrm{\hslash }S(E{)}^{\ast }{S}^{\mathrm{\prime }}(E),E\in I\mathrm{.}$

Then $Q(E)$ is Hermitian and

$\mathrm{tr}Q(E)=-i\mathrm{\hslash }\frac{(\det S{)}^{\mathrm{\prime }}(E)}{\det S(E)}=-i\mathrm{\hslash }\frac{d}{dE}\log \det S(E),$

where the last expression means the logarithmic derivative $(\det S{)}^{\mathrm{\prime }}\mathrm{/}\det S$, independent of a choice of logarithm branch. If the incoming and outgoing bases are changed by energy-independent unitary matrices $U_{\mathrm{i}\mathrm{n}}$ and $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$, so that $S^{\mathrm{♯}}=U_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\ast }S{U}_{\mathrm{i}\mathrm{n}}$, then

$Q^{\mathrm{♯}}=U_{\mathrm{i}\mathrm{n}}^{\ast }Q{U}_{\mathrm{i}\mathrm{n}}\mathrm{.}$

If $E$ is measured in joules and $\mathrm{\hslash }$ in joule-seconds, $Q$ has units of seconds. This is an algebraic scattering-matrix result; it does not by itself establish an operational arrival-time or dwell-time interpretation.

Wave, scattering and channel operators

Assume the Axiom of Choice and Countable Choice. Let $H_0$ be a self-adjoint operator on a complex Hilbert space $\mathcal{H}$ with inner product linear in its first variable, bounded below, let $g\in \mathcal{H}$, let $\lambda \in \mathbb{R}\setminus \{0\}$, and put

$P_{g}f=⟨f,g⟩g,H_{\lambda }=H_0+\lambda P_g\text{on }D(H_0),R_0(z)=(H_0-zI{)}^{-1}\mathrm{.}$

Adopt as an explicit conditional input the rank-one resolvent interface: $H_{\lambda }$ is self-adjoint, and whenever $z\in \rho (H_0)$ and $d(z):=1+\lambda m(z)\mathrm{\ne }0$, where $m(z)=⟨R_0(z)g,g⟩$, one has

$(H_{\lambda }-zI{)}^{-1}=R_0(z)-\frac{\lambda R_0(z)P_g{R}_0(z)}{d(z)}\mathrm{.}$

Spectral variables use the energy unit of $H_0$. If $\mathrm{\parallel }g\mathrm{\parallel }$ is dimensionless, $m(z)$ has inverse-energy units and $\lambda$ has energy units; in general the coupling is chosen so that $\lambda m(z)$ is dimensionless.

Let $E_{\star }\in \mathbb{R}$, take an open disk $D$ centered at $E_{\star }$, and let $E_{\star }$ lie in an open interval $I$ on which the spectral PVM $E_{H_0}(\cdot )$ of $H_0$ is absolutely continuous: $E_{H_0}(N)=0$ for every Lebesgue-null Borel set $N\subseteq I$. Suppose $m$ and, for specified test vectors $f,h$ in a subspace $X\subseteq \mathcal{H}$, the free matrix elements

$⟨R_0(z)f,h⟩,a_f(z):=⟨R_0(z)f,g⟩,b_h(z):=⟨R_0(z)g,h⟩$

have holomorphic continuations through $E_{\star }$ to $D$. Suppose $d_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}(z)=1+\lambda m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}(z)$ has a zero $z_0\in D$ with $\mathrm{Im}{z}_0<0$ of finite order $k\ge 1$, meaning that on some neighborhood of $z_0$ one has $d_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}(z)=(z-z_0{)}^{k}q(z)$ for a holomorphic function $q$ with $q(z_0)\mathrm{\ne }0$, and suppose $a_f(z_0)b_h(z_0)\mathrm{\ne }0$. Then the continued matrix element $⟨(H_{\lambda }-zI{)}^{-1}f,h⟩$ has a resonance pole of order $k$. In the simple-zero case its residue is

$-\frac{a_f(z_0)b_h(z_0)}{m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}^{\mathrm{\prime }}(z_0)}\mathrm{.}$

In particular, for the Gaussian model $H_0=M_{\mathrm{\mid }p{\mathrm{\mid }}^2}$, $g(p)={\pi }^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}2}$, and $X=\mathrm{span}\{g\}$, take $E_{\star }>0$, an open disk $D$ centered at $E_{\star }$ with $0\mathrm{\notin }D$, and the holomorphic square-root branch $k_D:D\to \mathbb{C}$ normalized by $k_D(E_{\star })=\sqrt{E_{\star }}>0$. (The subscript distinguishes this branch from the zero order $k$.) The continuation in the Gaussian lemma with precisely these disk and branch data supplies $m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$; every interval $I\subset (0,\mathrm{\infty })$ is absolutely continuous by the radial spectral measure formula. Every finite-order zero of $1+\lambda m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$ in the lower half of such a disk gives a pole of the continued scalar element $⟨(H_{\lambda }-zI{)}^{-1}g,g⟩$. These poles are not nonreal eigenvalues of the self-adjoint $H_{\lambda }$.

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Continued resolvent matrix elements and resonance poles

Projection valued measure

Integral of a measurable function against a projection-valued measure

Explicit Gaussian continuation of a free resolvent matrix element

Real multiplication operators and free plus bounded-potential Schrödinger operators

Resolvent of a self-adjoint operator: nonreal resolvents and the estimate

Spectral theorem for unbounded self-adjoint operators (PVM form)

Rank-one resolvent and bound-state equation

Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma

$\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets

A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra