Time-Domain Lippmann-Schwinger Equation

• Time-Domain Lippmann–Schwinger equation is a fundamental integral equation that describes transient wave scattering in complex, inhomogeneous media.
• It extends to both acoustic and quantum systems, enabling analysis via Fourier–Laplace transforms and establishing well-posedness and stability.
• Numerical schemes using convolution quadrature and trigonometric collocation yield convergence guarantees and robust error estimates in practical simulations.

The time-domain Lippmann–Schwinger equation is a fundamental integral equation that characterizes time-dependent wave phenomena in inhomogeneous media. In the context of acoustic scattering, it provides a framework for describing the scattering of transient waves by penetrable obstacles with variable material properties. The approach extends naturally to quantum mechanical systems, including the Dirac equation in relativistic scattering, establishing deep connections between the time-dependent and stationary scattering formalisms. The time-domain Lippmann–Schwinger equation underpins robust numerical schemes via convolution quadrature and spectral collocation methods, and its analysis yields rigorous results on well-posedness, stability, and convergence.

1. Formulation in Acoustic Scattering

Consider the scalar acoustic wave equation with spacetime variables $(x,t)$ in $\mathbb{R}^d$ ($d = 2,3$), governing the pressure field $u(x,t)$: $$\frac{1}{c(x)^2} u_{tt}(x,t) = \Delta u(x,t), \quad x \in \mathbb{R}^d,\, t > 0,$$ where $c(x)$ is the (possibly variable) sound speed. Outside a bounded Lipschitz domain $D \subset \mathbb{R}^d$, $c(x) = c_0 > 0$; within $D$, $0 < c(x) < c_0$. The total field decomposes as $u(x,t) = u^i(x,t) + u^s(x,t)$, with $u^i$ the incident field satisfying the homogeneous background wave equation, and $u^s$ the scattered field (causal for $t \leq 0$).

Defining the contrast

$$q_c(x) = \frac{c_0^2}{c(x)^2} - 1, \qquad q_c|_{\mathbb{R}^d \setminus D} = 0,$$

the scattered field satisfies

$$\frac{1}{c^2} u^s_{tt} - \Delta u^s = -\frac{1}{c_0^2} q_c (u^s_{tt} + u^i_{tt}),$$

with vanishing initial data. Introducing the retarded volume potential operator

$$(Vf)(x,t) = \int_0^t \int_D k(x-y, t-\tau) f(y, \tau)\, dy d\tau,$$

where $k(x,t) = \frac{\delta (t - \|x\|/c_0)}{4\pi \|x\|}$, the canonical time-domain Lippmann–Schwinger integral equation reads

$$u(x,t) = u^i(x,t) + \int_0^t \int_D G(x,y, t-\tau) m(y) u(y, \tau)\, dy d\tau,$$

with $$G(x,y,t) = \frac{\partial^2}{\partial t^2} k(x-y, t)$$ and $m(y) = q_c(y)/c_0^2$ (Lechleiter et al., 2014).

2. Fourier–Laplace Transform and Connection to Helmholtz Theory

Passing to the Fourier–Laplace domain, for any causal function $f(t)$ define

$$\hat{f}(s) = \int_0^\infty e^{-st} f(t)\, dt, \qquad s = \sigma + i\omega,\, \sigma > 0.$$

The convolution structure becomes algebraic, yielding the volume integral formulation: $$\hat{u}^s + \frac{s^2}{c_0^2} \hat{V}(q_c \hat{u}^s) = -\frac{s^2}{c_0^2} \hat{V}(q_c \hat{u}^i), \quad \text{in } L^2(D),$$ where

$$(\hat{V}\hat{f})(x) = \int_D \Phi(x,y,s) \hat{f}(y)\, dy, \qquad \Phi(x,y,s) = \frac{e^{-s\|x-y\|/c_0}}{4\pi\|x-y\|}.$$

This equation is a Helmholtz-type integral equation for each $s$ in the right half-plane, establishing the analytic dependence of the solution on $s$. Coercivity properties of the operator $I + \frac{s^2}{c_0^2} \hat{V}_{q_c}$ in the weighted space $L^2_{q_c}(D)$ ensure stability and invertibility (Lechleiter et al., 2014).

3. Existence, Uniqueness, and Well-Posedness

For each $s$ with $\Re s > 0$, the operator $I+\frac{s^2}{c_0^2}\hat{V}_{q_c}$ is invertible and analytic in $s$, with norm bounds

$$\left\| \left(I+\frac{s^2}{c_0^2}\hat{V}_{q_c}\right)^{-1} \right\|_{L^2_{q_c}\to L^2_{q_c}} \leq \frac{|s|}{\sigma},$$

and

$$\left\| \left(I+\frac{s^2}{c_0^2}\hat{V}_{q_c}\right)^{-1} \frac{s^2}{c_0^2} \hat{V}_{q_c} \right\| \leq C|s|^2,$$

as established via the Lax–Milgram lemma (Lechleiter et al., 2014). Upon Laplace inversion and use of the Paley–Wiener theorem, the time-domain equation is shown to possess a unique causal solution $u^s \in H^0_\sigma(\mathbb{R}_+; L^2_{q_c}(D))$ for any $u^i \in H^2_\sigma(\mathbb{R}_+; L^2_{q_c}(D))$, with the energy estimate

$$\| u^s \|_{H^2_\sigma} \leq C \| u^i \|_{H^2_\sigma}.$$

4. Time Discretization by Convolution Quadrature

A stable and accurate time discretization is achieved by convolution quadrature using an $A$-stable $k$-step method (e.g., BDF2). For time increments $t_n = n\Delta t$, one approximates $u^s(t_n)$ by $u_n^{s,\Delta t}$. The semi-discrete sequence satisfies a discrete convolution equation whose $z$-transform mirrors the Laplace-domain integral equation with $s = \delta(\xi)/\Delta t$, $\delta(\xi)$ being the generating function of the multistep method. Lubich’s convolution quadrature theory applies, yielding the convergence result: for a method of order $p$,

$$\left( \Delta t \sum_{n=0}^M \| u^s(t_n) - u_n^{s, \Delta t} \|^2_{L^2_{q_c}(D)} \right)^{1/2} \leq C (\Delta t)^p \| u^i \|_{H^{p+3}_\sigma}.$$

This provides a rigorous guarantee on the temporal discretization error (Lechleiter et al., 2014).

5. Spatial Discretization and Fully Discrete Analysis

Efficient spatial discretization is attained by trigonometric collocation after periodizing the contrast $q_c$ and kernel over a computational box $G_{2\rho} = (-2\rho, 2\rho)^d$. The solution is represented in the finite-dimensional space $T_N$ of trigonometric polynomials, collocated at a uniform grid. The periodic volume operator diagonalizes on the Fourier basis, enabling fast inversion via discrete Fourier transforms.

Error analysis based on estimates of Fourier coefficients yields

$$\| \hat{u}_{p,N}(s) - \hat{u}_p(s) \|_{H^2(G_{2\rho})} \leq C |s|^{18 + 4(k-1)} N^{-k} \left( \| q_{c,p} \hat{u}^i_p(s) \|_{H^{1+k}} + \cdots \right)$$

for smooth ($C^2 \cap H^{1+k}$) $q_{c,p}$ and data. The fully discrete error, combining time and space discretization for $u_m^{s, \Delta t, N}$, obeys: $$\left( \Delta t \sum_{m=0}^M \| u^s(t_m) - u_m^{s, \Delta t, N} \|^2_{L^2_{q_c}(D)} \right)^{1/2} \leq C \left( (\Delta t)^p + N^{-1} \right)$$ provided $\Delta t \gtrsim N^{-1/13}$. This constraint keeps the complex Laplace frequencies in a stable region for the numerical integrators (Lechleiter et al., 2014).

6. Numerical Validation and Robustness

Representative numerical experiments in two dimensions, such as acoustic scattering from disks with discontinuous $c(x)$, demonstrate:

• Second-order convergence in time until spatial error dominates (BDF2: $p=2$).
• $O(N^{-1})$ convergence in the spatial parameter $N$ until temporal error dominates.
• No observed instability even when the stability restriction $\Delta t \gtrsim N^{-1/13}$ is violated and $q_c$ is discontinuous. This suggests that the collocation method, while lacking a global guarantee, is effective and robust in practical scenarios with strong material contrast (Lechleiter et al., 2014).

7. Relativistic (Dirac) Time-Domain Lippmann–Schwinger Equation

In the relativistic Dirac framework, the time-domain Lippmann–Schwinger equation takes the Duhamel integral form for the solution $\psi(r,t)$ of

$$(i\partial_t - H) \psi(r,t) = 0, \qquad H = H_0 + V,$$

where $H_0 = \alpha \cdot (-i \nabla) + \beta m$, and $V(r)$ is a Hermitian $4 \times 4$ potential. The equation reads

$$\psi(r, t) = e^{-i H_0 t} \varphi_{\mathrm{in}}(r) - i \int_{-\infty}^t ds\, e^{-i H_0 (t-s)} V(r) \psi(r,s),$$

constituting the dynamical Lippmann–Schwinger equation (Sakhnovich, 2019).

Wave operators $W_\pm$ and the scattering operator $S = W_+^* W_-$ are defined via the strong limits of the interaction-picture propagators. Rigorous existence and completeness are established under suitable decay and regularity conditions on the potential.

A key result is the connection between the scattering operator in the time domain and the stationary (energy-resolved) scattering amplitude. In particular, the “quantum ergodic formula” relates the eigenstructure of the on-shell scattering matrix $S_p(E)$ to the stationary scattering amplitude. For central potentials, this recovers the familiar partial-wave expansion. All relativistic integral kernels retain matrix-valued structure and respect Lorentz covariance in their integral formulations (Sakhnovich, 2019).