Dynamical Formulation of Stationary Scattering

• Dynamical formulation of stationary scattering is a method that recasts fixed-energy scattering problems into spatial evolution equations, enabling extraction of scattering data via transfer matrices or wave operators.
• It employs non-Hermitian Hamiltonians and rigorous projection techniques to bridge stationary and time-dependent representations, ensuring detailed analysis in one-dimensional and multidimensional systems.
• This approach underpins practical applications in potential and Helmholtz scattering, inverse design in optics, and regularization of singular interactions, offering a unified framework for quantum and acoustic scattering.

A dynamical formulation of stationary scattering is a family of methods that replace a fixed-energy boundary-value problem by an evolution problem, or else identify stationary scattering data directly with objects defined from asymptotic dynamics. In one prominent line of work, a stationary Schrödinger, Helmholtz, Bergmann, or Dirac equation is rewritten as a first-order evolution equation in a spatial coordinate treated as an effective “time,” with a non-Hermitian Hamiltonian and a transfer matrix given by a time-ordered exponential. In another line, stationary wave matrices, generalized Fourier transforms, and fiberwise scattering matrices are shown to coincide with time-dependent wave operators and scattering operators. These formulations occur in one-dimensional potential scattering, multidimensional Helmholtz scattering, TE/TM and acoustic slab scattering, Stark Hamiltonians, long-range $N$-body systems, relativistic Dirac theory, and unitary scattering for quantum walks (Mostafazadeh, 2013, Loran et al., 2021, Loran et al., 12 Sep 2025, Adachi et al., 2019, Skibsted, 2021, Aldecoa, 2019).

1. Conceptual scope

The phrase denotes two closely related but technically distinct programs. The first treats one spatial coordinate as an evolution parameter and rewrites a second-order stationary equation as a first-order Schrödinger-like system for a two-component state. In that setting, the transfer matrix is literally an evolution operator, usually non-unitary because the effective Hamiltonian is non-Hermitian; scattering amplitudes become matrix elements or operator entries of that evolution (Mostafazadeh, 2013, Loran et al., 2021).

The second program starts from the stationary side—resolvents, limiting absorption principles, generalized eigenfunctions, and fiber decompositions—but proves that the resulting stationary wave operators or wave matrices coincide with time-dependent wave operators. The scattering matrix at fixed energy is then the fiberwise representation of the dynamical scattering operator. Abstract constructions of this type were developed for self-adjoint operator pairs with a rigging $F$, for unitary operators in a two-Hilbert-space setting, and for long-range $N$-body Hamiltonians (Azamov, 2013, Aldecoa, 2019, Skibsted, 2021).

A useful distinction follows from this split. In transfer-matrix formulations, the “time” variable is not physical time but a longitudinal coordinate such as $x$ or $z$. In wave-operator formulations, the dynamical content is literal large-$t$ asymptotics. This suggests that the same label covers both a spatial-evolution representation of stationary equations and a rigorous identification of stationary and dynamical scattering objects.

2. One-dimensional spatial evolution and transfer matrices

For the stationary Schrödinger equation

$-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$

one introduces asymptotic coefficients by

$$\psi(x)\sim A_\pm e^{ikx}+B_\pm e^{-ikx}\qquad (x\to\pm\infty),$$

and defines the transfer matrix $M(k)$ through

$$\begin{bmatrix}A_+\ B_+\end{bmatrix} = M(k) \begin{bmatrix}A_-\ B_-\end{bmatrix}.$$

The dynamical formulation replaces the second-order scalar equation by a first-order $F$0 system for

$F$1

which satisfies

$F$2

with

$F$3

The transfer matrix is then the evolution operator $F$4 (Mostafazadeh, 2013).

This recasting imports standard evolution-operator structure into stationary scattering. Composition of transfer matrices for concatenated potentials becomes the composition law for evolution operators, and the tracelessness of the effective Hamiltonian yields $F$5 in the one-dimensional prototype (Loran et al., 23 Aug 2025). Reflection and transmission amplitudes are recovered algebraically from $F$6, and the reciprocity statement $F$7 follows from the same determinant identity in that setting (Loran et al., 23 Aug 2025).

A further refinement treats a truncated potential $F$8 and evolves the transfer matrix with respect to the truncation point. This gives a first-order dynamical equation for $F$9, $N$0, and induces a Riccati equation for the right reflection amplitude. After introducing an auxiliary function $N$1, $N$2, the scattering problem reduces to a second-order linear ODE with universal initial conditions, equivalent to an initial-value formulation of the stationary Schrödinger equation for a Jost solution (Mostafazadeh, 2013).

Low-energy analysis fits naturally into the same framework. For exponentially decaying potentials, the transfer matrix admits a Laurent expansion in $N$3, and the coefficients are determined by two solutions of the zero-energy Schrödinger equation. A zero-energy transfer matrix $N$4 can itself be written as the evolution operator of another effective two-level non-Hermitian system, and zero-energy resonances are characterized by zeros of specific entries of $N$5 or of the trivially extended half-line problem (Loran et al., 2021).

3. Higher dimensions, fundamental transfer matrices, and evanescent sectors

In two and three dimensions, one chooses a longitudinal coordinate and Fourier-transforms in the transverse variables. The result is an effective first-order evolution equation in the longitudinal variable for a two-component state whose components are functions of transverse momentum. The transfer matrix becomes a $N$6 matrix of operators on an infinite-dimensional function space rather than an ordinary finite matrix (Loran et al., 2021, Loran et al., 2024).

For scalar Helmholtz scattering in $N$7, the higher-dimensional analogue is the fundamental transfer matrix

$N$8

where $N$9 is a $x$0 matrix of integral operators in transverse momentum space and $x$1 projects to the propagating band $x$2 (Loran et al., 2024). The entries of $x$3 determine the reflected and transmitted momentum amplitudes, from which the scattering amplitude is reconstructed (Loran et al., 2021).

A central correction to earlier multidimensional transfer-matrix attempts is the treatment of evanescent modes. The proper object is not a single transfer matrix but a pair of intertwined matrices: an auxiliary matrix obtained from the full non-unitary evolution and a fundamental matrix obtained by projection onto propagating modes. This resolves the role of $x$4 sectors and shows that evanescent waves influence the evolution even though they are absent from asymptotic scattering data (Loran et al., 2021).

The same mechanism extends to low-frequency asymptotics. In two and three dimensions, the Dyson expansion of the effective Hamiltonian provides low-frequency expansions of the scattering amplitude for scalar Helmholtz problems, and the method is used both on exactly solvable models and in outlining a low-frequency cloaking scheme (Loran et al., 2024). For TE and TM waves in planar media, Bergmann’s equation

$x$5

is rewritten as

$x$6

where $x$7 is a non-Hermitian $x$8 Hamiltonian depending on $x$9 and $z$0. The transfer matrix is the evolution operator of this effective two-level system, and its Dyson series yields low-frequency expansions of reflection and transmission coefficients, a generalized Brewster angle, and analytic conditions for transparency and reflectionlessness of PT-symmetric and non-PT-symmetric slabs (Loran et al., 12 Sep 2025).

4. Stationary wave operators, generalized eigenfunctions, and fixed-energy fibers

Not all dynamical formulations pass through spatial transfer matrices. For the one-body Stark Hamiltonian

$z$1

stationary scattering is developed in Besov-type spaces adapted to parabolic coordinates. The limiting resolvents $z$2 exist as operators $z$3, stationary wave operators are proved to exist and to be complete, generalized Fourier transforms are constructed, and the asymptotics of generalized eigenfunctions of minimal growth are characterized by the stationary scattering matrix (Adachi et al., 2019). Here the dynamical content lies in the wave-operator structure and generalized Fourier representation rather than in a transfer matrix.

An abstract constructive version begins with self-adjoint operators $z$4 and a rigging $z$5 such that $z$6 and $z$7 is Hilbert–Schmidt on bounded energy windows. One then constructs a sheaf of fiber Hilbert spaces $z$8, evaluation operators $z$9, and stationary wave matrices $t$0. The direct integrals of these wave matrices define stationary wave operators, and these are proved to coincide with the classical time-dependent wave operators. The stationary scattering matrix has an explicit fiberwise formula in terms of $t$1, $t$2, and the evaluation operators (Azamov, 2013).

For long-range $t$3-body Hamiltonians in the Dereziński–Enss class, including Coulomb interactions, all entries of the stationary scattering matrix are shown to be well defined at non-threshold energies and weakly continuous in the energy parameter. Channel wave matrices are strongly continuous, and away from a measure-zero exceptional set the scattering and channel wave matrices form a complete stationary scattering theory: the scattering matrix is unitary, strongly continuous, and characterized by the asymptotics of minimum generalized eigenfunctions (Skibsted, 2021). This places fixed-energy channel scattering on the same footing as the one-body theory.

5. From time evolution to stationary amplitudes

A complementary theme is the derivation of stationary amplitudes from genuinely time-dependent scattering processes. One route is the limiting-amplitude principle. For the Schrödinger equation in $t$4 driven by a harmonic source,

$t$5

the solution converges at large time to a stationary harmonic regime with amplitude $t$6. Sending the source to infinity turns spherical limiting amplitudes into the plane-wave Lippmann–Schwinger solution, and the resulting flux computation yields the usual differential cross section formula $t$7 (Komech, 2012).

A second route identifies the stationary scattering amplitude directly with the integral kernel of the fixed-energy dynamical scattering operator. For the three-dimensional Schrödinger equation with short-range potential, the on-shell $t$8 operator is related to the unitary scattering operator $t$9 by

$-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$0

and the stationary scattering amplitude satisfies

$-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$1

so the spatial asymptotics at $-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$2 are encoded in the angular kernel of the dynamical operator $-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$3 (Sakhnovich, 2019). The relativistic Dirac analogue goes further: the stationary amplitude is reconstructed from the eigenvalues and eigenfunctions of the energy-fiber scattering operator $-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$4, giving an explicit channel expansion in the spectral data of the dynamical scattering operator (Sakhnovich, 2019).

Generalized wave-operator theory makes this equivalence precise for long-range systems by inserting deviation factors. In that framework, generalized wave operators are defined by

$-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$5

and generalized stationary scattering uses large-$-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$6 asymptotics modified by a stationary deviation factor $-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$7 (Sakhnovich, 2016). For the radial Dirac equation with Coulomb-type potential, the generalized dynamical scattering operator is proved to coincide with the corresponding generalized stationary scattering operator, an identification presented as a quantum-mechanical analogue of ergodic formulas in classical mechanics (Sakhnovich, 2016). A related unitary theory for two-Hilbert-space scattering shows that stationary wave operators defined from resolvent boundary values coincide with strong wave operators, and the stationary scattering matrix is the fiber decomposition of the dynamical scattering operator; anisotropic quantum walks provide the main application (Aldecoa, 2019).

6. Applications, design principles, and caveats

Once stationary scattering is written as an evolution problem, inverse design becomes unusually explicit. In one dimension, prescribing an auxiliary function $-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$8 with universal initial conditions determines a finite-range potential through the linearized ODE, and this is used to construct optical potentials with threshold lasing, anti-lasing, and unidirectional invisibility at a prescribed wavelength (Mostafazadeh, 2013). In two and three dimensions, support conditions on the transverse Fourier transform of the potential give omnidirectional invisibility below a chosen cutoff and, in a complementary regime, exactness of the first Born approximation (Loran et al., 2022, Loran et al., 2021).

The formalism also changes how singular interactions are handled. For delta-function potentials in two and three dimensions, the fundamental transfer matrix yields finite scattering amplitudes without introducing the divergent $-\psi''(x)+v(x)\psi(x)=k^2\psi(x),$9 term of the standard Lippmann–Schwinger treatment. The projection onto propagating modes acts as an implicit regularization, and the resulting amplitudes match the usual renormalized answers (Loran et al., 2022, Loran et al., 2021). In higher dimensions, the same operator framework underlies recent reciprocity results: reciprocity is identified with an operator identity satisfied by the fundamental transfer matrix, and this leads to a multidimensional analogue of $$\psi(x)\sim A_\pm e^{ikx}+B_\pm e^{-ikx}\qquad (x\to\pm\infty),$$0 together with a generic anti-pseudo-Hermiticity statement for the scattering operator (Loran et al., 23 Aug 2025).

Several common misconceptions are clarified by the literature. First, in transfer-matrix formulations the effective “time” is a spatial coordinate, not physical time; physical large-$$\psi(x)\sim A_\pm e^{ikx}+B_\pm e^{-ikx}\qquad (x\to\pm\infty),$$1 asymptotics enter only in wave-operator and scattering-operator formulations. Second, in multidimensional transfer-matrix theory the auxiliary evolution operator is not itself the physical transfer matrix; one must account for evanescent sectors and project back to propagating modes (Loran et al., 2021). Third, stationary scattering states do not always form an orthogonal continuum basis in the naive sense. For generic finite-range potentials, overlap integrals can contain nondiagonal terms in addition to the $$\psi(x)\sim A_\pm e^{ikx}+B_\pm e^{-ikx}\qquad (x\to\pm\infty),$$2 contribution, and superpositions of exact stationary states may then have time-dependent norms and finite probability currents. Exceptional cases such as free theory, the $$\psi(x)\sim A_\pm e^{ikx}+B_\pm e^{-ikx}\qquad (x\to\pm\infty),$$3-potential, and the linear potential do not exhibit these nondiagonal terms (Ishikawa et al., 2024). This suggests that the dynamical formulation is not merely a calculational convenience; it is often the correct framework for identifying which stationary objects correspond to physically isolated scattering states.