Stationary potential scattering admits a formulation in terms of the quantum dynamics generated by a non-Hermitian effective Hamiltonian. We use this formulation to give a proof of the reciprocity theorem in two and three dimensions that does not rely on the properties of the scattering operator, Green's functions, or Green's identities. In particular, we identify reciprocity with an operator identity satisfied by an integral operator M, called the fundamental transfer matrix. This is a multi-dimensional generalization of the transfer matrix M of potential scattering in one dimension that stores the information about the scattering amplitude of the potential. We use the property of M that is responsible for reciprocity to identify the analog of the relation, det M = 1, in two and three dimensions, and establish a generic antipseudo-Hermiticity of the scattering operator. Our results apply for both real and complex potentials.

For a diagonalizable linear operator H:H -> H acting in a separable Hilbert space H, i.e., an operator with a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of eigenvectors that form a Reisz basis of H, the pseudo-Hermiticity of H is equivalent to its generalized parity-time-reversal (PT) symmetry, where the latter means the existence of an antilinear operator X:H -> H satisfying [X,H]=0 and X-2=1. The original proof of this result makes use of the anti-pesudo-Hermiticity of every diagonalizable operator L:H -> H, which means the existence of an antilinear Hermitian bijection tau:H -> H satisfying L-dagger = tau L tau(-1.) We establish the validity of this result for block-diagonalizable operators, i.e., those which have a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of generalized eigenvectors that form a Jordan Reisz basis of H. This allows us to generalize the original proof of the equivalence of pseudo-Hermiticity and generalized PT-symmetry for diagonalizable operators to block-diagonalizable operators. For a pair of pseudo-Hermitian operators acting respectively in two-dimensional and infinite-dimensional Hilbert spaces, we obtain explicit expressions for the antlinear operators tau and X that realize their anti-pseudo-Hermiticity and generalized PT-symmetry at and away from the exceptional points.

For a diagonalizable linear operator $H:\mathscr{H}\to\mathscr{H}$ acting in a separable Hilbert space $\mathscr{H}$, i.e., an operator with a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of eigenvectors that form a Reisz basis of $\mathscr{H}$, the pseudo-Hermiticity of $H$ is equivalent to its generalized PT-symmetry, where the latter means the existence of an antilinear operator $X:\mathscr{H}\to\mathscr{H}$ satisfying $[X,H]=0$ and $X^2=1$. We show that this equivalence is generally valid for block-diagonalizable operators, i.e., those which have a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of generalized eigenvectors that form a Jordan Reisz basis of $\mathscr{H}$. In particular, we establish the equivalence of pseudo-Hermiticity and generalized $PT$-symmetry for arbitrary linear operators acting in a finite-dimensional Hilbert space.

Stationary scattering of TE and TM waves propagating in an isotropic medium with planar symmetry is described by Bergmann's equation in one dimension. This is a generalization of Helmholtz equation which allows for developing transfer matrix methods to deal with the corresponding scattering problems. We use a dynamical formulation of stationary scattering to study the low-frequency scattering of these waves when the inhomogeneities of the medium causing the scattering are confined to a planar slab. This formulation relies on the construction of an effective two-level non-Hermitian quantum system whose time-evolution operator determines the transfer matrix. We use it to construct the low-frequency expansions of the transfer matrix and the reflection and transmission coefficients of the medium, introduce a generalization of Brewster's angle for inhomogeneous slabs at low frequencies, and derive analytic conditions for transparency and reflectionlessness of PT-symmetric and non-PT-symmetric slabs at these frequencies. We also discuss the application of this method to deal with the low-frequency scattering of TE and TM waves when the carrier medium occupies a half-space and the waves satisfy boundary conditions with planar symmetry at the boundary of the half-space. Because acoustic waves propagating in a compressible fluid with planar symmetry are also described by Bergmann's equation, our results apply to the low-frequency scattering of these waves.

For the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions, we identify a condition on the scattering interaction under which the Nth order Born approximation gives the exact solution of the scattering problem for some N >= 1.

For the scattering of plane electromagnetic waves by a general possibly anisotropic stationary linear medium in three dimensions, we give a condition on the permittivity and permeability tensors of the medium under which the first Born approximation yields the exact expression for the scattered wave whenever the incident wavenumber k does not exceed a pre-assigned value α. We also show that under this condition the medium is omnidirectionally invisible for k≤α/2, i.e., it displays broadband invisibility regardless of the polarization of the incident wave.

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.

The study of the scattering of electromagnetic waves by a linear isotropic medium with planar symmetry can be reduced to that of their transverse electric (TE) and transverse magnetic (TM) modes. For situations where the medium consists of parallel homogeneous slabs, one may use the standard transfer matrix technique to address the scattering problem for these modes. We extend the utility of this technique to inhomogeneous permittivity and permeability profiles by proposing a dynamical formulation of the scattering of TE and TM waves in which the transfer matrix for the medium is given in terms of the evolution operator for an effective nonunitary quantum system. This leads to a system of dynamical equations for the reflection and transmission amplitudes. Decoupling these equations, we reduce the solution of the scattering problem for TE and TM modes to that of an initial-value problem for a Riccati equation. We discuss the application of this observation in identifying media that do not reflect TE or TM waves with a given wavenumber and incidence angle.

We develop a fundamental transfer-matrix formulation of the scattering of electromagnetic (EM) waves that incorporates the contribution of the evanescent waves and applies to general stationary linear media which need not be isotropic, homogenous, or passive. Unlike the traditional transfer matrices whose definition involves slicing the medium, the fundamental transfer matrix is a linear operator acting in an infinite-dimensional function space. It is given in terms of the evolution operator for a non-unitary quantum system and has the benefit of allowing for analytic calculations. In this respect it is the only available alternative to the standard Green's-function approaches to EM scattering. We use it to offer an exact solution of the outstanding EM scattering problem for an arbitrary finite collection of possibly anisotropic nonmagnetic point scatterers lying on a plane. In particular, we provide a comprehensive treatment of doublets consisting of pairs of isotropic point scatterers and study their spectral singularities. We show that identical and $\mathcal{P}\mathcal{T}$-symmetric doublets do not admit spectral singularities and cannot function as a laser unless the real part of their permittivity equals that of vacuum. This restriction does not apply to doublets displaying anti-$\mathcal{P}\mathcal{T}$-symmetry. We determine the lasing threshold for a generic anti-$\mathcal{P}\mathcal{T}$-symmetric doublet and show that it possesses a continuous lasing spectrum.

Numerical transfer matrices have been widely used in the study of wave propagation and scattering. These may be viewed as descretizations of a recently introduced fundamental notion of transfer matrix which admits a representation in terms of the evolution operator for an effective non-unitary quantum system. We use the fundamental transfer matrix to develop a general method for the solution of the problem of radiation of an oscillating source in an arbitrary, possibly non-homogenous, anisotropic, and active or lossy linear medium. This allows us to obtain an analytic solution of this problem for an oscillating source located in the vicinity of a planar collection of possibly anisotropic and active/lossy point scatterers such as those modeling a two-dimensional photonic crystal.

The discovery of unidirectional invisibility and its broadband realization in optical media satisfying spatial Kramers–Kronig relations are important landmarks of non-Hermitian photonics. We offer a precise characterization of a higher-dimensional generalization of this effect and find sufficient conditions for its realization in the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions. More specifically, given a positive real number α and a continuum of unit vectors Ω, we provide explicit conditions on the interaction potential (or the permittivity and permeability tensors of the scattering medium in the case of electromagnetic scattering) under which it displays perfect (non-approximate) invisibility whenever the incident wavenumber k does not exceed α (i.e., k∈(0,α]) and the direction of the incident wave vector ranges over Ω. A distinctive feature of our approach is that it allows for the construction of potentials and linear dielectric media that display perfect directional invisibility in a finite frequency domain.

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

In two and three dimensions, the standard treatment of the scattering problem for a multi-delta-function potential, v(r)= n-expressionry sumexpressiontion (N)(n=1)z(n)delta(r-a(n)), leads to divergent terms. Regularization of these terms and renormalization of the coupling constants 3(n) give rise to a finite expression for the scattering amplitude of this potential, but this expression has an important short-coming; in the limit where the centers a(n) of the delta functions coincide, it does not reproduce the formula for the scattering amplitude of a single-delta-function potential, i.e., it seems to have a wrong coincidence limit. We provide a critical assessment of the standard treatment of these potentials and offer a resolution of its coincidence-limit problem. This reveals some previously unnoticed features of this treatment. For example, it turns out that the standard treatment is incapable of determining the dependence of the scattering amplitude on the distances between the centers of the delta functions. This is in sharp contrast to the treatment of this problem offered by a recently proposed dynamical formulation of stationary scattering. For cases where the centers of the delta functions lie on a straight line, this formulation avoids singularities of the standard approach and yields an expression for the scattering amplitude which has the correct coincidence limit. (C) 2022 Elsevier Inc. All rights reserved.

In [J. A. Rebou\c{c}as and P. A. Brand\~{a}o, Phys. Rev. A 104, 063514 (2021)] the authors compute the scattering amplitude for a $\mathcal{P}\mathcal{T}$-symmetric double-delta-function potential in three dimensions by invoking the far-zone approximation and summing the resulting Born series. We show that the analysis of this paper suffers from a basic error. Therefore its results are inconclusive. We give an exact closed-form expression for the scattering amplitude of this potential.

We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials.The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator H which develops an exceptional point for a discrete set of incident wavenumbers.We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem.We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space.The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of H and its exceptional points.Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.

In two dimensions, the standard treatment of the scattering problem for a delta-function potential, $v(\mathbf{r})=\mathfrak{z}\,\delta(\mathbf{r})$, leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant $\mathfrak{z}$. Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors' screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.

We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator \widehat{\mathbf{H}} 𝐇̂ which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space. The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of \widehat{\mathbf{H}} 𝐇̂ and its exceptional points. Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.

We examine the consequences of ignoring the contribution of evanescent waves to the scattering amplitude of a general finite-range potential v in two dimensions. This corresponds to an approximation scheme where v is approximated by an associated energy-dependent nonlocal potential V k which does not couple to evanescent waves. The scattering solutions ψ ( r ) of the stationary Schr¨odinger equation, ( −∇ 2 + V k ) ψ ( r ) = k 2 ψ ( r ), has the remarkable property that their Fourier transform ˜ ψ ( p ) vanishes unless p corresponds to the momentum of a classical particle whose magnitude is given by | p | = ~ k . We construct a fundamental transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We identify an infinite class of potentials for which the above approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schrödinger equation, ( − ∇ 2 + v ) ψ = k 2 ψ for a local potential v , they arise in dimensions greater than one and are generally present regardless of the details of v . The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing v with a certain energy-dependent nonlocal potential V k ˆ . We present a dynamical formulation of the stationary scattering for V ˆ k in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on v , we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for V ˆ k as a densely-defined operator acting in C 2 ⊗ L 2 ( − k , k ) .

We review a recently developed transfer matrix formulation of the stationary scattering in two and three dimensions where the transfer matrix is a linear operator acting in an infinite-dimensional function space. We discuss its utility in circumventing the ultraviolet divergences one encounters in solving the Lippman-Schwinger equation for delta-function potentials in two and three dimensions. We also use it to construct complex scattering potentials displaying perfect omnidirectional invisibility for frequencies below a freely preassigned cutoff.