In one dimension one can dissect a scattering potential v(x) into pieces vi(x) and use the notion of the transfer matrix to determine the scattering content of v(x) from that of vi(x). This observation has numerous practical applications in different areas of physics. The problem of finding an analogous procedure in dimensions larger than one has been an important open problem for decades. We give a complete solution for this problem and discuss some of its applications. In particular we derive an exact expression for the scattering amplitude of the delta-function potential in two and three dimensions and a potential describing a slab laser with a surface line defect. We show that the presence of the defect makes the slab begin lasing for arbitrarily small gain coefficients.Pacs numbers: 03.65.Nk, 42.25.Bs

We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a 2 × 2 matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for δ-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.

We introduce "chain by chain" method for constructing the constraint structure of a system possessing both first and second class constraints. We show that the whole constraints can be classified into completely irreducible first or second class chains. We found appropriate redefinition of second class constraints to obtain a symplectic algebra among them. 1

We offer a simple route to perfect omnidirectional invisibility in a spectral band of desired width. Our approach is based on the observation that in two dimensions a complex potential v(x,y) is invisible for incident plane waves with a wavenumber not exceeding a pre-assigned value α, provided that its Fourier transform with respect to y, which we denote by v˜(x,K), vanishes for K≤2α. We can fulfill this condition for potentials modeling the permittivity profile of an optical slab. Such a slab is perfectly invisible for any transverse electric wave whose wavenumber is in the range [0,α]. Our results also apply to transverse magnetic waves propagating in a medium with a relative permittivity ϵ^(x,y) that is a smooth bounded function with a positive real part.