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We propose a conceptually different method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology, and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential V from external scattering data. Although we seek a local (diagonally dominated) V as the solution to the posed problem, we allow V to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between V and the T matrix of the problem. Here it is important to realize that not every T corresponds to a diagonal V and we, therefore, relax the usual condition of strict diagonality (locality) of V. An iterative algorithm is proposed in which we seek T that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential V that is as diagonally dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion. This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II [H. W. Levinson and V. A. Markel, Phys. Rev. E 94, 043318 (2016)]. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation. [ABSTRACT FROM AUTHOR]

Motivated by recent developments in the experimental study of superconducting graphene and transition metal dichalcogenides, we investigate superconductivity of the Kane-Mele (KM) model with short-range attractive interactions on the two-dimensional honeycomb lattice. We show that intravalley spin-triplet pairing arises from nearest-neighbor (NN) attractive interaction and the intrinsic spin-orbit coupling. We demonstrate this in two independent approaches: We study superconducting instability driven by condensation of Cooperons, which are in-gap bound states of two conduction electrons, within the T-matrix approximation and also study the superconducting ground state within the mean-field theory. We find that Cooperons with antiparallel spins condense at the K and K' points. This leads to the emergence of an intravalley spin-triplet pairing state belonging to the irreducible representation A1 of the point group C6v. The fact that this pairing state has opposite chirality for K and K' identifies this state as a "helical" valley-triplet state, the valley analog to the ³He-B phase in two dimensions. Because of the finite center of mass momentum of Cooper pairs, the pair amplitude in NN bonds exhibits spatial modulation on the length scale of lattice constant, such that this pairing state may be viewed as a pair-density wave state. We find that the pair amplitude spontaneously breaks the translational symmetry and exhibits a p-Kekulé pattern. We also discuss the selection rule for pairing states focusing the characteristic band structure of the KM model and the Berry phase effects to the emergence of the intravalley pairing state. [ABSTRACT FROM AUTHOR]

We consider a superconductor proximity coupled to a two-dimensional ferromagnetic film with a skyrmion texture. Using the T-matrix calculations and numerical modeling we calculate the spin-polarized local density of states in the superconductor in the vicinity of the skyrmion. We predict the skyrmion bound states that are induced in the superconductor, similar to the well-known Yu-Shiba-Rusinov states. The bound-state wave functions have spatial power-law decay. It is suggested that superconductivity could facilitate an effective long-range interaction between skyrmions when bound-state wave functions overlap. [ABSTRACT FROM AUTHOR]