We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the generating function. The method allows us to choose the cyclic operators and corresponding generating function selectively, depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the perturbation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classical oscillator, Schrodinger equation, and wave equation in dispersive medium.

Abstract:
We present a closed-form solution for n-th term of a general three-term recurrence relation with arbitrary given n-dependent coefficients. The derivation and corresponding proof are based on two approaches, which we develop and describe in detail. First, the recursive-sum theory, which gives the exact solution in a compact finite form using a recursive indexing. Second, the discrete dimensional-convolution procedure, which transforms the solution to the non-recursive expression of n, including a finite number of elementary operations and functions.

Abstract:
We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the generating function. The method allows us to choose the cyclic operators and corresponding generating function selectively, depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the perturbation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classical oscillator, Schr\"odinger equation, and wave equation in dispersive medium.

Abstract:
We propose and develop a general method of numerical calculation of the wave function time evolution in a quantum system which is described by Hamiltonian of an arbitrary dimensionality and with arbitrary interactions. For this, we obtain a general n-order single-step propagator, which could be used for the numerical solving of the problem with any prescribed accuracy. We demonstrate an applicability of the proposed approach by considering a propagation of an electron in focused electromagnetic field with vortex electric field component.

Abstract:
We present an approach to numerically solving the time-dependent Schroedinger equation and other parabolic equations by the split-step technique with fast Fourier transform, which suppresses the backreflection of waves from the grid boundaries with any specified accuracy. Most importantly, all known methods work well only for a narrow region of incident waves spectrum, and the proposed method provides absorption of any wave whose length is large enough in comparison with the size of absorption region.

Abstract:
We model the emission of high energy photons due to relativistic charged particle motion in intense laser-plasma interactions. This is done within a particle-in-cell code, for which high frequency radiation normally cannot be resolved due to finite time steps and grid size. A simple expression for the synchrotron radiation spectra is used together with a Monte-Carlo method for the emittance. We extend previous work by allowing for arbitrary fields, considering the particles to be in instantaneous circular motion due to an effective magnetic field. Furthermore we implement noise reduction techniques and present validity estimates of the method. Finally, we perform a rigorous comparison to the mechanism of radiation reaction, and find the emitted energy to be in excellent agreement with the losses calculated using radiation reaction.

Abstract:
We perform a numerical study of the interaction of a high-intensity laser pulse with a nano-structured target. In particular, we study a target where the nano-structuring increases the absorption rate as compared to the flat target case. The transport of electrons within the target, and in particular in the nano-structure, is analysed. It is shown that it is indeed possible, using a terawatt class laser, to light up a nano-scale Christmas tree. Due to the form of the tree we achieve very strong edge fields, in particular at the top where the star is located. Such edge fields, as here located at ion rich spots, makes strong acceleration gradients possible. It also results in a nice, warm glow suitable for the holiday season.

Abstract:
Although, for current laser pulse energies, the weakly nonlinear regime of LWFA is known to be the optimal for reaching the highest possible electron energies, the capabilities of upcoming large laser systems will provide the possibility of running highly nonlinear regimes of laser pulse propagation in underdense or near-critical plasmas. Using an extended particle-in-cell (PIC) model that takes into account all the relevant physics, we show that such regimes can be implemented with external guiding for a relatively long distance of propagation and allow for the stable transformation of laser energy into other types of energy, including the kinetic energy of a large number of high energy electrons and their incoherent emission of photons. This is despite the fact that the high intensity of the laser pulse triggers a number of new mechanisms of energy depletion, which we investigate systematically.

Abstract:
We study nonperturbative pair production in intense, focused laser fields called e-dipole pulses. We address the conditions required, such as the quality of the vacuum, for reaching high intensities without initiating beam-depleting cascades, the number of pairs which can be created, and experimental detection of the created pairs. We find that e-dipole pulses offer an optimal method of investigating nonperturbative QED.

Abstract:
We study nonperturbative pair production in intense, focused laser fields called e-dipole pulses. We address the conditions required, such as the quality of the vacuum, for reaching high intensities without initiating beam-depleting cascades, the number of pairs which can be created, and experimental detection of the created pairs. We find that e-dipole pulses offer an optimal method of investigating nonperturbative QED.