We study the Birman-Schwinger operator for a self-adjoint realisation of the one-dimensional Hamiltonian with the Coulomb potential. We study both the case in which this Hamiltonian is defined on the whole real line and when it is only defined on the positive semiaxis. In both cases, the Birman-Schwinger operator is Hilbert-Schmidt, even though it is not trace class. Then, we have considered some approximations to the Hamiltonian depending on a positive parameter, under given conditions, and proved the convergence of the Birman-Schwinger operators of these approximations to the original Hamiltonian as the parameter goes to zero. Further comments and results have been included.

We provide of a method to integrate first order non-linear systems of differential equations with variable coefficients. It determines approximate solutions given initial or boundary conditions or even for Sturm-Liouville problems. This method is a mixture between an iterative process, a la Picard, plus a segmentary integration, which gives explicit approximate solutions in terms of trigonometric functions and polynomials. The segmentary part is particularly important if the integration interval is large. This procedure provide a new tool so as to obtain approximate solutions of systems of interest in the analysis of chemical reactions. We test the method on some classical equations like Mathieu, Duffing quintic equation or Bratu's equation and have applied it on some models of chemical reactions.

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some non-linear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindel\”of method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in Physics, in which we show that our method possesses an excellent precision even with low iterations.

We develop a one step matrix method in order to obtain approximate solutions of first order non-linear systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the exact solution. We apply the method to different well known examples and check its precision, in terms of local error, comparing it with the error produced by other methods. The advantage of the method over others widely used lies on the great simplicity of its implementation.