Bound states differential equation

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

In Eq. (32), we have included the full spectrum of a second-order ordinary differential equation with negative bound state eigenvalues and the continuum being the positive real axis. The free-particle background is mbee = i /k. In this case, the full m-function becomes [in the equation below we have introduced a natural generalization of Js, the "jump" or imaginary part, see Eq. (36) for the general case]... 

In the previous sections the potential scattering problem has been defined in terms of a Schrodinger differential equation with outgoing spherical-wave boundary conditions. The description and computational methods are analogous to those used for one-electron bound-state problems. In this section we see that the whole problem in the coordinate representation can be written in terms of a single integral equation, which in many ways is easier to understand physically than the differential equation. 

These two boundary conditions. Equations (14-19) and (14-20J. first stated by Dandcwerts, have become known as the famous Dcinckwerfs bound-ciiy conditions. Bischoff has given a rigorous derivation of them, solving the differential equations governing the dispersion of component A in the entrance and exit sections and taking the limit as Z) in entrance and exit sections approaches zero. From the solutions he obtained boundary conditions on the reaction section identical with those Danckwerts proposed. 

When the whole information over all the oi)cn states is needed in a single fixed energy QM cafeufation, a method based on a time-independent approach is used. In this a.j)j)roach, the time variable is factored out and the stationary wavefunction is exj)anded in terms of a set of one-dimension-less functions of the bound coordinates. This expansion and the subsequent integration over all the bound variables leads to a set of coupled differential equations on the coordinate connecting reactants to products (reaction coordinate) [25]. 

The most accurate method for multilevel curve crossing problems is, of course, to solve the close-coupling differential equations numerically. This is not the subject here, however instead, we discuss the applications of the two-state semiclassical theory and the diagrammatic technique. With these tools we can deal with various problems such as inelastic scattering, elastic scattering with resonance, photon impact process, and perturbed bound state in a unified way. The overall scattering matrix 5, for instance, can be defined as... 

The discrete variable method can be interpreted as a kind of hybrid method Localized space but still a globally defined basis function. In the finite element methods not only the space will be discretized into local elements, the approximation polynomials are in addition only defined on this local element. Therefore we are able to change not only the size of the finite elements but in addition the locally selected basis in type and order. Usually only the size of the finite elements are changed but not the order or type of the polynomial interpolation function. Finite element techniques can be applied to any differential equation, not necessarily of Schrodinger-type. In the coordinate frame the kinetic energy is a simple differential operator and the potential operator a multiplication operator. In the momentum frame the coordinate operator would become a differential operator and hence due to the potential function it is not simple to find an alternative description in momentum space. Therefore finite element techniques are usually formulated in coordinate space. As bound states x xp) = tp x) are normalizable we could always find a left and right border, (x , Xb), in space beyond which the wave-functions effectively vanishes ... 

Therefore, bound state-like solution methods can be employed in calculating the Cx- place of the differential equations for 4, ... 

In the context of the numerical solution of the Boltzmann equation and the numerical solution of partial differential equations, stability ensures that roundoff and other errors in the calculation are not amplified to cause the calculation to blow up. An algorithm for solving an evolutionary (e. g., a time-dependent) partial differential equation is stable if the numerical solution at a fixed time remains bounded as the time stepsize goes to zero. An important theorem in mathematics, the so-called Lax equivalence theorem, states that an algorithm converges if it is consistent and stable. 

The most regular behaviour of a solution of a differential equation is if it tends to a constant in the state-space. It is a bit more complicated if, after an initial transient period, the solution tends to a periodic orbit. In more than two dimensions it may also happen that the trajectories remain in a bounded set, but they neither tend to an equilibrium point, nor to an oscillatory solution. This is a loose definition of chaos. 

We begin with a word on the general properties of these equations and how they can be derived. Even though the FD representation is not variational in the sense of bounds on the ground-state energy, the iterative process by which we obtain the solution to the partial differential equations (PDFs) can be viewed variationally, i.e., we minimize some functional (see below) with respect to variations of the desired function until we get to the lowest action or energy. This may seem rather abstract, but it turns out to be practical since it leads directly to the iterative relaxation methods to be discussed below. (See Ref. 98 for a more complete mathematical description of minimization and variational methods in relation to the FE method.)... 

---