The Sturm-Liouville Equation

for any discrete set of values /3 , then corresponding solutions y = p will be obtained. Suppose for two distinct values, say )8 and we have solutions p and p . Each of these must satisfy Eq. 10.185, so write 

Our aim is to derive the orthogonality condition (Eq. 10.184) and suitable boundary conditions. To do this, multiply the first of the above equations by p  

The range of independent variables is given as a physical condition, so integrate the above between arbitrary points a and b 

It is clear that the LHS is almost the same as Eq. 10.184. We inspect ways to cause the RHS to become zero (when n is different from m). Each of the integrals on the RHS can be integrated by parts, for example, the first integral can be decomposed as 

When these are subtracted, the integrals cancel, so that we finally have from Eq. 10.190 

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First we consider the Sturm-Liouville equation over the interval a z /8 ... 

At times t < f0 w [where f0 ° is an infinitesimal amount less than f0 ], the density is zero. Only after the pair is formed can there be any probability of its existence [499]. This is cause and effect, but strictly only applicable at a macroscopic level. On a microscopic scale, time reversal symmetry would allow us to investigate the behaviour of the pair at time and so it reflects the inappropriateness of the diffusion equation to truly microscopic phenomena. The irreversible nature of diffusion on a macroscopic scale results from the increase of entropy, and should be related to microscopic events described by the Sturm—Liouville equation (for instance) and appropriately averaged. 

The effective eigenvalue may be defined in the context of the Sturm-Liouville equation as [78]... 

The physical description of the functional derivative Vee (r) requires knowledge of the wavefunction 4 for the determination of the electron-interaction component W e(r) = Wnlr) -i- W (r), and knowledge of both the wavefunction P and the Kohn-Sham orbitals < i(x) for the correlation-kinetic-energy component W, (r). The corresponding Kohn-Sham wavefunction is then a single Slater determinant. It has, however, also been proposed [42,52,53] that the wavefunction V be determined by solution of the Sturm-Liouville equation... 

The weighting function for this particular orthogonality condition defined with reference to the Sturm-Liouville equation is... 

This appears to be a formidable task. To accomplish it, we shall need some new tools. Under certain conditions, it may be possible to compute C without a trial-and-error basis. To do this, we shall need to study a class of ODE with homogeneous boundary conditions called the Sturm-Liouville equation. We shall return to the coated-wall reactor after we gamer knowledge of the properties of orthogonal functions. 

Because they depend on the Sturm-Liouville equation, the separation of variables method and the integral transform yield exactly the same solution, as you would expect. But the advantage of the integral transform is the simplicity of handling coupled PDEs, for which other methods are unwieldy. Moreover, in applying the finite integral transform, the boundary conditions need not be homogeneous (See Section 11.2.3). 

Ag)l2 Cf is the normalized intensity of the lines J and 7 and p is the probability of hopping per time unit. In Brown s model (see Section D.3.2), the relaxation mode given by the smallest nonvanishing eigenvalue, Aj, of the Sturm-Liouville equation is the only significant mode. ... 

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

Volumes have been written about the red herring known as Schrodinger s cat. Without science writers looking for sensation, it is difficult to see how such nonsense could ever become a topic for serious scientific discussion. Any linear differential equation has an infinity of solutions and a linear combination of any two of these is another solution. To describe situations of physical interest such an equation is correctly prepared by the specification of appropriate boundary conditions, which eliminate the bulk of all possible solutions as irrelevant. Schrodinger s equation is a linear differential equation of the Sturm-Liouville type. It has solutions, known as eigenfunctions, the sum total of which constitutes a state function or wave function, which carries... 

Equation (8.79) is of the Sturm-Liouville type, and with boundary conditions Eq. (8.84), solutions may be written as R (r ), cylindrical eigenfunctions or Bessel functions. Therefore, the solution to Eq. (8.75) may be written as... 

The roots h = Hi of this equation are the eigenvalues of the problem, which depend on the Biot number. As Fig. 2.29 shows, there is an infinite series of eigenvalues Hi < /r2 < /U3. .. which is in full agreement with the Sturm-Liouville theory. Only the following eigenfunctions... 

The formal solution of the fractional rotational diffusion equation is obtained just as that of Eq. (135) from the Sturm-Liouville representation [7,57]... 

By applying the Sturm-Liouville theorem, the coefficient A for partial differential equations with nonhomogeneous boundary conditions is obtained as ... 

Fuoss and Kirkwood32 have obtained equations identical with Eqs. 31 and 32 without introducing, explicitly, the exponential decay function. Like Debye they reasoned as if the problem were mainly one of diffusion by Brownian motion under the influence of an external force. Treating this problem as a Sturm-Liouville equation, they developed /lt into a complete set of orthogonal functions yx. A relaxation time rx is associated with each of these functions. 

It is clear that the heat balance produces an inhomogeneous equation, owing to the (assumed) constant heat source term Q. This means of course that a separation of variables approach will not lead to a Sturm-Liouville equation. However, the following expedient will produce a result that is homogeneous let temperature be decomposed into two parts a steady state (the future steady state) and a deviation from steady state... 

The strategy for using Sturm-Liouville transforms is, first, to carefully lay out the algebraic rules for this class of operator. Obviously, the defining equation and boundary conditions must be of the Sturm-Liouville type, as discussed in Chapter 10. 

One can see that the integral transform indeed facilitates the resolution of ODE boundary value problems and also partial differential equations comprised of Sturm-Liouville operators (e.g., Eq. 11.45). The simplicity of such operational methods lead to algebraic solutions and also give a clearer view on how the solution is represented in Hilbert space. Moreover, students may find that the Sturm-Liouville integral transform is a faster and fail-safe way of getting the solution. Thus, Eq. 11.52 represents the solution to an almost infinite variety of ordinary differential equations, as we see in more detail in the homework section. 

This new set of equations for Y now can be readily solved by either the method of separation of variables or the Sturm-Liouville integral transform method. We must also find u(x), but this is simply described by an elementary ODE (Lu = 0), so the Inhomogeneous boundary conditions (11.74) are not a serious impediment. ... 

To overcome these limitations, Payo et alJ proposed a quasi-analytic solution for small breaking angles, expressed as an expansion of the orthogonal set of functions of the Sturm-Liouville problem that arises from the resolution of the homogeneous equation by separation of variables. 

This results from the fact that the separation leads to a Sturm-Liouville equation for each coordinate, for which the theorem mentioned above is valid. Also in the cases where the boundary conditions do not require 4 to vanish, but only require that 4 remain finite, there are no significant changes. 

Coulomb Sturmians (CSs) are an exponential-type complete set of basis functions which satisfy a Sturm-Liouville equation [2]. The main objective of the present work is to derive an ADT for the Slater-type orbitals (STOs), which are the fundamental ETO, and thereby for the CSs, which are a linear combination of STOs. The expression for the two-center overlap integral is then worked out for the CSs as an illustration and numerical results and conclusions are presented. 

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