Generalized Sturm-Liouville transform

In Chapters 7 to 12 we deal with numerical solution methods, and partial differential equations (PDE) are presented. Qassical techniques, such as combination of variables and separation of variables, are covered in detail. This is followed by Chapter 11 on PDE transform methods, culminating in the generalized Sturm-Liouville transform. This allows sets of PDEs to be solved as handily as algebraic sets. Approximate and numerical methods close out the treatment of PDEs in Chapter 12. 

In this section, we will apply the finite integral transform to a general Sturm-Liouville system, and the integral transform is therefore called the Sturm-Liouville integral transform. Thus, all finite integral transforms are covered at once Fourier, Hankel, and so forth. 

In the previous section, we developed the finite integral transform for a general Sturm-Liouville system. Homogeneous boundary conditions were used in the analysis up to this point. Here, we would like to discuss cases where the boundary conditions are not homogeneous, and determine if complications arise which impede the inversion process. 

The application of the Sturm-Liouville integral transform using the general linear differential operator (11.45) has now been demonstrated. One of the important new components of this analysis is the self-adjoint property defined in Eq. 11.50. The linear differential operator is then called a self-adjoint dijferential operator. 

Before we apply the Sturm-Liouville integral transform to practical problems, we should inspect the self-adjoint property more carefully. Even when the linear differential operator (Eq. 11.45) possesses self-adjointness, the self-adjoint property is not complete since it actually depends on the type of boundary conditions applied. The homogeneous boundary condition operators, defined in Eq. 11.46, are fairly general and they lead naturally to the self-adjoint property. This self-adjoint property is only correct when the boundary conditions are unmixed as defined in Eq. 11.46, that is, conditions at one end do not involve the conditions at the other end. If the boundary conditions are mixed, then the self-adjoint property may not be applicable. 

As we have observed with the Sturm-Liouville integral transform and we will observe later for this generalized integral transform, there will arise an infinite set of eigenvalues and an infinite set of corresponding eigenfunctions. We then rewrite Eqs. 11.191 to 11.193 as follows to represent the nth values... 

Abstract In this work an algorithm based on the point canonical transformation method to convert any general second order differential equation of Sturm-Liouville type into a Schrodinger-like equation is applied to the position-dependent mass Schrodinger equation (PDMSE). This algorithm is next applied to find potentials isospectral to Morse potential and associated to different position-dependent mass distributions in the PDMSE. Factorization of worked PDMSE are also obtained. 

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