Sturm-Liouville Problems

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

The most general solution to a Sturm-Liouville problem is a function... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

In this addendum, we will derive the spectral function from Weyl s theory and in particular demonstrate the relationship between the imaginary part of the Weyl-Titchmarsh m-function, mi, and the concept of spectral concentration. For simplicity we will restrict the discussion to the spherical symmetric case with the radial coordinate defined on the real half-line. Remember that m could be defined via the Sturm-Liouville problem on the radial interval [0,b] (if zero is a singular point, the interval [a,b], b > a > 0), and the boundary condition at the left boundary is given by [commensurate with Eq. (5)]... 

M.S.P. Eastham, On the Location of Spectral Concentration for Sturm-liouville Problems with Rapidly Decaying Potential, Mathematika 45 (1998) 25. 

D.J. Gilbert, B.J. Harris, Bounds for the Points of Spectral Concentration of Sturm-liouville Problems, Mathematika 47 (2000) 327. 

It is interesting to note that the Gottingen school, who later developed matrix mechanics, followed the mathematical route, while Schrodinger linked his wave mechanics to a physical picture. Despite their mathematical equivalence as Sturm-Liouville problems, the two approaches have never been reconciled. It will be argued that Schrodinger s physical model had no room for classical particles, as later assumed in the Copenhagen interpretation of quantum mechanics. Rather than contemplate the wave alternative the Copenhagen orthodoxy preferred to disperse their point particles in a probability density and to dress up their interpretation with the uncertainty principle and a quantum measurement problem to avoid any wave structure. 

These lemmas and their counterpart for v (labeling the eigenvalue as /io) establish the existence of the rest points on the boundary of C+ x C+. As before, we label these rest points Eq,Ei,E2- As with the gradostat, the condition for coexistence is tied to the question of invasiveness. Now, however, the conditions take the form of comparison with the eigenvalues of certain Sturm-Liouville problems rather than with the stability modulus of matrices, as was the case in Chapter 6. We describe just enough of this to show the parameters on which the result depends. 

Think of nii as a parameter and let Xinxi) be the largest eigenvalue of the Sturm-Liouville problem just displayed. The eigenvalue X m2) is a strictly increasing function of m2 satisfying... 

Mikhailov M. D. and N. F. Vulchanov, 1983, A computational procedure for Sturm-Liouville problems, J. Comp. Phys, 50, 323-336. 

Among the many problems encountered in chemical engineering science, those that appear very frequently involve the Sturm-Liouville boundary value problems. Following is an illustration of what is meant by a Sturm-Liouville problem. 

Sturm-Liouville problems are categorized according to the type of boundary conditions that the differential equation must satisfy. 

Generally, there are three classes of Sturm-Liouville problems ... 

For the case with A < 0, it can be shown that only trivial solutions result. Therefore, the periodic Sturm-Liouville problem... 

This results in the class of singular Sturm-Liouville problems having type 3 boundary condition. Further, since... 

A knowledge of the properties of the eigenvalues and eigenfunctions can dramatically reduce the labor typically required to solve a Sturm-Liouville problem. These properties also provide a check on whether or not one s solution is reasonable. 

Properties of the Eigenvalues and Eigenfunctions OF A Sturm-Liouville Problem... 

For the regular and periodic Sturm-Liouville problems, there exist an infinite number of eigenvalues. Further, these eigenvalues can be labeled Ai, Aa,. .. so that X < n < m and Jim A = oo. 

For a regular Sturm-Liouville problem, any two eigenfunctions corresponding to a given eigenvalue are linearly dependent. 

The Laplace transform, F(s), of a solution to a Sturm-Liouville problem is analytic for all finite s except for poles, which correspond to the eigenvalues of the system. 

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