Sturm-Liouville eigenvalue

If R(k) = E equals a Sturm-Liouville eigenvalue Ek then the normalized eigenfunction uk(r) must be proportional to i/rk(Ek,r) (note that the dependence on the right-hand boundary is not explicitly indicated in ukand Ek), i.e.,... 

Brown [47] shaped up those semiqualitative considerations into a rigorous Sturm-Liouville eigenvalue problem by deriving the micromagnetic kinetic equation... 

The boundary value problem posed by the differential equation (2.166) and the two boundary conditions (2.168) and (2.169) leads to the class of Sturm-Liouville eigenvalue problems for which a series of general theorems are valid. As we will soon show the solution function F only satisfies the boundary conditions with certain discrete values /q of the separation parameter. These special values /q are called eigenvalues of the boundary value problem, and the accompanying solution functions Fi are known as eigenfunctions. The most important rules from the theory of Sturm-Liouville eigenvalue problems are, cf. e.g. K. Janich [2.33] ... 

But from the definition of the Sturm-Liouville eigenvalue problem (Eq.11.48), the LHS of Eq. 11.51 gives... 

V. Ledoux, M. Van Daele and G. Vanden Berghe, Efficient Computation of High Index Sturm-Liouville Eigenvalues for Problems in Physies, Computer Physics Communications, 2009, 180(2), 241-250. 

X here represents various variables and the equation is therefore a partial differential equation. L[ j represents a linear, homogeneous, self-adjoint differential expression of second order, ip is the desired function, p x) the density function and A the eigenvalue parameter of this Sturm-Liouville eigenvalue problem. ... 

The difference eigenvalue problem for X can be viewed as the Sturm-Liouville difference problem ... 

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,... 

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... 

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... 

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 Sturm-Liouville representation (136) is a formal solution as a knowledge of all eigenfunctions )( 4>), and corresponding eigenvalues Xp is required. However, this representation is very useful because it allows one to obtain a formal solution for the longitudinal complex susceptibility = x i ) According to... 

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

The eigenvalue problem of a general Sturm-Liouville system must be of the following general form, in analogy with the Sturm-Liouville relation given by Eq. 10.185... 

This proves that the eigenvalues of the Sturm-Liouville system are real, not complex numbers. 

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... 

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 all three classes of Sturm-Liouville problems, all eigenvalues are real. 

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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