Sturm-Liouville theory

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form 

A version of Green s theorem follows from partial integration of the symmetric integral 

For homogeneous boundary conditions, the logarithmic derivatives f[ /f and fj/f2 are equal at both end-points xo, xi. Hence the integrated term vanishes, and the differential expression L[f] is self-adjoint with these boundary conditions. The weighting function p can be eliminated by converting to u = p - f. Then A[m] = (Pit ) — Qu, where 

A Green function is defined as a solution of the inhomogeneous equation (using a Dirac delta-function) 

If the full set of eigenfunctions is complete, any function y in the Hilbert space that is orthogonal to all eigenfunctions must vanish identically. To prove this, suppose that some function y exists such that (ui y) = 0, i n, but (y y) = 1 and /,(y G y) = 1 for finite positive X. The construction given above develops the expansion 

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In a molecular-orbital-type (Hartree-Fock or Kohn-Sham density-functional) treatment of a three-dimensional atomic system, the field-free eigenfunctions ir e can be rigorously separated into radial (r) and angular (9) components, governed by respective quantum numbers n and l. In accordance with Sturm-Liouville theory, each increase of n (for... 

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... 

We obtain the coefficients An by applying initial condition (3-215) using the orthogonality properties of the eigenfunctions (as guaranteed by the general Sturm-Liouville theory). Again, numerical values are reported in Slattery.17... 

First, let us see what we can say about stability for the inviscid fluid. The key is to note that (12-128) and (12 129) are problems of the so-called Sturm Liouville type. This means that we can characterize the sign of the growth-rate factor a2 based on the sign of F. Before drawing any conclusions, it may be useful to briefly review the general Sturm-Liouville theory. The latter relates to the properties of the general second order ODE,... 

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)]... 

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] ... 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

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