By Carl M. Bender
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Extra resources for Advanced mathematical methods for scientists and engineers
For a discussion of the convergence of these expansions and their applications see the Rererences. In Example 5 of Sec. 1 we show how to use WKB theory to find approximate formulas for the eigenvalues E. (x). Example 7 Schriidinger eigenvaille problem. 11) has an infinite number of real positive eigenvalues Eo, E " E Physically, the eigenvalues are the allowed energy levels of a particle In the potential V(x). 4) is an example of a Schrodinger eigenvalue problem. 11) exists only for very special choices of V(x).
13. 5) for the Wronskian implies that if one solution to an Nthorder difference equation is known, then we can find an (N - l)th-order equation for the remaining unknown solutions. For example, if N = 2 and one solution a. is known, a second solution b. satisfies the first-order equation a. bn+ 1 b. an + 1 = There is a general procedure, called reduction of order, for lowering the order of any difference equation once one solution An of the equation is known. The object is to seek additional solutions a.
The Airy function Ai (- E) is a transcendental function whose zeros may be computed numerically. 088. 521, £, *6787,E. 023, .. The graph of the Airy function in Fig. 11 may be used to determine the approximate values of the first few zeros. Asymptotic methods give accurate approxImations to the larger zeros (see Sec. 7). [xample 5 Eigenvalue problem having a fillite "timber of eigenvalues. Consider the eigenvalue problem y" + (E + v sech' x)y = O. 1:(211 where V (j'I-';4;' - 1)/2 and n is an integer (see Prob.
Advanced mathematical methods for scientists and engineers by Carl M. Bender