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1. Consider the Sturm-Liouville equation dz (p(x)du) — q(x)u + \p(x)u = 0 dx on a < x 0 and q > 0 for a < x satisfies sin λ = p(t) p(t) n²π² p(t) (Sa √ de with n an integer. Ideally, you use the WKB ansatz. 2' dt 7

1. Consider the Sturm-Liouville equation dz (p(x)du) — q(x)u + \p(x)u = 0 dx on a < x 0 and q > 0 for a < x satisfies sin λ = p(t) p(t) n²π² p(t) (Sa √ de with n an integer. Ideally, you use the WKB ansatz. 2' dt 7

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Consider the Sturm-Liouville equation dz (p(x)du) — q(x)u + \p(x)u = 0 dx on a < x <b, with p > 0 and q > 0 for a < x <b. The boundary conditions are u(a) = u(b) = 0. (a) Show that the change of variables u = w/√p leads to the equation 2 1 d + [ - ²2 (1) - () ² + x² - 1 - = 0 (² w=0 dx P P d² w dx² with w(a) = w(b) = 0. (b) Show that for A large, the solution of the problem in part (a) is (VTS) -1/4 W = () where the eigenvalue > satisfies sin λ = p(t) p(t) n²π² p(t) (Sa √ de with n an integer. Ideally, you use the WKB ansatz. 2' dt 7
Transcribed Image Text:1. Consider the Sturm-Liouville equation dz (p(x)du) — q(x)u + \p(x)u = 0 dx on a < x <b, with p > 0 and q > 0 for a < x <b. The boundary conditions are u(a) = u(b) = 0. (a) Show that the change of variables u = w/√p leads to the equation 2 1 d + [ - ²2 (1) - () ² + x² - 1 - = 0 (² w=0 dx P P d² w dx² with w(a) = w(b) = 0. (b) Show that for A large, the solution of the problem in part (a) is (VTS) -1/4 W = () where the eigenvalue > satisfies sin λ = p(t) p(t) n²π² p(t) (Sa √ de with n an integer. Ideally, you use the WKB ansatz. 2' dt 7
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