Question 1. (15 marks) d2 (a) Show that a is a hermitian linear transformation, with respect to the inner product L (f, g> = a [0 f(x)*g(x) dx (a...
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How to show

data-value="-d^2/dx^2">d2/dx2 is hermitian linear transformation with respect to given inner product?

Also how to justify the given eigenfuction expansion?

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Question 1. (15 marks) d2
(a) Show that — a is a hermitian linear transformation, with respect to the inner product
L
(f, g> = a [0 f(x)*g(x) dx (a e Rm,
when acting on the vector space of (nice) functions satisfying f ’(0) = f ’(L) = 0. (2 marks) (b) Use this to explain how to justify the following eigenfunction expansion: 1 0° mrx
f(x) = 5610 + gun COS T. [You need not justify the actual form of the eigenfunctions.] (2 marks)

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