Let act on complex-valued functions on

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Inner products, orthogonality and the adjoint

Problem

Let T=d2/dx2T = \mathrm{d}^2/\mathrm{d}x^2 act on complex-valued functions on [0,1][0,1], with inner product ⟨f,g⟩=∫01f∗(x)g(x) dx\langle f,g\rangle = \int_0^1 f^*(x)g(x)\,\mathrm{d}x. Integration by parts twice gives ⟨f,Tg⟩=[f∗g′−f∗′g]01+⟨Tf,g⟩.\langle f, Tg\rangle = \big[f^*g' - f^{*\prime}g\big]_0^1 + \langle Tf, g\rangle. ((a)) Evaluate the boundary term [f∗g′−f∗′g]01\big[f^*g' - f^{*\prime}g\big]_0^1 for f(x)=xf(x) = x and g(x)=x3g(x) = x^3. ((b)) Does ⟨f,Tg⟩=⟨Tf,g⟩\langle f,Tg\rangle=\langle Tf,g\rangle hold for this pair? State what a domain for TT must do to this boundary term, for every pair of functions in it, before that identity can hold throughout the domain.

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