The dimensionless function on the whole line is to be expanded in the Hermite…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Special functions and orthogonal polynomials · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

The dimensionless function f(x)=x2f(x)=x^2 on the whole line is to be expanded in the Hermite polynomials Hn(x)H_n(x) (H0=1H_0=1, H1=2xH_1=2x, H2=4x2−2H_2=4x^2-2), the eigenfunctions of a singular Sturm-Liouville problem on (−∞,∞)(-\infty,\infty) with weight w(x)=e−x2w(x)=e^{-x^2} and normalisation ∥Hn∥w2=2nn!π\|H_n\|_w^2=2^nn!\sqrt\pi. You may use the standard Gaussian integrals ∫−∞∞e−x2 dx=π\int_{-\infty}^\infty e^{-x^2}\,\mathrm{d}x=\sqrt\pi, ∫−∞∞x2e−x2 dx=π/2\int_{-\infty}^\infty x^2e^{-x^2}\,\mathrm{d}x=\sqrt\pi/2, ∫−∞∞x4e−x2 dx=3π/4\int_{-\infty}^\infty x^4e^{-x^2}\,\mathrm{d}x=3\sqrt\pi/4. Compute the first three coefficients c0c_0, c1c_1, c2c_2 in f(x)=∑ncnHn(x)f(x)=\sum_n c_nH_n(x), using cn=⟨Hn,f⟩w/∥Hn∥w2c_n=\langle H_n,f\rangle_w/\|H_n\|_w^2.

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