Complex Methods | Part IB, 2003

(a) Show that if ff satisfies the equation

f(x)x2f(x)=μf(x),xR,f^{\prime \prime}(x)-x^{2} f(x)=\mu f(x), \quad x \in \mathbb{R},

where μ\mu is a constant, then its Fourier transform f^\widehat{f} satisfies the same equation, i.e.

f^(λ)λ2f^(λ)=μf^(λ).\widehat{f}^{\prime \prime}(\lambda)-\lambda^{2} \widehat{f}(\lambda)=\mu \widehat{f}(\lambda) .

(b) Prove that, for each n0n \geq 0, there is a polynomial pn(x)p_{n}(x) of degree nn, unique up to multiplication by a constant, such that

fn(x)=pn(x)ex2/2f_{n}(x)=p_{n}(x) e^{-x^{2} / 2}

is a solution of ()(*) for some μ=μn\mu=\mu_{n}.

(c) Using the fact that g(x)=ex2/2g(x)=e^{-x^{2} / 2} satisfies g^=cg\widehat{g}=c g for some constant cc, show that the Fourier transform of fnf_{n} has the form

fn^(λ)=qn(λ)eλ2/2\widehat{f_{n}}(\lambda)=q_{n}(\lambda) e^{-\lambda^{2} / 2}

where qnq_{n} is also a polynomial of degree nn.

(d) Deduce that the fnf_{n} are eigenfunctions of the Fourier transform operator, i.e. fn^(x)=cnfn(x)\widehat{f_{n}}(x)=c_{n} f_{n}(x) for some constants cn.c_{n} .

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