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## Homework Statement

Consider the set of functions [itex] {f(x)} [/itex] of the real variable [itex] x [/itex] defined on the interval [itex] -\infty< x < \infty [/itex] that go

to zero faster than [itex] 1/x [/itex] for [itex]x\rightarrow ±\infty[/itex] , i.e.,

[tex]

\lim_{n\rightarrow ±\infty} {xf(x)}=0

[/tex]

For unit weight function, determine which of the following linear operators is Hermitian when acting upon [itex] {f(x)} [/itex]:

[tex] (a) \frac{d}{dx} + x[/tex] [tex](b) -i \frac{d}{dx}+x^2 [/tex][tex](c) ix \frac{d}{dx} [/tex][tex](d) ix \frac{d^3}{dx^3} .[/tex]

## Homework Equations

[itex] Hf(x)=λf(x) [/itex] has real values of [itex] λ [/itex] where [itex] H [/itex] is a Hermitian operator and [itex] λ [/itex] are it's eigenvalues

## The Attempt at a Solution

[tex] a) \frac{df(x)}{dx} + xf(x) = λf(x) [/tex][tex] \frac{df(x)}{dx} + (x-λ)f(x) = 0 [/tex] [tex]\text{Integrating factor is }e^{\int (x-λ)dx}=e^{\frac{1}{2} x^2-λx}[/tex]

[tex]e^{\frac{1}{2} x^2-λx}f(x)=constant[/tex]

I've done a similar thing for parts a), b), and c) but I'm not sure what to do with this or if it even helps. For d) I've tried to work out the eigenfunctions but get to mess and didn't really want to continue down the route I was going without knowing if this was useful or not.

[tex]\text{b) leads to } e^{ix(\frac{x^2}{3}-λ)}f(x)=constant[/tex]

[tex]\text{c) leads to } x^{iλ}f(x)=constant[/tex]

[tex]\text{d) } ix \frac{d^3f(x)}{dx^3}=λf(x) [/tex]

[tex] x \frac{d^3f(x)}{dx^3}+iλf(x)=0 [/tex]

[tex]\text{let }x=f(t)[/tex]

[tex]\frac{d^3f(x)}{dx^3}=\frac{d^3f(x)}{dt^3}\frac{d^3t}{dx^3} [/tex]

[tex]\text{want }\frac{d^3t}{dx^3}=\frac{1}{x}[/tex]

[tex]\text{(After integrating 3 times }t=\frac{1}{2}(x^2(ln(x)-\frac{3}{2})) [/tex]

This is were I decided not to continue until I knew whether or not I was actually doing anything right.

Thanks in advance for any help.