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Topic: zero eigen function
Replies: 3   Last Post: Jun 7, 2014 12:32 AM

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Roland Franzius

Posts: 418
Registered: 12/7/04
Re: zero eigen function
Posted: Jun 7, 2014 12:32 AM
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exdreaming@gmail.com wrote:
> Dear all, I'd like to know whether there's a condition for a linear
> operator such that it only have eigenfunction f=0.
> Thank you!



Since any linear operator would map f=0 to f=0 with eigenvalue 0 this
trivial eiegnfunction is excluded form the notion "egenfunction".
Sometimes one uses the term "nontrival eigenfunction" in the context of
nonlinear differential equations.

Eg for L^2(R) the typical unbounded linear operators like X and iP

X: f[x] -> x f(x)
iP: f[x] -> f'[x]

have no nontrivial eigenfunctions.

The generalized eigenfunctions are the distributional kernels
delta(x-x_0) and e^(i (k-k0) x) from space of linear functionals dual
to
eg. the space of smooth test functions with rapid decay in space
|x|->oo
and Fourier space |k| ->oo, such that the map  f -> f* X(f) is in L^1.

A dense set in L^2(R) on which on can define X, iP and limits is given
by the oscillator eigenfunctions of (d/dx + x )(-d/dx + x)

f_n(x) = exp(-x^2/2) HermiteH_n(x)

but for almost all functions f in L^2(R) the images of X f and f' are
not in L^2.

In l^2, the space of absolute square summable sequences, the analogon
are operators of the form

X_ik = 1/sqrt(i) delta_(i,k+1) +-Transpose

It is the matrix representations of X and iP in the Hermite polynomial
base { f_n }  of L^2(R).

--

Roland Franzius



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