Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Re: zero eigen function
Posted:
Jun 7, 2014 12:32 AM


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(xx_0) and e^(i (kk0) 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



