# Questions tagged [eigenfunctions]

For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.

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### Square-integrable eigenfunctions of the Schrödinger operator decay $\pm \infty$

Why is it necessary for an eigenfunction of $H=\frac{d^2}{dx^2}+u(x)$ that is square-integrable that it tends to zero at $\pm \infty$?
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### Method to find eigenfunctions?

I have a linear operator $L(p(x)) = -p(x+1) + p(x) + \frac6x \int_0^x p(y) dy$ with eigenvalues of it's representative matrix found to be 6, 3, and 2. From this the respective eigen vectors are (1,0,0)...
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### How to find the optimal ${\bf Q}$ that maximizes $\left|{\bf I}+{\bf AQA^HB^{-1}}\right|$

I have the following question, which has stumped me for a long time. ${\max}_{\bf Q \succeq 0}~~~~{\rm det}\left({\bf I}+{\bf AQA^HB^{-1}}\right)$ s.t.$~~~{\rm Tr}\left({\bf Q}\right) \leq 1$. ...
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### Relation eigenfunction of linearized PDE and solution of the original PDE

Consider $$\frac{\partial^2}{\partial x^2}u+\mu \sin(u) = 0 \\ u(0) = 0 = u(1)$$ The linearized version is for small $u$ $$\frac{\partial^2}{\partial x^2}u+\mu u = 0$$ This gives for the general ...
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### What functions can be represented as a series of eigenfunctions

Consider the differential equation: $y'' = \lambda y$ with the boundary conditions $y(0) = y(2\pi) = 0$. This equation has eigenfunctions $\mu_n(x) = \sin(\frac{nx}{2})$ with the corresponding ...
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### Why is a matrix multiplied by an eigenvector not parallel to that eigenvector?

If $\lambda$ is a non-zero eigenvalue with a corresponding eigenvector $v$, then $A v$ is parallel to $v$. This statement is false. Why is that? Would it be parallel to $\lambda v$?
I am solving an exercise concerning the Airy eigenvalue problem $$-y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*)$$ which (among other things) asks me to prove that all eigenvalues are positive. I ...