# Tagged Questions

For questions on eigenfunctions.

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### Neumann Series for integral equation with inhomogeneous term zero

Consider the method described in the following article: http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html In this notation, what happens when $f(x)=0$? All the terms seem to be zero ...
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### Can the zero vector be within the eigenspace

I have a matrix that looks like this: $$\begin{pmatrix} 1 & 2 & 4 \\ 2 & 4 & 2 \\ 4 & 1 & 1 \end{pmatrix}$$ Now the calculated eigenvalues are: $-3$, $2$ and ...
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### Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy$ Does anyone know for which functions (or types of ...
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### For what operators are $sin(ax)$ and $cos(bx)$ eigenfunctions?

So it is clear that the operator $\frac{d^{2}}{dx^{2}}$ is an eigenfunction of $sin(ax)$ and $cos(ax)$. For what other operators are sins and cosines eigenfunctions?
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The solution of the symmetric integral equation below: $$g(s) = f(s) + \lambda \int_{-1}^{1} (st +s^2t^2)g(t)dt \tag{*}$$ with separable kernels method is $$g(s) = f(s) + \lambda \int_{-1}^{1} ... 0answers 28 views ### Simple random walk on the N-cycle I am considering the following example: In my lecture notes we noted that "the functions (\phi_j)_j form a basis". I think they refer to the space \mathbb{C}^G where G is the above ... 1answer 66 views ### Least positive eigenvalue of the BVP y''-\lambda y'+\frac{2\lambda-1}{x}y=0, y(0) = y(1/2) = 0 Find the first positive eigenvalue \lambda of the boundary value problem over x\in [0,\frac{1}{2}].$$y''-\lambda y'+\frac{2\lambda-1}{x}y=0, \quad y(0)=y(\tfrac{1}{2})=0.$$My approach: I ... 0answers 20 views ### What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds? Consider the operator \Box=g^{\mu\nu}\nabla_\mu\nabla_\nu acting on a function space \mathbf{F}(M), given by the set of functions \phi:M\to\mathbb{R} whose values go to zero at infinity (at the ... 1answer 28 views ### How do I find an A_0 and A_n which satisfy the initial conditions of this heat equation? Let's say I have the heat equation \frac {\partial u}{\partial t} = k\frac {\partial^2 u}{\partial x^2}, 0 \lt x \lt L, t \gt 0, subject to the boundary conditions$$\begin{cases} \frac ...
I came across the following statement in a paper, The hyperbolic Laplacian is a real smooth operator, and the system of nodal lines determines $f$ up to a constant multiple. Here, $f$ is an ...