# 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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### What's the name of this “shooting”-like method for numerical solving of PDE eigenproblems?

Consider a PDE of the following form: $$\left(\partial_r^2+\frac5r\partial_r+\frac4{r^2} \hat L\right)\Psi(r,p)+(E- V(p)U(r))\Psi(r,p)=0,\tag1$$ where $\hat L$ is a differential operator ...
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### Finding eigenfunctions and eigenvalues from a differential equation

Consider the differential equation $$X''(x)+\lambda X=0$$ on $0 \leq x \leq 1$with boundary conditions $$X'(0)+X(0)=0 \ \ \ \ \text{and} \ \ \ \ X(1)=0.$$ I have a few problems here that I think ...
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I came across this: Theorem 1 The eigenfunctions of Sturm-Liouville BVP above satisfy the integral relationship: $$\int_a^b r(x)\phi_n(x) \phi_m(x) \ dx = 0$$ if $m \not= n$, where $\... 3answers 59 views ### Finding eigenfunction to this operator I have the operator $$A : -e^{-2ax} \frac{\partial}{\partial x} \left(e^{2ax}\frac{\partial}{\partial x}\right)\\ D_A = \left( v \in C^2[0,L] \quad | \quad v(0) = v(L) = 0 \right)$$ ... 2answers 73 views ### how to make use of t' in tx = Ax vs general case I ran into this problem on my differential equations homework set. Previously, the questions were in the form x' = Ax, and they were relatively straight forward and easy to complete. I was gifted a ... 4answers 72 views ### Find eigenvalues & eigenvectors for an integral. Can anyone please explain me how to solve it? Find the nonzero eigenvalues and the corresponding eigenvectors:$T:[-1,1]\rightarrow[-1,1]$$T((f(x))=\int_{-1}^1(x^2 y + y^2 x) f(y) \, dy$$ 1answer 103 views ### How to find eigenvalues and eigenfunctions of simple-looking differential operator What process can be used to solve for the eigenvalues and eigenfunctions of the following differential operator? $$H=A\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\cos(... 0answers 25 views ### Fixed points of cubic transformation Let f: \mathbb{R}^n \rightarrow \mathbb{R}^n be a Lipschitz continuous operator and let fix(f) denote the set of fixed points of f. Define the operator g = (1-a) f + a(1-b) f^2 + a b f^3, for ... 0answers 69 views ### Operator acting on a product of functions Let f \in \mathcal{H} be a function f: \mathbb{R} \rightarrow \mathbb{C} in a Hilbert Space \mathcal{H}. Now suppose that for some linear operator T that acts on \mathcal{H} and for some f \... 0answers 117 views ### Solve this integral equation. Let h(x) be a known well-behaved function, I have to solve for \sigma(t):$$ \phi(x) = \int_a^b\log\left[\left(x-t\right)^2 + \left(h(x) - h(t)\right)^2\right]\sigma(t)dt $$Where, b>a>0, ... 0answers 27 views ### Cylindrical Operator Maybe you can help me solve this (simple?) problem I'm too stupid to tackle :-( I want to find the eigenfunctions to the Operator$$ \widehat{O} = -\partial_z^2 - \frac{1}{r}\partial_r r\partial_r $$... 1answer 90 views ### Eigenvalues of second order ordinary differential equation The question is find all the eigenvalue of the following equation -\frac{d^2y}{dx^2}+x^2y=\lambda y I have found the first function which is y=e^{\frac{-x^2}{2}}, however I have no clue on how ... 0answers 182 views ### Eigenvectors of discrete Laplace matrix for 2D unit square under Neumann boundary condition Eigenvectors of discrete Laplace matrix for 2D unit square with free boundary is simply$$ \phi(x,y)= \cos(\frac{\pi}{n} kx) \cos(\frac{\pi}{m} ly) $$It is easy to see that its 2nd order derivative ... 0answers 31 views ### Scale-dependent isoperimetric inequalities (e.g. with heat kernel?) Suppose \Omega\subseteq\mathbb R^2 is a compact, connected planar region with a smooth connected boundary \partial\Omega. Take \lambda_0,\lambda_1,\lambda_2,\ldots to be the Laplacian spectrum ... 0answers 54 views ### Eigenvalue solution of a 2nd order ODE of a geophysical fluid dynamic problem community. I am working on a project with a professor of mine and he suggested me to numerically solve a geophysical fluid dynamic instability ODE equation of a paper of Boccaleti et al. (2007) (... 0answers 35 views ### Eigenvalues of the Black-Scholes operator The Black-Scholes operator is given by$$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)I want to prove that this operator has ... 1answer 31 views ### How to find the eigenvalues of this operator? The Question: Consider the system \begin{align} & Ly(x) \equiv y''(x)+y(x)=f(x),\qquad 0<x<1 \\ & y(0)=1, \qquad y(1)=0 \end{align} (i) Find the eigenvalues and eigenfunctions of L ... 1answer 101 views ### Solution Space of Schrodinger's Equation We are given Hermitian operator of the form H(x) =(-\hbar^2/2m) \partial^2/\partial x^2 + V(x) (where \hbar and m are real constants) which has orthogonal eigenfunctions corresponding to a ... 0answers 25 views ### The Proper Form of Eigenfunctions and Linear Operators So I'm learning about eigenfunctions and eigenvalues and there appear to be 2 main forms 1) L[y]=\lambda y: This is intuitive to me as a direct extension of what I learned in Linear Algebra. 2) L[... 1answer 87 views ### Trick to prove orthogonality for eigenfunctions? Is there a magic trick to see that\int_0^LX_m(x)X_n(x)dx=0$$for$$X_n(x) = c_n\Big[\big[\sin(\omega_n L) - \sinh(\omega_n L)\big]\big[\cos(\omega_n x) - \cosh(\omega_n x)\big] \\ - \big[\cos(\... 1answer 89 views ### Self-Adjoint Operator on Even Periodic Functions I am attempting to show the following operator, which acts on even2 \pi$-periodic functions, is self-adjoint and find its eigenfunctions and eigenvalues.$Ly=\frac{d^2y}{dx^2}, \:\: -\pi\leq x \leq ...
In this question Understanding solution of PDE using method: separation of variables. (in which I put a bounty..) how and why the eigenfunctions were computed like that? The origin of the $T_n$ ...
Find the eigenvalues and eigenfunctions of the operator $$Ly=\frac{d^2y}{dx^2},-\pi\le x \le \pi,$$ which operates on even-2$\pi$ periodic functions. I am unsure of where to start. Any help would ...