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 ...