# 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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### Determine the eigen vectors by dividing a normal vector

I have a matrix [[1, 2] [3, 4]] I had to create a formula in python to determine the eigen values and eigen vectors of the matrix. Determining the eigen values ...
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### Numerical solution of first-order linear PDEs

Consider a linear, first order partial differential operator $L$ with: $$Lu = \sum^{N}_{i=1} a_{i}(x_1,\ldots,x_{N})\frac{\partial}{\partial x_{i}} u$$ For some---lets say Lipschitz continuous---...
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### Perturbed Eigenvalue Problem

Consider the nonlinearly perturbed eigenvalue problem $$\dfrac{d^2\phi}{dx^2}+\lambda \phi= \varepsilon \phi^3$$ $$\phi(0)=0, \ \phi(L)=0$$ Determine the perturbation of the eigenvalue $\lambda_1$. ...
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Heads up Im a physicist! I want to know explicitly the eigenfunctions of the 1D gaussian kernel $$K(x,y) = e^{-(x-y)^2/\sigma^2}$$ when it is integrated, that is $$(Kf)(n,x)=\int_{-\infty}^{\infty}... 0 votes 0 answers 32 views ### Approaches to find eigenfunctions of integral kernel? Heads up I’m a physicist! I have seen many questions on the site on how to find the igenfunctions of specific examples of integral kernels and I haven’t go the gist of it yet. I’m interested in the ... 0 votes 0 answers 14 views ### How Can One Express u(xy) as a Diagonalized Transform Kernel, K(x,y)? Consider a projection operator P_{u}g(x)=<g(x),u(x)>, where u(x) is an eigenfunction normalized under an inner product, <u_{m}(x),u_{n}(x)>=\delta_{m,n}. (ASIDE: Inner products may ... • 39 2 votes 0 answers 183 views ### Solution to the Heat Equation: Time-dependent Source, Spherical Coordinates, Insulating Boundary Condition I am looking to solve the PDE that takes the form of$$\frac{\partial C}{\partial t}-Q(t)=D\nabla^2C$$where$$\nabla^2C = \frac{1}{r^2}\frac{\partial}{\partial r}r^2\frac{\partial C}{\partial r}$$I ... 0 votes 0 answers 28 views ### Do time-invariant self-adjoint operators have locally orthogonal eigenfunctions? Let T be some self-adjoint, time-invariant (in that it commutes with any shift) operator on L^2(\mathbb{R}). Let u, v be generalized eigenfunctions of T. Is it true that the product uv ... • 1,025 0 votes 0 answers 45 views ### Time-Variant Eigenvalue Dilemma I've got this problem where it is the diffusion equation with a source term in  x : [0,L]. \frac{\partial v}{\partial t} = D_V \frac{\partial^2 v}{\partial x^2} + \frac{\partial \bar{c}}{\partial t}... • 19 0 votes 0 answers 70 views ### Solving Diffusion Equation with Non-Homogeneous Mixed Conditions - Robin & Neumann I've been grappling with this problem for a while now and am at a loss. Here is the problem statement for a function c(x,t) as defined by the following parabolic equation: \frac{\partial c}{\... • 19 2 votes 2 answers 35 views ### Eigenfunction differential equation with boundary values Consider the differential equation$$f'' + 2f' + (\lambda + 1)f = 0, \ \ \ \ f(0) + f'(0) = 0, f(L) = 0.$$We can make g(x) = e^x f(x), so our differential equation becomes$$g'' + \lambda g = 0.$$... • 1,197 4 votes 0 answers 70 views ### Do Laplace-Beltrami eigenfunctions vary continuously with the metric? I'm interested in Laplace Beltrami operators$$-\Delta_g:\ \ D(-\Delta_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric ... • 41 2 votes 0 answers 44 views ### Hermitian operator: Eigenfunction conjugate for the same eigenvalue In section 17.3.4 of the textbook Mathematical Methods for Physics and Engineering, the author claims that since \begin{align*} \mathcal{L}y_i = \lambda_i \rho y_i \end{align*} for an Hermitian linear ... • 117 0 votes 0 answers 35 views ### Radial Wave Eigenfunctions of the Radial Laplacian, with Centers off the Origin of Coordinates The purely radial version (without \theta or \phi angles) of the Laplacian is, \nabla ^2 = \frac{\partial^2}{\partial r^2} + r^{-1} \frac{\partial}{\partial r}. It has eigenfunctions of the ... • 313 3 votes 0 answers 57 views ### Eigenfunctions of Laplacian in Different Coordinate Systems The Laplacian, \nabla ^2 = \frac{\partial^2}{\partial x^2 }+ \frac{\partial^2}{\partial y^2 }, in 2D cartesian coordinates, has eigenfunctions of the form Ae^{-i(k_x x + k_y y)} + Be^{i(k_x x + k_y ... • 313 0 votes 0 answers 57 views ### Need help finding the eigenvalues of this differential equation I have the following second-order ordinary differential equation p(1-p)X''(p)+(ap(1-p)+bp+c)X'(p)+(d*p+e+f\lambda)X(p)=0 where a,b,c,d,e,f are arbitrary constants, and of which Mathematica gives ... 4 votes 0 answers 73 views ### Extending Ky Fan's eigvenalues inequality to kernel operators Base result The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \lambda_i(B)$$where A, B are Hermitian matrices and \... • 3,106 1 vote 1 answer 113 views ### Eigenvalue equation - angular bracket notation How do you read this formula?$$\left\langle \dfrac{1}{\sqrt{2}} ( \alpha\beta - \beta\alpha) \bigg\rvert \hat{S^2} \bigg\rvert \dfrac{1}{\sqrt{2}} ( \alpha\beta - \beta\alpha)\right\rangle$$\alpha ... • 289 3 votes 1 answer 140 views ### Confusion about orthogonality for eigenfunctions of separable PDE, where eigenvalues satisfy \tan \lambda = \lambda I am having a go at solving the following simple problem: \begin{array}{rllc} \dfrac{\partial^{2}T}{\partial x^{2}}+\dfrac{\partial^{2}T}{\partial y^{2}} & =0 & \text{in }[0,1]... • 173 0 votes 0 answers 36 views ### Conditions on operator defined on real valued separable Hilbert space Consider the separable real valued Hilbert space  H = W^{2,2}([0,1], \mathbb{R}) and a linear map A defined on a dense subset of H, A : D_A \to H. I want to know, what properties should A ... • 297 3 votes 1 answer 65 views ### Computing the eigenvalues of the precision operator C_0^{-1}=\eta(-\triangle)^p+KI Consider L_2(\mathbb{T}) with the basis$$\phi_{2k}(x)=\sqrt{2}\cos(2\pi k x)\\ \phi_{2k-1}(x)=\sqrt{2}\sin(2\pi k x) for $k\in\mathbb{N}$. The functions $\phi_k$ belong to the domain $H^{2p}$ of ...
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How can i solve this laplace operator? $\nabla^2u(x,y)=\lambda u(x,y)$ in rectangular region $0\le x\le\pi$, $0\le y\le\pi$ and boundary conditions $u_x(0,y)=u_x(\pi,y)=0$ and $u_y(x,0)=u_x(x,\pi)=0$ ...
### If $e_1$ is the first eigenfunciton on $Ω$, then $0<\inf \frac{e_1(x)}{d(x,\:∂Ω)}<\sup \frac{e_1(x)}{d(x,\:∂Ω)}<\infty$?
Let $(e_1,\lambda_1)$ denote the first eigenpair of the Dirichlet Laplacian on a bounded open set $\Omega$ with smooth boundary such that $\max_{\overline\Omega}e_1=1$. We can show that there is a \$c&...