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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Spherical harmonics as simultaneous eigenfunctions

If I consider operators $L^2$ and $L_z$ in spherical coordinates where $L^2$ is the angular momentum squared operator and $L_z$ is the $z-axis$ component of the angular momentum, is a function like $\...
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The connections between the eigenfunctions of SOPDE and eigenvector in Linear Algebra

In my mathematical physics course, we were introduced to the separation of variable techinique to solve second order partial differential equations. After the separation of variables, we solve each ...
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Graph Laplacian for weight matrix with negative edges

How can I normalize my weight matrix to get a positive semi-definite Laplacian, if I am using a weight matrix with negative edges?
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An eigenfunction inequality for integral operators

I have an integral operator $T$ defined with respect to a positive semidefinite kernel function $k: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ probability measure $\mu(dx) = p(x) dx$ defined ...
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Eigenfunction and partial derivative (linearity)

After noting that $x \to e^{ixp}$ is an eigenvector (with eigenvalue $p$) of the endomorphism $\hat p:u \to \frac{1}{i} \frac{\partial u}{\partial x}$, my teacher goes one step beyond and states that ...
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Matching Coefficients of Fourier Series with Separation of Variables Solution for Discontinuous Boundary Conditions: 2D Slab Conduction

I am trying to find the temperature profile in a 2D domain with steady heat conduction. The non-dimensional domain is shown below. Domain dimensions, coordinate system, boundary conditions, and ...
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Is $x+\frac{d}{dx}$ differential operator diagonalizable?

I would like to prove or disprove the following statement: Let $T=x+\frac{d}{dx}$ be a differential operator. Then $T$ is diagonalizable under some initial/boundary condition. Here is my attemp to ...
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The domain and range of eigenfunctions

If I have a known function $k(x,y)$, with eigenvalues $\lambda_n$ and eigenfunctions $f_n(r)$, what can I say about the domain of these eigenfunctions? Is it the same as the range of $k()$?
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Brownian Motions, Integrals, and Orthonormal Functions

I am out of my element with a topic I am working on. I think I have dwindled down the part I am stuck on to the following. If $\phi_n(t)$ is a sequence of orthonormal (eigenfunctions) functions and $...
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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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What is a bilinear concomitant (or conjunt) with regards to linear differential operators

Context I am studying self-adjoint eigenfunction problems using [1]. I am working through Example 1 on page 54 in [1]. Example 1 (page 54 in [1]) Suppose we have the linear differential operator $$ L[...
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Second order Sturm-Liouville ODE with difficult boundary conditions

How do you solve the problem with these boundary conditions. The equation $u_{xx}-(4-\lambda)u=0$ is clearly in Sturm-Liouville form, and the characteristic equation is $z^2-(4+\lambda)=0$ which has ...
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Evolution of the eigenfunctions of a Lax operator

Let $L:=L(t)$ be a Lax operator, i.e., there exists an operator $P:=P(t)$ such that the pair (L,P) satisfy the Lax equation $$ \frac{dL}{dt}=PL-LP\,.$$ This operator satisfies the isospectral property,...
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Quantum Graphs and the Gross-Pitaevskii equation

I'm a final year mathematical physics student and I've been tasked with researching into the gross-pitaevskii equation and it's solutions on quantum graphs. I have a pretty good understanding but I'm ...
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Two eigenfunctions with parallel gradients

Let $(M^2,g)$ be a compact connected Riemannian surface without boundary. We denote by $\Delta = -div(\nabla)$ the Laplace-Beltrami operator. Suppose $u$ and $v$ are eigenfunctions of $\Delta$ ...
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Eigenvalues and eigenfunctions of a discrete problem

Do you know which are the eigenvalues and eigenfunctions of the $T$-periodic discrete problem $$-\Delta_2x(k-1)=\lambda x(k), x(k)=x(k+T) (k\in\mathbb{Z}),$$ where $-\Delta_2x(k-1)=x(k+1)-2x(k)+x(k-1)$...
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Conditions for $A^2x=\lambda^2 x \implies Ax=\lambda x$ w/ $A$ hermitian (Hilbert space $L^2$)

I have found that the function $$f(z)=e^{iqz}-\frac{a}{2iq+a}e^{iq|z|}$$ satisfies $$-\partial_z^2f(z)=(q^2+a\delta(z))f(z).$$ Note this looks like an eigenvalue equation, but I am not sure if we can ...
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Some problems in Cheeger's "A lower bound for the smallest eigenvalue of the Laplacian"

Pictures below is from Cheeger's "A lower bound for the smallest eigenvalue of the Laplacian" $f$ is the eigenfunction of smallest eigenvalue of Laplacian on $M$, where $M$ is Riemannian ...
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Partial Differential Equations With Two Solution Paths

When solving a $2$D Heat Equation, suppose I separate the solution into time and space, i.e., $f_1(t,\ T(t),\ T_t(t),\ ...) = f_2(x,\ y,\ Z(x,\ y),\ Z_x(x,\ y),\ Z_y(x,\ y),\ ...) = \lambda$, and then ...
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Is the eigen-composite under differentiation unique, at least up to a scalar addition or coefficient?

Let's suppose you have two differentiable functions $f(x)$ and $g(x).$ First of all, is there a name to a situation where $\frac{d}{dx}[ f \circ g(x)] = g(x)$, or, are there any articles that go into ...
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Finding Eigenvalues of First Order ODEs with Periodic BC?

Suppose I have the eigenproblem: $$A(x)\frac{df(x)}{dx} + B(x) f(x) = \lambda f(x), x\in[a,b]$$ $$f(a)=f(b)$$ where $A$, $B$ are some smooth real functions for simplicity. If we would have also an ...
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Kernel matrix: eigen-functions, eigenvalues

The kernel is defined as $K(x,y) = (\langle x,y\rangle + 1)^d$ for $x,y\in R^{p}$. For, example for $p=2, d=2$ $K(x,y) = 1+2x_1y_1+2x_2y_2+x_1^2y_1^2+x_2^2y_2^2+2x_1x_2y_1y_2 = \sum_{m=1}^{M}h_m(x)...
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Solving $u_t=4u_{xx}+\sin{(\pi x)}+\sin{(2\pi x)}$ using an eigenfunction expansion

I would like to solve $\begin{cases} u_t=4u_{xx}+\sin{(\pi x)}+\sin{(2\pi x)} & 0<x<1, t>0 \\ u(0,t)=0 \\ u(1,t)=0 \\ u(x,0)=0\end{cases}\tag*{}$ using an eigenfunction expansion. My ...
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Trying to find eigenfunctions for the D.E. $y''+λy=0$ with B.C. $y'(0)=y'(π)=0$

The way of solution which I followed is: Setting $\lambda=k^2$ I get $y'' + k^2y=0$ which gives $y(x)=Asin(kx)+Bcos(kx)$ Taking the 1st derivative $y'(x)=kAcos(kx)-kBsin(kx)$ I apply the BC $y'(0)=kA=...
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How are $X_0(x)=1$ and $X_1(x)=x$ eigenfunctions for $v''(x)=\lambda x=0*x$ s.t. $v_{x}(0)=v_{x}(l)=\frac{v(l)-v(0)}{l}$?

How to conclude that for $\lambda =0$, $X_0(x)=1$ and $X_1(x)=x$ are the only eigenfunctions? I know there are two. B.C.: $v_{x}(0)=v_{x}(l)=\frac{v(l)-v(0)}{l}$
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Sturm-Liouville problem - Partial Differential Equations

I tried looking on the Mathematica forum for a similar solution but unfortunately haven't found one. I was given the following Sturm-Liouville differential equation problem: $x^2y''+5xy'+\lambda y=0$ $...
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Finding Eigen Functions/Values for arbitrary integral equations?

I cannot figure this out from my text book. All the solutions look to me to be specialized to particular cases and I want to know what the general approach is. Is there a simple(ish) analogue of ...
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When solving a S-L eigenvalue problem like $y'' + (\lambda + 1)y = 0$, is it better to set $\lambda =/>/< 0$, or $\lambda + 1 =/>/< 0$

So if I'm given a Strum Liouville equation in a form similar to the one I've mentioned above, which method gives me the correct eigenvalues and eigenfunctions? Should I be using $$\lambda= 0, \lambda&...
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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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Eigenfunctions of 1D gaussian kernel

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}...
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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 ...
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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 ...
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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 ...
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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 $uv$ ...
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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}...
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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}{\...
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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 $$g'' + \lambda g = 0.$$ ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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Eigenvalue equation - angular bracket notation

How do you read this formula? $$\left\langle \dfrac{1}{\sqrt{2}} ( \alpha\beta - \beta\alpha) \bigg\rvert \hat{S^2} \bigg\rvert \dfrac{1}{\sqrt{2}} ( \alpha\beta - \beta\alpha)\right\rangle$$ $\alpha$ ...
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3 votes
1 answer
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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{equation} \begin{array}{rllc} \dfrac{\partial^{2}T}{\partial x^{2}}+\dfrac{\partial^{2}T}{\partial y^{2}} & =0 & \text{in }[0,1]...
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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 $A$ ...
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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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solving Laplace operator and its expansion

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 $u_y(x,0)=u_x(x,\pi)=0$ ...
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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&...
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