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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2
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0answers
73 views

Discrete theta functions and periodicity

I'm doing quantum mechanics and I have an eigenfunction which is a theta function. I then discretised it, since I want see if I can find the eigenvalues for the discrete case by finding the ...
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1answer
215 views

Singular linear systems of ODEs

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0. $$ Then, to find ''bound states'', you solve on the right and find the ...
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1answer
21 views

related to eigen value eigen function

$\frac{d}{dx}(x\frac{dy}{dx}+\frac{λ}{x}y)=0$ where $λ>0$ Find the eigen value and eigen function using the condtion $y(0)+y_1(0)=0$ and $y(1)+y_1(1)=0$ May be it is easy, I try this but did ...
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1answer
532 views

Number of similarity classes with same characteristic equation.

Let say I have a characteristic polynomial e.g. $(\lambda-1)^4(\lambda-2)^3$. How can I find the number of non-similar matrices with this characteristic equation?
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1answer
28 views

Finding eigenfunctions in a polynomial space

I'm not sure how to do this, especially since the answer contradicts with mine. Question: Let $V$ be the vector space of all real functionson $[0,\pi]$ which are arbitrarily often differentiableon $(0,...
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2answers
6k views

When eigenvectors for a matrix form a basis

It is well known that if n by n matrix A has n distinct eigenvalues, the eigenvectors form a basis. Also, if A is symmetric, the same result holds. Consider $ A =\left[ {\begin{array}{ccc} 1 &...
4
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1answer
94 views

An approximate eigenvalue that is not an eigenvalue

Could you please help me understand why $\lambda$ in the example below is not an eigenvalue? It's easy to see that each $\lambda_n$ is an eigenvalue but I am having difficulty ascertaining that their ...
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0answers
19 views

Changing frequency is a linear transformation?

Does changing the frequency of sinusoidal curves count as linear transformation? Im studying about eigen values and functions in PDE. There it sin (lambda.x) is a eigen function with eigen value ...
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1answer
306 views

Orthonormal set for the set of compactly supported functions

How to find an orthonormal set in the space of compactly supported smooth functions on $\mathbb{R}$? Moreover, for the operator $\frac{d^2}{dt^2}$, what are the eigenfunctions in the space of ...
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1answer
431 views

Eigenvalues and eigenfuctions of Laplacian in an annulus, with certain boundary condition

Given $\Sigma_\varepsilon = \left\lbrace (x,y)\in\mathbb{R}^2:\left\lvert\sqrt{x^2+y^2}-1\right\rvert\le \varepsilon \right\rbrace$ for some small $\varepsilon>0$. I want to determine analytically ...
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66 views

Analytic solution for partial differential equation (Eigenvalues and eigenfunctions)

Given $\Sigma_\epsilon = \{(x,y)\in\mathbb{R}^2:|\sqrt{x^2+y^2}-1|\le \epsilon \}$ for some small $\epsilon>0$ I want to determine(analytically) all the functions $u$ and constants $\lambda$ (...
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124 views

How will covariance matrix converge to the covariance function, especially in the aspect of eigenvalues and eigenvectors

In spatial statistics, people want to estimate the covariance function $C(x,y)$, $x,y\in [0,1]$. Because this covariance function is positive definite, we have the following decomposition $$C(x,y)=\...
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1answer
367 views

Sturm-Liouville eigenvalues and eigenvectors

Hi i was wondering if someone could give me the following definitions. The First part of the question is to explain the terms eigenvalue and eigenfunctions in relation to the strum liouville problem ...
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92 views

Normal derivative of the First Dirichlet eigenfunction

Good afternoon. My question is related to the Maximum principle. In fact: We consider $\Omega\subset(M,g)$ a bounded domain in a Riemannian manifold and we consider the first Dirichlet ...
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1answer
30 views

Making an argument regarding the minimum characteristic polynomial rigorous.

The exercise I'm doing is: Let $a, b, c$ be elements of a field $F$ and let $A$ be the following matrix over $F$. Prove that the characteristic polynomial of $A$ is $x^3-ax^2-bx-c$, and that this is ...
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1answer
1k views

Eigenvalues and eigenvectors of a reflection about a plane

Consider the linear transformation $T: \mathbb{R^3 \rightarrow R^3}$ given by the reflection about the plane $\textit{P}:x+2y-z=0$. In other words, $T(\textbf{v})=\textbf{v}-2\text{proj}_n\textbf{v}$, ...
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1answer
183 views

Book suggestions about eigenfunction expansion method in PDE solutions

I want to solve non-homogenous PDE's (or PDE's system) by the method. I am looking for the notes, books etc. including theory and examples with solution about the method. If I find right book, I can ...
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2answers
750 views

Find eigenvalues and eigenfunctions for $y'' + λy = 0$

$y'' + λy = 0$ BC (Boundary Conditions): $y'(0) = 0, y(π) = 0$ My Work I set $\lambda = \mu^2$ and $y=e^{rx}$ $$ y^2 + \mu^2y = 0 \\ r^2 + \mu^2 = 0 \\ r = \pm\mu i \\ y=c_1\cos(\mu x) + c_2\sin(\...
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2answers
66 views

Ordinary differential equation eigenvalues/eigenfuntions

Consider the DE $$y''+\lambda y=0$$ where $\lambda$ is a constant subject to the boundary conditions $$y(0)=0$$ and $$y(a)=0$$ where $a$ is a positive constant I want to find the eigenvalues and ...
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52 views

Sturm–Liouville problem: approximating potential.

Let one-dimentional Sturm–Liouville problem: $$ \frac{d^2f(x)}{dx^2}+U(x)f(x)=\lambda f(x) $$ with some appropriate boundary conditions, have a set of solutions $\{\lambda_i,\phi_i\}$. Let further ...
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1answer
319 views

Proof for Sturm Liouville eigenfunction expantion pointwise convergence theorem

In "Elementary Partial Differential Equation" by Berg and McGregor, the following theorem is given without proof: Let $f(x)$ be piecewise smooth on the interval $[a,b]$ and let $\{\varphi_n(x)\}$ ...
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1answer
163 views

Doubt relating to Laplace transform, eigenfunction and LTI systems

The complex exponential e^(st) is an eigenfunction of Linear time invariant (LTI) systems. This implies that if the input is a complex exponential e^(st), the output will be the same complex ...
5
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1answer
243 views

Orthogonality between eigenfunctions of euler-bernoulli differential equation for a simple beam

The Euler-Bernoulli differential equation without load is: $$ EI{\frac {\partial ^{4}w}{\partial x^{4}}}+\mu {\frac {\partial ^{2}w}{\partial t^{2}}}=0 \tag{1}$$ Using $w(x,t)=\phi(x)q(t)$ the ...
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1answer
127 views

How does putting this DE in Sturm-Liouville form help me to solve this integral problem?

Consider the following question I have worked out that the Eigenvalues for this DE are $\lambda_n = 4 - \frac{n^2 \pi^2}{4}$ and that the corresponding Eigenfunctions are $y_n = B \cdot Sin(n \pi)$ ...
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36 views

Integral Equation with Non convergent Eigenfunction Expansion

Let $K(x,y)=K(y,x)$ be a continuous symmetric function. The integral equation $$\varphi(x)=\lambda\int_a^b K(x,y)\varphi(y)dy$$ has eigenvalues $\lambda_n$ and eigenfunctions $\varphi_n(x)$. It is ...
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22 views

How far off can a variational test function be if it results in very precise estimate of an eigenvalue?

I'm reading about variational principle in quantum mechanics, which basically states that smallest (algebraically) eigenvalue $E_0$ of an Hermitian operator $H$ can be calculated as the minimum of the ...
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1answer
458 views

Bessel Functions Eigenfunctions

I've been asked to find the bounded eigenvalues and eigenfunctions of this DE in this form: $$\frac{d}{dx}\left(x\frac{dy}{dx}\right) = \lambda xy$$ where $x\in[0,1]$ and $y(1)=0$ The hint given is ...
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28 views

What are current state-of-the art methods of calculating eigenfunctions of 3-body Coulombic Hamiltonian to high precision?

I'm trying to implement a numeric solver of quantum 3-body problems, which would be able to calculate the wavefunctions (not only energies!) to high precision (15 decimal places) on a simple home PC. ...
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1answer
127 views

The eigenfunction of a regular Sturm–Liouville problem is either a real valued function or a complex constant multiplied by a real valued function.

The eigenfunction $u$ of a regular Sturm–Liouville problem is either a real valued function or a complex constant multiplied by a real valued function. Given hint: Observe that if $u$ is an ...
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1answer
362 views

What is the relation between completeness and the closure relation?

Why the closure relation is equivalent to the completeness of a set of eigenfunctions: $\{\psi _k\}$ which is complete: $\{\psi _k\}$ is a complete basis $\iff \sum_k \psi^*(x) \psi(x') = \delta(x-...
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1answer
54 views

Maximum and minimum value

The transfer function of a system is given by $$ \frac{V_o(s)}{V_i(s)} = \frac{1-s}{1+s} $$ Let the output of the system be $v_o(t)= V_m \sin(\omega t + \phi)$ for the input $v_i(t) = V_m \sin(\omega ...
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1answer
222 views

Determining if $\lambda$ is an eigenvalue given a boundary value problem

I'm given a BVP, $y''+2y'=\lambda y$; $\ y(0)=0,\ y(1)=0.$ Is $\lambda=-1$ an eigenvalue? If not why? If it is, how do we find the corresponding eigenfunction? I'm also supposed to solve the above ...
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58 views

Eigenvalues and eigenfunctions of a seemingly simple operator.

I would be very thankful for any hints concerning the finding the eigenvalues and eigenfunctions $f(x)$ of the following operator defined on the segment $[0,2\pi]$: $$ -\frac{d^2}{dx^2}+V\cos mx, $$ $...
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1answer
218 views

Diagonalizing differential operator

I would like to diagonalize the differential operator $D=-\partial^2_t+a^2$ with Dirichlet boundary conditions $x(0)=x(T)=0.$ So far I have tried to find the eigenfunctions of $D$, $$Df = \lambda f$$ ...
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51 views

Eigen functions/value for SL-operator

I've been stuck on finding the eigenfunctions/values for the radial part of the Sturm Liouville laplace operator in spherical coordinates for two given boundry conditions $u(r1)=u(r2)=0$. I have the ...
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1answer
56 views

How to calculate $\int_{0}^{1}-\frac{1}{2\pi }\ln(\left \| x-y \right \|)\cdot e^{i2\pi kt_{x}}dt_{x}$

I came across the integral $$\int_{0}^{1}-\frac{1}{2\pi }ln(\sqrt{(r\cdot\cos(2\pi t_{x})-r\cdot \cos(2\pi t_{y}))^{2}+(r\cdot\sin(2\pi t_{x})-r\cdot\sin(2\pi t_{y}))^{2}})\cdot e^{i2\pi kt_{x}}dt_{x}$...
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0answers
94 views

Eigenfunction and Eigenvalues of an Integral Operator

Given the following integral operator: $(Au)(y)=\int \limits_{\Gamma}u^*(x,y) u(x) \,ds_x$ with $\Gamma=\{x\in\mathbb{R}^2:x=r\begin{pmatrix}\cos(2\pi t) \\ \sin(2\pi t) \end{pmatrix},0 \leq t < ...
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27 views

Formulate the linear operator in infinite dimension using its eigenfunctions

I know that in LA, you can represent a square matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ with its eigenvalues and eigenvectors. Why is it not possible to have an equivalent equation for infinite ...
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1answer
3k views

eigenfunctions $y''+\lambda y=0$ and $y^{\prime} (0)=0$ , $y^{\prime} (1)=0$

Find the eigenvalues and eigenfunctions for $y^{\prime \prime}+\lambda y=0$ with the boundary conditions $y^{\prime} (0)=0$ , $y^{\prime} (1)=0$ then prove that the eigen functions are orthogonal, ...
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0answers
80 views

Using an eigenfunction expansion to solve $y''(x) + y(x) = \cos x$

Hello people im new to this site and have a problem i cant seem to solve: Use the method of eigenfunction expansions to solve $y''(x) + y(x) = \cos x$ $∀x ∈ (0, π)$ subject to $\left\{ \begin{array}{...
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2answers
1k views

Eigen signals and eigen functions

Can anyone show me that both $\cos t$ and $\sin t$ are eigen signals. Here is a little bit background of eigen-function. The output of a continuous-time, linear time-invariant system is denoted ...
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1answer
98 views

A representation of scaling operator for functions

Let us define a "scaling operator" $\hat{\Sigma}(\lambda)$ such that when it acts on any function $f(x)$, it gives $$\hat{\Sigma}(\lambda)f(x)=f(\lambda x).$$ Is it possible to represent the operator $...
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0answers
58 views

Struggling to find the eigenfunctions of an equation.

I have the equation, $$(1+\lambda)\xi(r,\theta)=\gamma\int_{0}^{2\pi}w(r,\theta, R, \theta')\xi(R,\theta')d\theta'$$, where $w=w(r,\theta, r', \theta')$ I know that, $$\xi_{n}(R,\theta)=cos(n\...
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1answer
167 views

Modification of Power Method [closed]

How to devise a simple modification of the power method to handle the following case: λ1 = -λ2 > |λ3| ≥ |λ4| ≥ |λ5| ≥ ... ≥ |λn| ? Could you please help me?
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16 views

Eigenvalues and Eigenvectors of continous variable function

Basically I am given a function $P_{l,k}$ where P is continuous and hermitian in $l$ and $k$ i.e. $P_{l,k}^*=P_{k,l}$. What is the general procedure to find its eigenvalues and eigenfunctions? There ...
2
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1answer
155 views

Normalizing a Sturm-Liouville theory problem

I have the following D.E.: $$y''(x) + \lambda y(x) = 0$$ on $[0,L]$ I've solved for 1) $\lambda =0$ 2) $\lambda > 0$ 3) $\lambda <0$ And found that $\lambda > 0$ gives me the only non-...
4
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0answers
495 views

Eigenvalues and eigenfunctions of differential operator.

I'm trying to find the eigenvalues (atleast the lowest) and eigenvectors of $$\alpha \frac{\partial^2}{\partial r^2} + \beta V(r) $$ with $\alpha$ and $\beta$ constant, $V(r) = \frac{a}{r}$ and for ...
3
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0answers
51 views

Polar Laplace equation on partially bounded domain - what am I doing wrong?

I am trying to solve the following problem: Let $\Omega$ denote the region $\{ (r, \theta) : r>1, 0 < \theta < \pi \}$. Find the bounded solution to the problem: $\begin{align}\nabla^{...
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1answer
40 views

Find the non-trivial eigen functions

I am asked to find the non-trivial eigen values for the following differential equation $y''+2y'+\lambda y=0$ given $$y'(0)-y(0)=0,y'(1)=0$$ So the characteristic equation is $r^2+2r'+\lambda =...
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1answer
72 views

Need help putting $-Y^{\prime\prime} - \mu Y^{\prime} = 0$ into Sturm-Liouville form

This question is related to the question I asked in this post. I'm trying to find an integrating factor or some way to turn $-Y^{\prime\prime}(y) - \mu Y^{\prime}(y) = 0$ into Sturm-Liouville form. ...