# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...