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Questions tagged [eigenfunctions]

For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.

Let $X$ be a function space and $T$ an operator on $X$. A eigenfunction of $T$ is a nonzero function $f \in X$ such that the following holds $$ T f = \lambda f $$ where $\lambda$ is a scalar and called the eigenvalue.

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Can we construct Sturm Liouville problems from an orthogonal basis of functions?

Given a sequence of functions orthogonal over some interval, which satisfy Dirichlet boundary conditions at that Interval, can we construct a Sturm Liouville problem that gives these as its eigenfunctions? For example, if we take the sequence of…
Manishearth
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Positivity of principal eigenvalue for $L\phi=-\triangle \phi + \nabla \cdot( u \phi )$

EDIT: This question is still unresolved as of April 18, 2014. The two answers provide useful work in the right direction, but neither resolves the question. A counterexample should have $u \in C^1(\overline{\Omega})$. I am reading this paper and in…
nullUser
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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 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1…
user155214
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Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the functional $\mathcal{F} ( f ) = \frac{\displaystyle…
Miloslav Capek
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the boundary and outside of some compact region). 1 -…
user219526
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What is the difference between eigenfunctions and eigenvectors of an operator?

What is the difference between the eigenfunctions and eigenvectors of an operator, for example Laplace-Beltrami operator?
Fei Zhu
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$L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator

Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$. We know $w_k$ are smooth functions. Is such a bound true: $$\lVert w_k…
Upin
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physical meaning of laplace-beltrami eigenfunctions?

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-Beltrami operator and the natural vibration analysis…
Fei Zhu
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Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$. EDIT: These were not given to be zero !! Maybe this helps... The text (Nagle Saff and Snider, end of…
James S. Cook
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How is the Laplace Transform a Change of basis?

This question is primarily based on the following answer's way of reasoning, https://math.stackexchange.com/a/2156002/525644 If you want to write a new answer to the question; "How is the Laplace Transform a Change of basis?" Please do. In jnez71's…
Aravindh Vasu
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Does every representation of the harmonic oscillator Lie algebra necessarily admit a basis of eigenfunctions?

It is well-known in quantum mechanics that the harmonic oscillator Hamiltonian given by $\mathcal{H} = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}x^2 - \frac{1}{2}$ admits a basis of eigenfunctions on $L^2(\Bbb R,dx)$. There are many proofs of this…
Cameron L. Williams
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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 converging solution as $x\rightarrow \infty$, then solve…
Gateau au fromage
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Why does the set of an hermitian operator's eigenfunctions spans the functions space

During a discussion about linear hermitian operators, my professor claimed that if a linear operator $M$ is hermitian under a certian set of conditions, then genrally any function that fulfills these conditions can be expressed as an infinite sum of…
user1691388
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Proof that Legendre Polynomials are Complete

Can somebody either point me to, or show me a proof, that the Legendre polynomials, or any set of eigenfunctions, are complete?
anon
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Can exponential functions be thought of as eigenfunctions for the derivative operator?

I.e. is the function $y=b^{kx}$ an eigenfunction for the derivative operator $\frac{dy}{dx}$, where k is a constant because the derivative of such a function is ${k\ln(b)}b^{kx}$, which is a constant ($k\ln(b)$) times the original function…
Baker12
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