Questions tagged [eigenfunctions]
For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.
782 questions
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What are the complex fixed points of the curl operator? ($\nabla \times \vec v = \vec v$)
Let $\vec{v}=(F_x,F_y,F_z)$, where the components are functions $\mathbb C^3 \to\mathbb C$ and the subscript simply denotes the coordinate. I am curious in finding non-trivial complex $\vec{v}$ such ...
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The discreteness of spectrum of linear 2nd elliptic operator $Lu=-\Delta u + \eta \cdot \nabla u + hu$.
I wonder if the spectrum of linear 2nd elliptic operator $$Lu=-\Delta u + \eta \cdot \nabla u + hu$$ on bounded domain
is discrete. Here $h$ is a smooth function without sign condition and $\eta$ is ...
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First eigenfunction of Laplace-Beltrami Operator
In the book of Jost (Riemannian Geometry and Geometric Analysis), in section 3.2, he is finding the eigenfunctions of the Laplace-Beltrami operator. He defines the first eigenvalue as the Rayleigh ...
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how to construct a volume element of a coordinate system warped by a kernel function
Take a grid in an arbitrary number of dimensions. I construct a graph kernel to define the connectivity of the grid, and apply that kernel to the grid to create a weighted digraph.
I Construct the ...
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Trace of Squared Integral Operator, and Integral of its Kernel squared
This is from the text Theoretical Foundations of Functional Data Analysis,with an Introduction to Linear Operators, pertaining to a proof of theorem 4.6.7.
First, following from an application of ...
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Non-local Schrodinger-like equation and its spectrum
Consider the following definition of non-local Laplacian operator,
$$\Delta^K u(x) = \int_{\Omega} dy\,K(x,y)\left(u(y)-u(x)\right),\tag{1}$$
where $K(x,y)=K(y,x)$ is measurable and "good" ...
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Generalization of Weyl asymptotic law
The wiki page for Weyl's law has the following formula for the asymptotics of the eigenvalue counting function for a Schroedinger operator $L = -\hbar^2\Delta + V$:
$$N(E) \sim (2\pi\hbar)^{-d}\iint_{\...
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$n$ orthogonal vectors span ${\Bbb R}^n$. Does this hold in the infinite-dimensional case?
I’ve been getting into Linear Algebra as it is applied to Quantum Mechanics and came to something rather beautiful. Let's define Hilbert space as the space of all functions $f : {\Bbb R} \to {\Bbb C}$ ...
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Why are normal mode frequencies discrete?
Consider a partial differential equation in a box $x \in [0,1]$ in $u \equiv u(t,x)$
$$ u_{t} = L(x, \partial_x, \partial_x^2, \ldots) u$$
with boundary condition $u(1)=0$. One can separate variables ...
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Bounds on eigenvalues of Fredholm integral equation of the first kind
Consider the Fredholm integral equation of the first kind \begin{equation} \mu_n^2\psi_n(\tau) = \int_0^T K(\tau,\tau')\psi_n(\tau')\mathrm{d}\tau',~n\in\mathbb{Z}^+\end{equation} where $K(\tau,\tau') ...
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Adjoint and spectrum of derivative operator $\hat{D}$
$\hat{D} = \frac{d}{dx}$
Calculate the adjoint in the space of the periodic functions of $L^2[a, b]$:
$$\begin{align} (g, \hat{D}f) = \int_a^b \overline{g(x)}f'(x) &= \left[\int_a^b \overline{g(x)...
- 174
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Numerical algorithms for minimal eigenvalue to a couple of non linear ordinary differential equation
Problem:
I have a problem of this form:
$$
\left( - \frac{\partial_r^2}{2} + h(r,f,g) - p(r,f,g) \right) f(r) = K f(r) \\
\left( - \frac{\partial_r^2}{2} + h(r,f,g) + p(r,f,g) \right) g(r) = K g(r)
$$...
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Axisymmetric Stokes flow field in cylindrical coordinates in a semi infinite domain
I am trying to solve the following simultaneous second-order PDEs for deriving the 2D and axisymmetric Stokes flow field (in cylindrical coordinates) $\mathbf{u}(r,z) = u_r(r,z)\hat{\mathbf{e}}_r + ...
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Is the Laplace-Beltrami metric tensor unique for eigenpairs on $S^2$?
Consider that I have found an eigenpair ($f(\theta,\varphi),\lambda$) which solves the Laplace-Beltrami eigen equation on the 3D unit sphere, $S^2$: $\Delta_gf+\lambda f=0$ with the standard metric ...
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Showing $Ker(u-\alpha_1 Id)\circ (u-\alpha_2 Id) = Ker(u-\alpha_1 Id)\oplus Ker(u-\alpha_2 Id)$ for derivative operator.
Let $\partial$ denote the derivative operator on a space of infinitely differentiable functions.
I am trying to prove that
$$\ker \left(a(\partial - \alpha_1 \operatorname{id})\circ (\partial - \...
- 1,367
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Help with derivation of dirac delta from divergent integral
I'm following this paper, and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy
\begin{equation}
\langle f_s|f_{s'}\rangle = \int_{0}^{1} \...
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A question on properties of Mass forms.
I am studying Proposition 3.3.3 from "Automorphic Forms and L-functions for the group $GL_n(\mathbb{R})$" by D. Goldfeld, and I need a clarification regarding the behavior of Maass forms as $...
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Solving for a vector field whose divergence is proportional to the field itself
I'm trying to solve the following pair of equations in two dimensions:
$$ \nabla \cdot \mathbf{E} = b E, $$
$$ \nabla \times \mathbf{E} = 0, $$
where $b$ is a real constant, $\mathbf{E} = E\, \hat{\...
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Is the absolute value of an eigenfunction again an eigenfunction?
Let $\Omega \subseteq \mathbb{R}^n$ be open. Suppose $u : \Omega \to \mathbb{R}$ is a $C^2$ function satisfying $-\Delta u = \lambda u$ for some $\lambda \ge 0$. I would like to show that $|u|$ also ...
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Divergent Tail Sums of Approximations of Non-trace Class Compact Operators
I'm working on approximations of compact operators that are not trace class, and I'm looking for ways to provide meaningful approximation error estimates for truncated eigenfunction expansions. I ...
user1328882
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Sign of the first eigenfunction of the Laplacian
I am trying to prove that the first eigenfunction of the Laplacian operator in an open domain $\Omega$ does not change sign and that the first eigenvalue $\lambda_1$ is simple (with Dirichlet-boundary ...
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Are the eigenvectors of the generalised Laplace operator always periodic?
In $\mathbb{R}$ the eignevectors / eigenfunctions of the Laplace operator yield the fourier series, which, among other things, is made up of exclusively periodic functions.
If you have a Riemannian ...
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What are the spaces whose linear operators admit eigen"vectors"?
I know that linear operators over $n$-dimensional, $\mathbb{R}-$vector spaces admit $n$ eigenvalues (in the algebraic closure of $\mathbb{R}, \mathbb{C}$) and eigenvectors. The proof of this is just ...
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Asymptotics of negative laplacian eigenvalues
I’ve come across in a paper the fact that the eigenvalues of the negative Laplacian $-\Delta v_k = \lambda_k v_k$ (dirichlet BC, bounded domain in $\mathbb{R}^d$) follow the asymptotic relation
$\...
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Eigenvalue problems $y''(x) + λy(x) = 0, 1 < x < 2, y(1) = y'(2) = 0$ [closed]
Consider the eigenvalue problem $y''(x) + λy(x) = 0, 1 < x < 2, y(1) = y'(2) = 0$. Given the fact that its eigenvalues are positive, find all eigenvalues $λ_n$ and the corresponding ...
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