All Questions
Tagged with differential-operators ca.classical-analysis-and-odes
24 questions
3
votes
1
answer
154
views
Deriving differential equation from difference of PDE solutions
This is an edited cross-post from Math SE because after several days it's received no good answer. I think it's less appropriate for a general QA Math site and is likely better for Overflow with ...
- 33
2
votes
0
answers
73
views
The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product
$$
\langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)}
$$
...
- 21
1
vote
0
answers
21
views
Regularity of solutions of a 2nd order singular integro-differential operator
I have trouble finding the regularity of the solutions to a particular equation. I define
$$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
- 41
1
vote
1
answer
95
views
Monotonicity of the top eigenfunction of the generator of a diffusion
Consider in 1D the operator given by
$$
\mathcal{L} = \frac{d^2}{dx^2} - V'(x)\frac{d}{dx},
$$
where $V(x)$ is a convex, sufficiently quickly growing potential, so that $\mathcal{L}$ has a complete ...
- 2,449
5
votes
1
answer
1k
views
Analytic functions where all derivatives vanish at infinity and which are bounded
Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$.
I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
- 749
1
vote
0
answers
119
views
Sturm-Liouville Problem: When does $w y^2$ vanish at a singular boundary point?
It is well known (e.g. Courant, Hilbert - Methods of Mathematical Physics) that solutions of the Sturm-Liouville problem on an interval $J=(a,b)$
\begin{equation} \tag{1}
\left(p y' \right)' - qy \; = ...
- 183
2
votes
0
answers
42
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
- 749
5
votes
2
answers
233
views
Analytic approximations of smooth vector fields
Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with
$$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$
on $\mathbb{R}^3$ for any $\alpha,K$.
Further, we ...
- 749
7
votes
1
answer
767
views
Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials
There are some known criteria for the Sturm-Liouville Problem
\begin{equation} \tag{1}
\frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y
\...
- 183
14
votes
1
answer
1k
views
Computing spectra without solving eigenvalue problems
There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
- 649
2
votes
0
answers
162
views
A question about whether an operator can be lipschitz or not
Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...
- 371
3
votes
1
answer
160
views
Differential operators and rules Ore polynomial
(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.)
When dealing with (nonlinear) dynamical systems, one often deals with state space ...
- 131
2
votes
0
answers
97
views
Ergodic type ODE problem
Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
- 133
4
votes
2
answers
410
views
Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$
The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function).
$$(x^2y')'-x^2y=\lambda \;y$$
Now for a higher-degree ...
- 1,199
1
vote
1
answer
94
views
a question about complex Hessians on complex tori
Suppose we have a real-valued smooth function on a complex torus:
$$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$
i.e., this $f$ is a real-valued smooth function on $\...
- 593
1
vote
0
answers
148
views
References for the Sturm oscillation theorem
What is the most general form of the Sturm oscillation theorem?
So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...
- 263
-2
votes
1
answer
712
views
The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]
In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
- 123
5
votes
1
answer
187
views
Getting out a system of linear ODEs by knowing the Magnus expansion
Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t_1)...
- 749
-4
votes
1
answer
571
views
Derivatives of infinite order [closed]
Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature?
For example, can one make sense of
$$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 \...
- 1,594
9
votes
1
answer
386
views
Differential operator simplification
Does anyone know the explicit formulation for the $q_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$
I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that $$(x+D)H_n(x)e^{-x^...
- 402
2
votes
1
answer
738
views
Leibniz rule for Pseudo-differential operators of negative order
Does anyone know of some good references for a fractional Leibniz rule for pseudo-differential operators of negative order? As a specific example, I would like to compute $\partial_{x}^{-1}(uv)$, ...
1
vote
1
answer
113
views
Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?
Consider the differential operator $D:$
$$
Du:=\frac{-d^2}{dx^2}u
$$
on the function space $$C=\{ u\in C^2([0,1]):u(0)=u(1)=0\}.$$
It's not hard to find the eigenvalues and eigenvectors(...
user14319
5
votes
0
answers
501
views
Monodromy of differential equations
Let $D$ an ordinary differential equation (let's assume $D$ only has regular singularities). Suppose $D$ = $D_0$$\partial$, where $\partial$ is $q\frac{d}{dq}$ My question is: Does there exist any ...
- 51
4
votes
5
answers
891
views
Analytic hypoellipticity of linear ordinary differential operators
Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...
- 1,412