All Questions
Tagged with sp.spectral-theory differential-equations
7 questions with no upvoted or accepted answers
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
4
votes
0
answers
410
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Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
- 325
2
votes
0
answers
150
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Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
- 155
1
vote
0
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69
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Show that an integral operator with Bessel function kernel is bounded on $L^2(0,\infty)$
Let $J_0$ denote the Bessel function of the first kind of order $\nu = 0$ (see DLMF 10.2),
$$
J_0(z) = \sum_{k = 0}^\infty (-1)^k \frac{(\tfrac{1}{4} z^2)^k }{k! \Gamma(k + 1)}.
$$
Put $u_0(r) = r^{1/...
- 481
1
vote
0
answers
119
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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
1
vote
0
answers
93
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inverse problem to resolution of the identity
Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
- 587
1
vote
0
answers
137
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Decay rate of Discrete Prolate Spheroidal Sequences in frequency
What is the decay rate of DPSS sequences in frequency?
Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...
