All Questions
Tagged with perturbation-theory functional-analysis
44
questions
1
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35
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How is changing the boundary conditions a finite rank perturbation?
I have a question about a statement I came across which I'd be happy to understand more.
On $L^2(0,1)$, we can consider two self-adjoint operators. The first operator $H_0$ acts as $H_0f=-f''$, with ...
- 4,333
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40
views
Convergence rate of eigenvectors for perturbed matrices
Let $f(s)= (f_{1}(s),\ldots,f_{d}(s)), s \in \mathbb{R}^d, \textbf{o} \le s \le \textbf{1}$, be vector of probability generating functions, were each entry has finite third moments. Consider matrix of ...
- 1
7
votes
2
answers
123
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Perturbing a measure $\mu$ so that the integral $\int fd\mu$ becomes nonzero
Let $X$ be a compact subset of $\mathbb{R}^d$, let $f\in L^2(X)$ be an unknown function with $\lVert f\rVert_2=1$ for which we may assume suitable regularity (e.g. Lipschitz, $C^1$), and let $\mu$ be ...
- 610
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24
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Question related to isolated eigenvalue of a Hermitian operators
This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
- 78
1
vote
1
answer
105
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Theorem (Rellich) - Perturbation Theory
I got stuck with part of a proof of: The steps are all clear to me, until it is said:
"Therefore $f_w$ is the eigenvector of $ A+wB$ associated with the eigenvalue lying within for all small ...
- 39
0
votes
0
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38
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Number of distinct discrete eigenvalues of a self adjoint operator after compact perturbation increases.
Let $T$ be a self adjoint operator on a complex separable Hilbert space $H$. Let $K$ be self adjoint compact operator. So, $T+K$ is also self adjoint operator. I want to know for what $K$ the number ...
- 78
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42
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Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator
Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator. In particular, is there any way we can say that the element of essential spectrum is an eigenvalue of ...
- 78
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votes
1
answer
90
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Perturbation of a maximal dissipative operator by a non-negative self-adjoint operator.
Let $A$ be a maximal dissipative operator in a Hilbert space $\mathcal{H}$, and consider $B$ a self-adjoint operator such that
$$ \langle B\xi,\xi \rangle \geq 0\ , \quad \xi\in \mathcal{H}\ .$$
Does $...
- 87
1
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0
answers
51
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Reference request: Let $T$ be a self-adjoint operator, and assume that $V$ is compact and self-adjoint. Then $\sigma_{ess}(T + V ) = \sigma_{ess}(T)$.
I am looking for a proof of the following result.
Let $T$ be a self-adjoint operator, and assume that $V$ is compact and self-adjoint. Then $\sigma_{ess}(T + V ) = \sigma_{ess}(T)$.
Here the context ...
- 520
2
votes
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answers
89
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Alternative reference request for Kato's Perturbation Theory
I am studying Analytic Perturbation Theory from Kato's book.But sometimes I find it really difficult to follow.(e.g. In Chapter 2 , he directly talks about algebraic singularities without defining it ...
1
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0
answers
30
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Approximate orthogonality between two sets of Hermite functions.
Consider the set of Hermite functions $\{\phi_{n}(x,\varepsilon_{1})\}_{n}:= A$ defined below.
\begin{equation}
\label{eqn:funcs}
\phi_{n}(x,\varepsilon_{1}) = \frac{\sqrt[8]{1+\big(\frac{2\...
- 71
3
votes
0
answers
75
views
The linear heat eqaution with potential $V\in L_{t,x}^{\frac{n+2}2}$ is well-posed for initial data in $L^p$ for any $p\in(1,\infty)$
Consider the linear heat equation with potential
$$u_t-\Delta u-V(x,t)u=0\qquad \text{in }\ \ \mathbb R^n\times[0,\infty),$$
where $V\in L_{t,x}^{\frac{n+2}2}$. Show that this equation is well-posed ...
- 13.7k
0
votes
1
answer
99
views
Number of negative eigenvalue of a perturbation by a bounded operator
Let $H=(H,(\cdot, \cdot))$ be a Hilbert space, $L:D(L) \subset H \rightarrow H$ be a self-adjoint operator (not necessarily bounded) and $A:H \rightarrow H$ be a bounded and symmetric operator. By ...
- 1,657
3
votes
0
answers
359
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Perturbation of the spectrum
In a lecture note, in the demonstration of a result it is said that "using analytic perturbation theory" it is possible to deduce certain things. The reference is Kato's classic book, "...
- 2,797
1
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1
answer
134
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Schrödinger operator whose potential analytically depends on a parameter – how does the spectrum change?
Let's say we have a self-adjoint operator $H_s$ on the Hilbert space $L^2(\Omega \subseteq \mathbb{R})$ defined by
$$
H_s \, \psi(x) := -\psi''(x) + V_s(x) \, \psi(x) \: ,
$$
where $s \in \mathbb{R}$...
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