All Questions
Tagged with perturbation-theory spectral-theory
22
questions
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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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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
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109
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Derivative of Spectral Radius of Matrix $\exp(A(t))$
I am faced with the practical problem of solving a system
$$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$.
...
- 569
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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
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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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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
3
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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
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93
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Reference for Perturbation theory
I need to study the eigenvalues and the eigenfunctions of a perturbated operator knowing the eigenvalues and the eigenfunctions of the initial operator, using the theory of perturbation and writing ...
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46
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Regular Perturbation theory for $L=L_0+\epsilon ix$, where $L_0$ is an unbounded self-adjoint operator, on $\mathbb R$
I am looking to understand the spectrum of the following operator:
$L_{\epsilon}[u](x)=L_0[u](x)+i\epsilon x u(x)$ on $\mathbb R$. Here $L_0$ is a negative semi-definite self-adjoint operator that has ...
0
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2
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530
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Convergence of eigenvalues and spaces of sequence of compact, szmmetric and positive-semi-definite operators in Hilbert spaces.
I have a compact operator, which is uniformly approximated by (finite-dimensional) compact operators and I am concerned with the question, how the eigenvalues and eigenspaces of the former are ...
- 1,439
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2
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197
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Computing the resolvent of a rank one projector
I'm reading through this paper in dynamical systems, that's using a bit of perturbation theory and operator theory. The authors make the following claim:
Now, since $P_0$ is a projection of rank one, ...
- 125
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168
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"Modern" reference in perturbation theory in quantum mechanics
I am interested in mathematical aspects of eigenvalue perturbation theory, in particular which arises in quantum mechanics. I have some knowledge in measure theory, functional analysis, etc. I am ...
- 2,108
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103
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Why should I look at the Resolvent formalism and think it is a useful tool for spectral theory?
Wikipedia calls Resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
- 447
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43
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Conditions for Boundedness of Spectral Measures of Perturbations of Self-Adjoint Operators?
Suppose $A$ is an unbounded self-adjoint operator in a Hilbert space $H$ with discrete spectrum $$\lambda_0 < \lambda_1 < \cdots$$ bounded below with lowest eigenvalue $\lambda_0$, lowest ...
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