Questions tagged [perturbation-theory]
Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.
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Approximating solution to vector recurrence relation with element-wise exponential
$$
\mbox{Let}\ A \in \mathbb{R}^{n \times n}\ \mbox{and}\ \gamma_{1}, \ldots, \gamma_{n} \in \mathbb{R}\ \mbox{with}\ \gamma_{i} > 0,\ \forall\ i.
$$
$$
\mbox{Define}\ \operatorname{f}: \mathbb{R}^{...
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Degenerate perturbation theory to nonlinear equation
I want to use perturbation theory to find the solution to the following nonlinear equation:
$$x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0,$$where $i=1\cdots N$ and $...
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How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix
If
D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
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Eigenvalue Spectrum of a non-Hermitian Matrix by a Hermitian Matrix Perturbation
Let $H_{eff}$ be a $n$ dimensional matrix defined by the eigenvalue spectrum $\Lambda$:
$$\Lambda(H)_n=\Lambda(H_n+H_{eff}),$$
Where $H$ is a infinite dimensional matrix, $\Lambda(H)_n$ are its lowest ...
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Degree with which a polynomial changes with some small change
Soft question: I was curious as to how one could measure the degree with which a polynomial is perturbed. More formally, let $P(x) \in \mathbb{C}$ be a polynomial and $\epsilon$ be a very small number,...
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Sensitive eigenvectors to small perturbations in the matrix?
I've encountered a mathematical issue in my research. To provide some context, I have a known density matrix that I am reconstructing numerically using quantum data. The rebuilt matrix has ...
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Eigenvalue/eigenvector sensitivity in multidimensional scaling
From classical multidimensional scaling, a Cartesian coordinate matrix can be obtained as $\mathbf{X} = \mathbf{V} \mathbf{\Lambda}^{1/2}$, where $\mathbf{\Lambda}$ is a diagonal matrix of eigenvalues ...
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How to approach this singular perturbation problem?
I have set myself the following singular perturbation problem:
For small values of $\varepsilon > 0$ find the two roots closest to $x=0$ for the equation.
$${x^4} - \,\,{x^2} + \,\,\varepsilon (x +...
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Solving a new system of PDEs using solutions of an old system
I got stuck in my research. Briefly speaking, the following is a system of 6 variables ($u,v,p,h_{11},h_{12},h_{22}$) I need to analyze:
\begin{equation}
g^2\frac{\partial u}{\partial X}+\frac{\...
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Find the condition number of a normal matrices.
Find the condition number of a normal matrices.
My attempt:-
I know condition number of $X\in \mathbb C^{n,n}$ is defined by $\kappa(X)=||X|| \cdot ||X^{-1}||.$ Definition of Normal matrix is given by ...
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Periodic perturbation of ODE
Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.5, the author mentioned that the following is well-known in the theory of ODEs.
If $x_0$ is an ...
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proving complex Integral relation from perturbation theory MQ
Someone can help me to prove this identity, it comes from a normalization in MQ. From perturbation theory time dependent we have
$$H(t)=H_0+H’(t)$$
$$|Ψ>=c_a(t)e^{-iE_at/\hbar}|Ψ_a>+c_b(t)e^{-...
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Converting an integral to hypergeometric function [closed]
I have encountered an integral as follows $$\int_{0}^{1}{\left(k^{2}x^{2}-k^{2}x+m_{2}^{2}+m_{1}^{2}x-m_{2}^{2}x \right)^{\frac{d-4}{2}}}dx$$ Any suggestion how to convert it into a hypergeometric ...
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A case problem about rank-1-perturbation of diagonal matrices
I have the following prediction for rank-1 perturbations of diagonal matrices, but I don't know how to prove (or disprove it).
Given $v:= [v_1,...,v_K] \in (0,1]^K$, we define $a:= \sum_{k=1}^K v_k = \...
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Asymptotic expansion of root of $\varepsilon x \tan(x)=1$
Indicate a range of roots of $\varepsilon x \tan(x)=1$ for which it is impossible to get an approximation using expansions.
Since $\varepsilon$ is small, I think for the equation to hold, we need ...
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