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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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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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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Perturbation Theory and Asymptotics
I have a question regarding the use of the words "Perturbation Theory" and "Asymptotics".
I assume that they can not be used interchangeably.
Is asymptotic analysis a perturbation ...
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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.
$${x^4} - \,\,{x^2} + \,\,\varepsilon (x +...
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proving complex Integral relation from perturbation theory MQ
Someone can help me to prove this identity, it comes from a normalization in MQ. From perturbation theory time dependent we have
$$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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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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Leading order perturbation to the solution of a dynamical system
I was reading the paper 'A Proposal on Machine Learning via Dynamical Systems', where I came across the following steps:
Consider a system-
$$\frac{dz}{dt} = f(A(t),z),$$ with $z(0) = x.$
So, the ...
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Birth-death : Always more than 1 bifurcation?
Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (ie, $f'(x) = 0$). I call this point degenerate if $\det \text{Hess}_x f = 0$ (so, equivalently, if the ...
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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 ...
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