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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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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Perturbation theory in math and quantum mechanics

This question is about my lets say lack of understanding. I can not make the connection between what i study and what i saw in lectures. For example in our QM lectures we saw perturbation theory like ...
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Lorenz - why is the solution with noise in initial condition on the same curve?

I want to use Lorenz'63 for the purpose of testing some algorithms I'm currently researching. When I defined a system and solved it in Matlab, plot of x-coordinates for explicit and perturbed initial ...
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Perturbation Theory for the Definite Generalized Eigenvalue Problem

I have become confused on some basic questions concerning the perturbation theory of the generalized eigenvalue problem. I am likely missing something simple, but I have a couple of questions about ...
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regular or singular perturbation?

Can the problem of solving for $x$ the equation $\varepsilon x +t =0$, where $1<t<\infty$ and $\epsilon>0$ is a small parameter be termed a (very simple case of) singular perturbation problem ...
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Integral asymptotic approximation

I am trying to attain a two-term approximation for the following integral as $m$ goes to $1$ from below: $$I=\int_{0}^{\pi /2}\frac{\mathrm d\theta }{\sqrt{1-(m^2)\cdot\sin(\theta)^2 }}.$$ So far I am ...
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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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WKB form manipulation

If I have the equation $$y'' +\sin\left(\frac{1}{x}\right)y = 0$$ and I approximate $\sin(\frac1x)$ as $\frac1x$, how would I get epsilon squared in front of the first term (which is required for WKB ...
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WKB form substitution for V

If I have the equation $$y'' +\sin(1/x)y = 0$$ how would I replace $\sin(1/x)y$ in order to get epsilon squared in front of the first term? I have tried defining $X= \delta\sin(1/x)$, but that doesn't ...
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Prove perturbation theory breaks down

I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ODE:...
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Expected Time of Absorption when Markov Chain is "Nearly" Irreducible

Suppose we have a Markov chain with states indexed $1$ through $n$ that is irreducible with a unique stationary distribution ${\bf \pi} \in \mathbb{R}^n$. The transistion probabilities of the Markov ...
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Finding mean of perturbed Gaussian using perturbation theory

I am trying to approximate the mean of the following distribution using perturbation theory: $$P(x) \propto \exp\left( \frac{1}{2\sigma^2}x^Tx + \alpha\frac{1}{2}(f(x)-y_0)^2 \right)$$ where $y_0$ ...
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Asymptotic expansion for $xe^{1/x}=e^{\lambda}$

I need to find a 3-term expansion as $\lambda\rightarrow\infty$ for each of the solutions to this transcendental equation. Plotting $xe^{1/x}$ I assume that there are 2 solutions, one around 0 and the ...
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Asymptotic expansion of $x\exp(1/x)=\exp(\lambda)$ as $\lambda\to\infty$

I am struggling to find a three-term expansion of the following equation $x\exp(1/x)=\exp(\lambda)$ as $\lambda\to\infty$ for each of the solutions for $x$. I graphed the functions $1/x$ and $\log(x)$ ...
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Deriving solution of a second order ode from a closely related one with a different scalar coefficient for the term in X(x)

I solved the wrong differential equation! I was looking at a parabolic pde in the style of a heat equations which upon separation of variables I wrote as \begin{equation} a x^2\left(\frac{d^2 X}{d x^...
If $x$ is a local minimum of the function $f$, then the minimum of $f+\lambda g$ at the first order in $\lambda$ is given by $x+\delta x$ with $\delta x=-\lambda g'(x)/f''(x)$. The proof is very ...
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 ...