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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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 $...
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Leading eigenpair of degenerate non-symmetric matrix
Consider a non-symmetric matrix ${\bf A}_0$ with one eigenvalue $\lambda$ and all other eigenvalues are zero. The corresponding left $\bf u$ and right $\bf v$ eigenvectors to the eigenvalue $\lambda$ ...
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Deriving a Hopf Bifurcation – Perturbation Method vs. Jacobian Matrix
When deriving a Hopf bifurcation of a dynamical system, the usual process is:
Find a fixed point $(x_0, y_0)$
Perturb the system about the fixed point $(x_0+\tilde{x}, y_0+\tilde{y})$
Linearize, ...
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Method to solve this ODE $x^{(6)}+2Ax^{(4)}+A^2x^{(2)}+B^2x = 0$
I have to solve this ODE: $$x^{(6)}+2Ax^{(4)}+A^2x^{(2)}+B^2x = 0$$
where the upper index in brackets () indicates the order of the time derivative, $A = 4(m^2-2eHs_z)$ and $B= 4meH$, both are ...
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How to do this perturbation expansion?
I got the following expansion in the context of studying $\phi^4 $ quantum field theory. This is the solution for exact 2-point propagator in the ladder-rainbow approximation. The expansion is -
$\mu^{...
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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 ...
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Perturbation theory with coupled nonlinear differential equations
I’ve got a problem with a set of differential equations for which I’m trying to find fixed points (or rather restrictions for the parameters).
The equations have the form
(1) $\frac{d}{dt} R(t) = -B ...
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Why there is a sign change when not writing in long division?
I have a question about understanding how this step
$$
\int_0^{\delta/\epsilon}\frac{du}{1+u+\epsilon^2u^3/3+O(\epsilon^4u^5)}
$$
becomes
$$
\int_0^{\delta/\epsilon}\frac{du}{1+u}\left(1-\frac{\...
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Solution to a nonlinear ODE
$$ a_1 y'''''+ a_2 y'''+\left(a_3 + y^2 \right) y' = 0 $$
where $a_1, a_2, a_3$ are constants with $a_1>0$ and $a_2,a_3 \in \mathbb{R}$.
Is there a general solution $y(x)$ to the above differential ...
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A question on perturbed ODE (bender and orszag)
This is from example 2 of chapter seven, consider the IVP
\begin{equation}
\begin{cases}
y''=f(x)y,
y(0)=1,
y'(0)=1
\end{cases}
\end{equation}
we then make a perturbation expansion $y(x)=\sum_{n=0}^{\...
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Singular perturbation theory involving exponentials
Suppose we're given the following second order ODE
\begin{align}
-\epsilon x''(t)=e^{x(t)}-1
\end{align}
with boundary conditions
\begin{align}
x(0)=0,\quad x(1)=1.
\end{align}
Suppose $\epsilon>0$ ...
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If $\int \bar \Psi \frac{1}{ r_1 - r_2} \Psi d\tau= \frac{20\pi k}{a^5}$, what is $\int \bar \Psi \left( \frac{1}{r_1 - r_2} -E_1 \right) \Psi d\tau$?
Suppose that
$$\Psi = ke^{ar_1}e^{ar_2}$$
and it is known that
$$ \left(E_0 + \frac{1}{\lvert r_1 - r_2 \rvert}\right)\Psi = E\Psi$$
for some constants $E, E_0$.
Assuming that $E$ can be expressed as ...
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How to integrate $\int \frac{e^{-x}e^{-y}}{\lvert \vec x - \vec y \rvert} dx dy$ for the helium atom?
What methods are used to integrate
$$\int \int \frac{e^{-x}e^{-y}}{\lvert \vec x - \vec y \rvert} dx dy$$
which comes up in perturbation theory calculations such as for the helium atom (8.2.6). The ...
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Approximating the solution to a system of 3 nonlinear ODEs with the KBM method?
Background
I have the following system of ODEs:
$\dfrac{\mathrm{d}x}{\mathrm{d}t}=x\dfrac{q}{Q}-x\dfrac{x+y}{M}\quad$ (Eq. 1)
$\dfrac{\mathrm{d}q}{\mathrm{d}t}=y(1-\dfrac{q}{Q})(1-c)(1-v)-aq-y\dfrac{q}...
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Approximating the solution to a system of two ODEs with the KBM method?
Background
I have the following system of ODEs:
$$ \begin{aligned} \dot x (t) &= \alpha - \beta x(t) y(t) \\ \dot y (t) &= \delta x(t) y(t) - \gamma y(t) \end{aligned} $$
where all parameters ...
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