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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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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Approximating solution to vector recurrence relation with element-wise exponential
$$
\mbox{Let}\ A \in \mathbb{R}^{n \times n}\ \mbox{and}\ \gamma_{1}, \ldots, \gamma_{n} \in \mathbb{R}\ \mbox{with}\ \gamma_{i} > 0,\ \forall\ i.
$$
$$
\mbox{Define}\ \operatorname{f}: \mathbb{R}^{...
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How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix
If
D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
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Eigenvalue Spectrum of a non-Hermitian Matrix by a Hermitian Matrix Perturbation
Let $H_{eff}$ be a $n$ dimensional matrix defined by the eigenvalue spectrum $\Lambda$:
$$\Lambda(H)_n=\Lambda(H_n+H_{eff}),$$
Where $H$ is a infinite dimensional matrix, $\Lambda(H)_n$ are its lowest ...
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Degree with which a polynomial changes with some small change
Soft question: I was curious as to how one could measure the degree with which a polynomial is perturbed. More formally, let $P(x) \in \mathbb{C}$ be a polynomial and $\epsilon$ be a very small number,...
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Sensitive eigenvectors to small perturbations in the matrix?
I've encountered a mathematical issue in my research. To provide some context, I have a known density matrix that I am reconstructing numerically using quantum data. The rebuilt matrix has ...
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Eigenvalue/eigenvector sensitivity in multidimensional scaling
From classical multidimensional scaling, a Cartesian coordinate matrix can be obtained as $\mathbf{X} = \mathbf{V} \mathbf{\Lambda}^{1/2}$, where $\mathbf{\Lambda}$ is a diagonal matrix of eigenvalues ...
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Solving a new system of PDEs using solutions of an old system
I got stuck in my research. Briefly speaking, the following is a system of 6 variables ($u,v,p,h_{11},h_{12},h_{22}$) I need to analyze:
\begin{equation}
g^2\frac{\partial u}{\partial X}+\frac{\...
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Find the condition number of a normal matrices.
Find the condition number of a normal matrices.
My attempt:-
I know condition number of $X\in \mathbb C^{n,n}$ is defined by $\kappa(X)=||X|| \cdot ||X^{-1}||.$ Definition of Normal matrix is given by ...
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How to approach this singular perturbation problem?
I have set myself the following singular perturbation problem:
For small values of $\varepsilon > 0$ find the two roots closest to $x=0$ for the equation.
$${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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Interpolation of perturbed rotations and approximating the linearized effect of doubly perturbed rotations
I am reading through "State Estimation for Robotics" by Timothy Barfoot and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:...
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Book suggestions for Perturbation Theory in Quantum Mechanics
I've been searching the web for rigorous books on Perturbation Theory, specifically as an undergraduate physics student. In my experience, many quantum mechanics books lack rigor in their explanations....
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On estimating $\exp(-iHt)$ when $H$ is perturbed
Let us assume that we have an unbounded Hamiltonian $H$ and we perturb it a bit to be $H'=H+ \varepsilon A$. I am sure that estimating $||\exp(-iHt) - \exp(-iH't)||$ belongs to the subject of ...
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Convergence rate of eigenvectors for perturbed matrices
Let $f(s)= (f_{1}(s),\ldots,f_{d}(s)), s \in \mathbb{R}^d, \textbf{o} \le s \le \textbf{1}$, be vector of probability generating functions, were each entry has finite third moments. Consider matrix of ...
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Approximate solution to ODE potentially using perturbation theory
On the wikipedia article for stokes drift (https://en.wikipedia.org/wiki/Stokes_drift) they show that for:
$\dot{\zeta} = u\sin(k\zeta - wt)$ has approximate solution (by perturbation theory):
$\zeta(\...
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Exact solution to Dirac delta perturbation for particle in a box
Using diagrammatic perturbation theory the energy of a particle in a box with a Dirac delta potential can be closely approximated. The following energy correction terms to the ground state energy ($\...
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Finding eigenvalues of a turning point ODE using WKB method
Question: Using the WKB method, provide an approximation for the eigenvalue, $\lambda$, of the problem:
$$y'' + \pi^2\lambda y(1+2\cos \pi x)\sin^2(\pi x/2) = 0,~0\le x\le 1,~y(0)=y(1)=0.$$
Compute ...
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Asymptotic expansion of $\int_{\xi}^{\infty}\frac{e^{-\alpha t}}{t}dt$
Question: Provided $\xi \ll 1$, find the first three terms of the asymptotic expansion of the integral $$I(\xi) = \int_{\xi}^{\infty}\frac{e^{-\alpha t}}{t}dt.$$
My approach: Assuming $\alpha>0$ ...
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Leading-order approximation of $\int_0^{\infty} e^{t-z(t^4-2t^2)}\sin^2(2\pi\nu t)~dt$
Question: For $z \gg 1$, find the leading-order approximation to the integral,
$$\int_0^{\infty} e^{t-z(t^4-2t^2)}\sin^2(2\pi\nu t)~dt,$$
allowing for any value of the parameter $\nu > 0$.
My ...
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Determine the two-term expansion for large roots of the transcendental equation $\tan(x) =\frac 1x$
For this problem, I was given a hint that when $x$ is large, $\frac 1x$ is nearly zero, and $x \sim n\pi$ where $n$ is a large integer.
Initially tried the taylor expansion, but that didn't work out.
...
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Perturbing a measure $\mu$ so that the integral $\int fd\mu$ becomes nonzero
Let $X$ be a compact subset of $\mathbb{R}^d$, let $f\in L^2(X)$ be an unknown function with $\lVert f\rVert_2=1$ for which we may assume suitable regularity (e.g. Lipschitz, $C^1$), and let $\mu$ be ...
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behavior of SDE as parameter goes to infinity (Ornstein-Uhlembeck?)
In a physics paper, I saw the following (weird) heuristic argument:
Let $\theta,v>0$ be constants. Starting from the SDE
\begin{equation}
dX_t=D(X_t)(U'(X_t) -\theta(X_t-vt))dt +\sqrt{2D(X_t)}dW_t
...
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Two Timing (Multiple Time Scales) with Coupled IVPs
Question: Find the leading-order approximation for times of order $\epsilon^{-1}$ to $$\ddot{x} + x = y,~~\dot{y} = \epsilon(xy - 2y^2),~~x(0) = 1,~~\dot{x}(0) = 0,~~y(0)=1.$$
My approach: Let the ...
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Why is this approximate solution correct?
Consider the following differential equation
$$ y''=-y + \alpha y |y|^2, $$
where $y=y(x)$ is complex in general and $\alpha$ is a real constant such that the second term is small compared to $y$ ($||^...
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Question related to isolated eigenvalue of a Hermitian operators
This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
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Invertibility of the product of matrices when the norm is less than 1
I am reading through a book about numerical linear algebra. It is challenging but I understand the concepts after some research. However, there is one thing that I can't figure out and it's about the ...
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Expansion of $y=\sqrt{1+x+\frac{\varepsilon}{\varepsilon+x}}$
I'm reading "Introduction to Perturbation Methods", Second Edition, by Mark H. Holmes.
In ch 2.2.5 "Matching Revisit" it explains the approach to match the outer and boundary layer ...
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Leading order matching of $\epsilon x^py'' + y' + y = 0$
Question: The function $y(x)$ satisfies $$\epsilon x^py'' + y' + y = 0,$$ in $x\in [0,1]$, where $p<1$, subject to the boundary conditions $y(0) = 0$ and $y(1)=1$. Find the rescaling for the ...
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Nondimensionalizing Fourth Order Differential Equation for an Elastic Beam Under Tension
I am going through the textbook A First Look At Perturbation Theory 2nd ed. by James G. Simmonds and James E. Mann Jr.
Exercise 1.14 states: "An elastic beam of section modulus $EI$, resting on ...
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Derivative of Spectral Radius of Matrix $\exp(A(t))$
I am faced with the practical problem of solving a system
$$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$.
...
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Can we say anything about how $\delta x$ and $\delta y$ are related to each other?
I have an equation of the form $$\frac{x^2}{y} = F(r), $$ where F is a function of $r$. This equation has a solution $r(x,y)$. Suppose we perturb this solution to $r(x + \delta x, y + \delta y)$ for ...
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Asymptotic expansion of $I(\alpha;\epsilon) = \int_0^{\infty}\frac{dx}{(\epsilon^2+x^2)^{\alpha/2}(1+x)}$
Question: Evaluate the first two terms of as $\epsilon \to 0$ of $$I(\alpha;\epsilon) = \int_0^{\infty}\frac{dx}{(\epsilon^2+x^2)^{\alpha/2}(1+x)},$$ for $\alpha = \frac{1}{2}, 1, 2$, if $$C(\alpha) = ...
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Asymptotic expansion of $\int_0^{\pi/2}\frac{\sin^2\theta}{(1-m^2\sin^2\theta)^{1/2}}d\theta$
Question: Evaluate the first two terms of
$$I(m) = \int_0^{\pi/2}\frac{\sin^2\theta}{(1-m^2\sin^2\theta)^{1/2}}d\theta$$ as $m\to 1^-$.
My approach: Setting $m = 1-\epsilon$ and $k = \sin\theta$, we ...
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Asymptotic approximation of an integral using splitting range
Question: Evaluate the first two terms as $\epsilon\to 0$ of $$\mathcal{N}(z)=\int_z^1\frac{dx}{\sqrt{x^3+\epsilon}}$$ where $0\le z <1$.
My approach: First, let us calculate $\mathcal{N}(z=0)$. It ...
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Bound on number of positive roots of deformed polynomial
In the comments for the following linked question Descartes rule of sign with positive real exponents, the following was stated:
" The positive roots depend continuously on the exponents. This ...
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Duffing equation with non-linearity factor greater than unity
I have been trying to solve the following non-linear equation taking help from the book regarding perturbative technique by H. Nayfeh (chapter -4)
$$\ddot{x}+\frac{x}{(1-x^2)^2}=0$$ where $x<1$ ...
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Applying WKB Method for a Fourth Order Schrodinger Like Equation
I was trying to apply the WKB method to analyze the approximate eigenvalue condition for the Schrodinger like equation
\begin{equation}
\varepsilon^{4} y^{(4)} = (E-V(x))y, \quad y(\pm \infty) = 0, V(...
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Given a perturbation to a symmetric matrix with multiple zero eigenvalues, seeking perturbation to eigenvalues
Let $A$ be a real, symmetric $n \times n$ matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ where at least two of the eigenvalues are zero. Let $V$ be a real, symmetric $n \times n$ ...
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Find the general solution of an ODE with a nonlinear perturbative term
Let's say I start with the linear differential equation
$$ y''=-y, $$
which has (for example) the two solutions $y=e^{\pm i x}$, therefore following the superposition principle the general solution is ...
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Bound for $\left\|\nabla^2 u\right\|_{L^2(\Omega)}$
Consider the elliptic equation
$$
\begin{aligned}
-\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\
u=0, & \text { on } \partial \Omega.
\end{aligned}
$$
where $\Omega$ is a bounded domain....
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Perturbed real roots of an exponential-polynomial equation
Question: Develop three terms of the perturbation solutions to the real roots of
$$(x^3 + 2x^2 + x)e^{-x} = \epsilon,$$
identifying the scalings in the expansion sequence $\delta_0(\epsilon)x_0 + \...
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Finding regular and singular roots of a cubic perturbed polynomial using rescaling
Question: Find the rescalings for the roots of
$$\epsilon^5 x^3 - (3 - 2\epsilon^2 + 10\epsilon^5 - \epsilon^6)x^2 + (30 - 3\epsilon -20 \epsilon^2 + 2\epsilon^3 + 24\epsilon^5 - 2\epsilon^6 - 2\...
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Develop perturbation solutions of a cubic polynomial
Question: Develop perturbation solutions to
$$x^3 + (3+4\epsilon + \epsilon^2)x^2 + (3 + 9\epsilon + 7\epsilon^2 + 2\epsilon^3)x + 1 + 5\epsilon + 8\epsilon^2 + 5\epsilon^3 + \epsilon^4 = 0$$
finding ...
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