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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