Questions tagged [perturbation-theory]

Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.

Filter by
Sorted by
Tagged with
0 votes
0 answers
14 views

The perturbation bound for the rank-1 matrix decomposition

Suppose $\mathbf B=\sigma^2 \bf uv^\top + Z$, where $\sigma^2 \bf uv^\top$ is an unknown rank-1 matrix, $\bf B$ is the observation, and $\bf Z$ is the perturbation. Let $\hat\sigma^2 \bf \hat{u} \hat{...
Jasper Cha's user avatar
2 votes
0 answers
49 views

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 ...
user19833's user avatar
3 votes
2 answers
103 views

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 ...
Azur's user avatar
  • 2,194
1 vote
0 answers
27 views

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 ...
GSofer's user avatar
  • 4,313
0 votes
0 answers
22 views

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:...
humble_torch_student's user avatar
0 votes
0 answers
35 views

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....
Alessandro Tassoni's user avatar
0 votes
0 answers
34 views

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 ...
Lwins's user avatar
  • 624
0 votes
0 answers
35 views

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 ...
Taras's user avatar
  • 1
0 votes
1 answer
42 views

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(\...
Jamminermit's user avatar
  • 1,923
7 votes
1 answer
367 views

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 ($\...
joerivan's user avatar
  • 108
0 votes
0 answers
59 views

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 ...
Sanket Biswas's user avatar
3 votes
1 answer
95 views

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 $\alpha>0$ ...
Sanket Biswas's user avatar
2 votes
1 answer
66 views

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 ...
Sanket Biswas's user avatar
1 vote
1 answer
65 views

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. ...
Nate's user avatar
  • 11
7 votes
2 answers
119 views

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 ...
Juno Kim's user avatar
  • 610
3 votes
0 answers
43 views

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 ...
Asasuser's user avatar
  • 305
1 vote
1 answer
71 views

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 ...
Sanket Biswas's user avatar
5 votes
1 answer
118 views

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 $y$ ($||^...
user655870's user avatar
0 votes
0 answers
23 views

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."...
Bikhu's user avatar
  • 78
0 votes
0 answers
29 views

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 ...
Jouenshin's user avatar
1 vote
0 answers
64 views

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 ...
athos's user avatar
  • 5,239
1 vote
1 answer
49 views

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 ...
Sanket Biswas's user avatar
1 vote
1 answer
43 views

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 ...
Alor'ad's user avatar
  • 11
1 vote
0 answers
89 views

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 $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$. ...
Paul Joh's user avatar
  • 697
1 vote
1 answer
45 views

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 ...
Geigercounter's user avatar
1 vote
2 answers
92 views

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) = ...
Sanket Biswas's user avatar
1 vote
1 answer
59 views

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 ...
Sanket Biswas's user avatar
1 vote
1 answer
100 views

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 ...
Sanket Biswas's user avatar
2 votes
0 answers
53 views

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 ...
Abady Kabbaj's user avatar
1 vote
0 answers
39 views

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 $x<1$ ...
R. Bhattacharya's user avatar
0 votes
1 answer
122 views

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(...
theo's user avatar
  • 3
1 vote
1 answer
86 views

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 $n \times n$ ...
Dawson Beatty's user avatar
1 vote
0 answers
77 views

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 ...
user655870's user avatar
0 votes
0 answers
37 views

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....
Chandler's user avatar
  • 445
0 votes
1 answer
60 views

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 + \...
Sanket Biswas's user avatar
0 votes
0 answers
31 views

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\...
Sanket Biswas's user avatar
1 vote
0 answers
36 views

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 ...
Sanket Biswas's user avatar
0 votes
0 answers
33 views

Perturbation with positive diagonal

Suppose I have some matrix $A$ and I perturb it by some small amount: $A \to A + \delta A$. I am interested in determining the sign of the quantity $a^\dagger \delta A a$. Here is what I know: $a$ is ...
redfive's user avatar
  • 101
1 vote
1 answer
91 views

Theorem (Rellich) - Perturbation Theory

I got stuck with part of a proof of: The steps are all clear to me, until it is said: "Therefore $f_w$ is the eigenvector of $ A+wB$ associated with the eigenvalue lying within for all small ...
X-man's user avatar
  • 19
2 votes
0 answers
95 views

Poisson Equation for a perturbed sphere - both exterior and interior solutions

I am trying to solve a Heat transfer problem in a slightly ellipsoidal geometry using the Poission Equation, and Cauchy matching conditions on the boundary between the interior (which is a finite ...
Prakash_S's user avatar
0 votes
1 answer
84 views

Asymptotics of a nonlinear PDE

Consider the partial differential equation with boundary conditions \begin{equation} \frac{\partial u}{\partial t} = \varepsilon \bigg( (u+1)\frac{\partial u}{\partial x}\bigg), \qquad \frac{\partial ...
Giraffes4thewin's user avatar
1 vote
0 answers
47 views

Where does this factor of $\pi$ come from in the period of small oscillations about equilibrium points?

I am working through some exercises in Arnold's Mathematical Methods of Classical Mechanics book, specifically the second problem on page 20. For context, $T(E)$ is the period of motion along a closed ...
masjgomz's user avatar
1 vote
0 answers
22 views

Perturbations of an integrable system with no resonant tori

I think the following is probably a trivial application of KAM theory, which I know little about, so I am hoping someone can give me an answer pointing me in the right direction. Suppose I have a ...
octonion's user avatar
  • 381
0 votes
0 answers
37 views

Number of distinct discrete eigenvalues of a self adjoint operator after compact perturbation increases.

Let $T$ be a self adjoint operator on a complex separable Hilbert space $H$. Let $K$ be self adjoint compact operator. So, $T+K$ is also self adjoint operator. I want to know for what $K$ the number ...
Bikhu's user avatar
  • 78
1 vote
1 answer
38 views

Roots of an equation (perturbation theory)

Consider $xe^{x-1}+x-2-\epsilon=0$ Assume the solution can be expanded in terms of $\epsilon$, i.e. $x=a_{0}+a_{1}\epsilon+a_2\epsilon^2+... (1)$ Substitute (1) into the equation, we have $(a_{0}+a_{1}...
tan1123581321's user avatar
0 votes
0 answers
33 views

Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator

Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator. In particular, is there any way we can say that the element of essential spectrum is an eigenvalue of ...
Bikhu's user avatar
  • 78
6 votes
0 answers
82 views

Method of Dominant Balance with high order system

This question comes from Bender and Orszag's Asymptotic Methods and Perturbation Theory. I'm practicing applying the method of dominant balance to study behavior as $x\to \infty$ for systems which ...
mwalth's user avatar
  • 1,106
0 votes
0 answers
22 views

Transversality of stable and unstable manifolds in a travelling wave equation

I am trying to understand the persistence of a heteroclinic orbit $(u^*,v^*,c^*)$ in the FitzHugh-Nagumo equation. I use geometric singular perturbation theory as described by Jones in "Geometric ...
Rumpsteakinator's user avatar
0 votes
1 answer
75 views

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 $...
Niser's user avatar
  • 87
0 votes
0 answers
40 views

Resolvent set of perturbed operator

Let $A_0:\mathcal{H} \supset \mathcal{D}(A_0) \to \mathcal{H}$ be a densely defined (by $\mathcal{D}(A_0)$ I denote a domain of an operator $A_0$), closed and self-adjoint operator acting on a ...
MI00's user avatar
  • 277

1
2 3 4 5
17