Questions tagged [perturbation-theory]

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2 votes
1 answer
71 views

Lipschitz continuity of eigenprojections

This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
1 vote
1 answer
310 views

Eigenvalue perturbation under sparse perturbations

Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
7 votes
1 answer
150 views

Is there an infinite combinatorics of common transseries expansions?

By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...
1 vote
0 answers
45 views

Bounds on the spectral radius of a perturbed directed graph

Suppose $(G_n)$ is a sequence of strongly connected directed graphs (without multiple edges) with $G_n$ having $n$ edges such that the adjacency matrix $A_n$ of $G_n$ is primitive, and let $(G_n’)$ be ...
1 vote
0 answers
43 views

How to get perturbation bounds of singular vectors

Let an adjacency matrix $A={A^\top}\in {\mathbb{R}^{n \times n}}$ (a binary matrix) of a simple undirected graph and its degree matrix $D$ be given. When adding $Q$ edges into the graph, which is ...
2 votes
0 answers
121 views

Linear elliptic problems: Are gradient estimates preserved after perturbation?

(This question is a duplicate from here) We start with the linear elliptic PDE $$ -\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\ u=0 \quad\text{on}\ \partial\Omega $$ We assume that $\...
1 vote
0 answers
98 views

Schrödinger equation approximation – continuity of eigenvalues with respect to potential

The question has been crossposted from Stackexchange after receiving no answers. Setup: the time-independent Schrödinger equation (eigenvalue problem): $(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$ (On ...
6 votes
3 answers
698 views

How do I solve the following definite integral (preferably by an asymptotic method)?

$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$ Note: $\mu$ here is an extremely small constant. I have tried: Estimating the integral by ...
3 votes
1 answer
614 views

Perturbed vs. unperturbed Hamiltonian system

Let's take a time-periodic Hamiltonian $H(t,x,y)$ on $\mathbb{R}^2$ and apply an arbitrarily small time-independent perturbation to $H$ via $$ \tilde H (t,x,y) = H(t,x,y) + \epsilon V(x,y), $$ where $...
7 votes
2 answers
231 views

Finding $\theta$ such that at least one eigenvalue of $A(\theta)$ is real

Is there a known method to find a set of $\theta$ such that at least one eigenvalue of $A(\theta)$ is purely real? Assume $A(\theta)$ is a real square matrix whose elements are linear functions of a ...
2 votes
2 answers
297 views

Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE) \begin{align} \label{eq:42} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...
4 votes
1 answer
103 views

Uniform decay of operator norm for smooth family of operators

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...
0 votes
1 answer
121 views

Could variable be still function in x and y after performing Reynolds averaging over area

All, Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$? Equations (1-...
0 votes
1 answer
187 views

Local differentiability of eigenvalues and eigenvectors of a real symmetric matrix

Let $A(x)\in\mathbb{R}^{n\times n}$ be a real symmetric matrix depending on the point $x\in\mathbb{R}^n$, where the eigenvalues are not necessarily simple. Can we say that for all $x$ there exists an ...
4 votes
1 answer
218 views

Asymptotics of integral representation of distribution

I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community. Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
2 votes
0 answers
135 views

Choosing the derivative of a flow

I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
0 votes
0 answers
42 views

Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?

Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed. Then, they are simultaneously ...
2 votes
0 answers
135 views

Perturbation theory for $UV^*$ in singular value decomposition

There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition' ...
2 votes
1 answer
84 views

Bounding eigenvalue/eigenspace perturbations for hermitian matrices

Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix. For real $t$, let us consider the one-parameter family $$ H(t) = H + t V$$ of Hermitian matrices. Kato's perturbation theory ...
12 votes
2 answers
1k views

Eigenvalue perturbation theory via Feynman diagrams

Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
3 votes
0 answers
238 views

Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?

In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...
5 votes
1 answer
119 views

Is Sun's spectral variation bound for normal matrices optimal?

In On the variation of the spectrum of a normal matrix, Sun proves the following result (Corollary 1.2): Let $A$ be an $n$-square normal matrix and $B$ an arbitrary $n$-square matrix. Then $$ \min_{\...
0 votes
1 answer
228 views

Essential spectrum under perturbation

Given a Banach space $X$ and a bounded linear operator $T$ on $X$. It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation. My question is about minimal ...
3 votes
0 answers
144 views

Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system $$Ax=a \qquad B(x)y=b(x), $$ where $a$ is a vector with scalar entries, $A$ is a ...
1 vote
2 answers
922 views

Lipschitz continuity of eigenvalues and eigenvectors of Hermitian matrices

It is well-known that the eigenvalues (in decreasing order) of a Hermitian matrix $A$ are Lipschitz continuous functions of $A$. Do there exist orthonormal eigenvectors that vary in a Lipschitz ...
3 votes
2 answers
252 views

Proof (or reference) about $\lambda_i(A+\epsilon e_je_j^*) = \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2).$

I'm looking for a proof (or a reference in a textbook) about the fact that $$ \lambda_i(A+\epsilon e_je_j^*) =_{\epsilon \to 0} \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2), $$ where $A$ is a ...
0 votes
2 answers
245 views

Convergence of the eigenvector matrix for an analytic perturbation of a singular matrix

Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H_1 + \epsilon^2 H_2 + \epsilon^3 H_3 + ... $$ All matrices are symmetric. Assume $...
0 votes
1 answer
156 views

Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates? Thank you in advance !
0 votes
1 answer
114 views

Perturbative approach starting from a probability distribution approximated form

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$, such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity. Consider the generic ...
2 votes
0 answers
43 views

Stability in coefficients for the rescaled radiative tranport equation

One form of the radiative transport equation is as follows: $$ v\cdot \nabla_x u + \left(\epsilon \sigma_a(x) + \frac{1}{\epsilon}\sigma_s(x)\right) u - \frac{1}{\epsilon}\sigma_a(x)\int_{S^{n-1}} p(v,...
1 vote
1 answer
103 views

Sum of positive self-adjoint operator and an imaginary "potential": literature request

To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
0 votes
0 answers
89 views

Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?

Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that $$\tag{1}S^{-1}\cdot A\cdot S = \...
19 votes
1 answer
2k views

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
1 vote
0 answers
117 views

Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation $$ (u=u_\epsilon)\\ \partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\ u(0,x)=u_0(...
3 votes
0 answers
183 views

How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
5 votes
1 answer
310 views

Analytic families of compact self-adjoint operators: eigenvalue extension

Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of $A(t)$ splits into ...
5 votes
0 answers
133 views

Series representation for unbounded perturbations of semigroup generators

Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then $A+B$...
1 vote
1 answer
153 views

Perturbation of the value of a general-sum game at a equilibirium

Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes ...
9 votes
2 answers
874 views

Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and ...
2 votes
0 answers
76 views

Two-variable singular perturbation analysis

I am having difficulty with a two-variable singular perturbation analysis on a set of ODE's. The key difficulty is also present in the following, embarrassingly simple problem: If $x\sim \mathcal{O}(...
6 votes
0 answers
221 views

Poking into a Lie group with your finger

I consider this as a differential geometry problem. I have asked some of my classmates who are more interested in that, and also looked into some literature, but none of what I've found seems to help. ...
12 votes
2 answers
968 views

A toy model in 0-d QFT

Questions For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$ The answer should directly relates to a counting problem about Feynman diagrams. Is there a tutorial for how ...
5 votes
1 answer
207 views

Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?

While analyzing a variational problem, I came to the following question: Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ ...
0 votes
0 answers
89 views

Coexistence of different solutions in a nonlinear matrix differential equation

I've faced a system of first-order nonlinear matrix differential equation, and I have tried to use perturbation method to approach the solutions. The differential equation has the form: \begin{...
7 votes
1 answer
483 views

Understanding Gillman's proof of the Chernoff bound for expander graphs

My question is about the proof of Claim 1 in this paper: Gillman (1993). At the end of the proof, the author says: The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
1 vote
0 answers
49 views

Perturbations of the eigen/singular directions

Let $A = U_A \Sigma_A V_A^\top$ and $B = U_B \Sigma_B V_B^\top$, and $A+B = U \Sigma V^\top$ be the respective singular vector decompositions. Is there some known relationship of the form $$\| U_A ...
2 votes
1 answer
736 views

$C^1$ perturbation of diffeomorphism is diffeomorphism?

if $f \in $ diff($M$), where $M$ is manifold, if $C^1$ perturbation $f_{\epsilon} $ of $f$ s.t. $||f_{\epsilon}-f||_{C^1} < \epsilon $. Can we prove $f_{\epsilon} \in $ diff($M$) if $\epsilon$ is ...
1 vote
0 answers
102 views

stochastical stable

Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...
2 votes
1 answer
274 views

Eigenvalue Argument Perturbation

Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g., $$ | \lambda_A - \lambda_B | \leq \| A - B \|_F, $$ where $A$ and $B$ are Hermitian ...
2 votes
0 answers
301 views

Determinant of a rank r perturbation

In the following paper: Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations on page 79, Golub et al. have the following set of equations: $f(\lambda) = \...