All Questions
Tagged with perturbation-theory reference-request
20
questions
3
votes
2
answers
107
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 ...
- 2,311
1
vote
0
answers
51
views
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 ...
- 520
2
votes
0
answers
89
views
Alternative reference request for Kato's Perturbation Theory
I am studying Analytic Perturbation Theory from Kato's book.But sometimes I find it really difficult to follow.(e.g. In Chapter 2 , he directly talks about algebraic singularities without defining it ...
3
votes
0
answers
75
views
The linear heat eqaution with potential $V\in L_{t,x}^{\frac{n+2}2}$ is well-posed for initial data in $L^p$ for any $p\in(1,\infty)$
Consider the linear heat equation with potential
$$u_t-\Delta u-V(x,t)u=0\qquad \text{in }\ \ \mathbb R^n\times[0,\infty),$$
where $V\in L_{t,x}^{\frac{n+2}2}$. Show that this equation is well-posed ...
- 13.7k
1
vote
0
answers
46
views
Regular Perturbation theory for $L=L_0+\epsilon ix$, where $L_0$ is an unbounded self-adjoint operator, on $\mathbb R$
I am looking to understand the spectrum of the following operator:
$L_{\epsilon}[u](x)=L_0[u](x)+i\epsilon x u(x)$ on $\mathbb R$. Here $L_0$ is a negative semi-definite self-adjoint operator that has ...
0
votes
0
answers
52
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 an ...
- 1,802
4
votes
1
answer
160
views
Characterizing (stationary) points by the number of valleys one can descent into
In non-convex optimizing of more than 2 times differentiable $f: \mathbb{R}^2 \mapsto \mathbb{R}$ we can encounter saddle points that have multiple valley one could descent into.
At $(0,0)$ there are ...
- 236
1
vote
0
answers
46
views
Bound eigenbasis error of perturbed matrix by eigenbasis error of perturbation
I've been searching without much luck for a reference that can help with the following problem:
Let $R$, $S$, and $E$ be symmetric matrices, with $R = S + E$, where $E$ is a small perturbation. Then ...
1
vote
1
answer
103
views
Books that provide justifications of perturbations and asymptotic methods.
I am looking for books which provide justifications (proofs of appropriate theorems ) of various perturbation methods.
In particular I would like to study about justification of matched asymptotic ...
- 69
2
votes
0
answers
58
views
Reference on double perturbation theory?
I have a system that depends on three perturbations:
$$\frac{dx}{dt}=f(x;\epsilon,\mu,\eta)$$
The $\epsilon$ perturbation is independent of the other ones, but $\mu,\eta$ have a joint interaction I ...
- 1,039
0
votes
0
answers
30
views
Big-O / smal-o for perturbed matrix-inversion: What is $(A_n + \mathcal O_n(n^{-\alpha}))^{-1}$?
Let $\alpha > 0$ and for each natural number, let $B_n$ be a square matrix $B_n = \mathcal O(n^{-\alpha})$ as $n \rightarrow \infty$. Suppose $A_n$ is ineverible for every $n$ and $A_n \rightarrow ...
- 9,575
0
votes
1
answer
152
views
Singular perturbation theory in non-standard form
Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form,
\begin{align}
\dot{x} &=f\left( x,z,\varepsilon \right) \\...
- 1,833
3
votes
1
answer
2k
views
Is there a book on the purely mathematical version of perturbation theory?
Is there a book on the purely mathematical version of perturbation theory, or all current references just in relation to applied fields like statistics and quantum mechanics? I remember first coming ...
- 131
5
votes
3
answers
1k
views
Books on Perturbation Methods
I am having problems finding descent books on perturbation methods. I am looking for a book which covers; asymptotic expansions, matched Asymptotic expansions, Laplace's Method, Method of steepest ...
2
votes
0
answers
79
views
Perturbation/Homotopy of injective map by injective map is injective?
The following is a thought that I am hoping is true. I haven't been able to prove it and cannot seem to find a reference either. Ideas and references are appreciated.
Suppose $F_i: \mathbb{R}^n \...
- 1,153