All Questions
Tagged with perturbation-theory mathematical-physics
20
questions
7
votes
1
answer
394
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 ($\...
- 108
2
votes
0
answers
87
views
Are there methods for capturing transcendentally small terms in perturbation theory?
By "transcendentally small terms" I mean corrections to a perturbation that are smaller than all polynomial orders. They are generally exponential in form. I would have linked to a Wikipedia ...
- 2,412
2
votes
0
answers
180
views
On the convergence of (eigen)values in perturbation theory ?!
Background: Let $H, H_0$ be real square matrices, we want to estimate the eigenvalues/vectors of
$$H = H_0 + \epsilon R$$
where $0 \leq \epsilon \leq 1$, those of $H_0$ are assumed to be known.
Let $\...
- 947
0
votes
0
answers
29
views
Existence of Taylor-like expansion in perturbation parameter
In perturbation theory, we assume that a Taylor-like series exists in the perturbation parameter. But is this always true? Or are there certain conditions? Also, can I get some resource ...
1
vote
1
answer
250
views
Pertubation theory in sagemath
I have seen the documentation on how to truncate polynomials using sage but I am stuck as to how I can actually apply this in my work...
I am currently trying to find the...say Ricci tensor for a ...
- 13
1
vote
0
answers
232
views
'Taylor Expansion' of Integral - Asymptotic expansion - Exponential function
I need to evaluate the following integral in the limit $\kappa \ll 1$
$$\int_0^\infty exp(-\kappa t) f(t)\, dt,$$
where
$$f(x) = (1+x)(1-2x)\frac{u(x) \ln(u(x))}{u(x)^2 - 1},$$
$$u(x) = \frac{\sqrt{1+...
- 121
3
votes
0
answers
104
views
Second order homogenous differential equation with variable coefficients
I have a complicated non-linear first order homogeneous differential equation for coherent states $\psi(t)$. Via perturbation theory I obtained a linear non-homogeneous first order recursive ...
- 331
0
votes
1
answer
81
views
perturbation theory $\&$ integrability
Suppose we are studying square-integrable eigenfunctions of a linear operator (e.g. an ordinary differential operator), doing perturbation theory in a small parameter.
Suppose first-order ...
- 351
3
votes
0
answers
405
views
Approximating definite integral over infinitesimal interval (reformulated)
Pursuant to helpful comments by user254433, I have decided to take another swing at this problem while reformulating it with a simplified example.
(Reformulated) General Problem: Generally speaking, ...
- 427
1
vote
0
answers
129
views
Property of Laplace Transform
I've been reading a physics book about non-perturbative contributions to the path integral, and there is an equality I'm having difficulties to understand; I thought it might relate to an identity I ...
- 375
1
vote
1
answer
59
views
Perturbation of the non linear Schrödinger equation
I'm studying the non linear Schrödinger equation:
$$A_t=i A_{xx} - i \vert A \vert^2 A$$
In the problem sheets that I'm working on my teacher writes that this equation has solutions $A=Q e^{i \Omega ...
- 3,631
0
votes
0
answers
190
views
Asymptotic Expansion of Integrals
I am considering the Integral $$\int_{0}^{\infty}dt\frac{e^{-t}t^{s}}{1+\lambda x t}$$
I am trying to have an asymptotic expansion of the above said integral. The process of integration by parts ...
- 11
2
votes
2
answers
1k
views
How to justify differentiating an asymptotic series in WKB method
Given a second-order linear ordinary differential equation,
\begin{equation}
\epsilon^2 \frac{d^2y}{dx^2} = Q(x) y(x),
\tag{1}\label{1}
\end{equation}
where $\epsilon$ is regarded as a small positive ...
- 290
3
votes
1
answer
345
views
How to solve an ODE with $y^{-1}$ term
My major is not Mathematics, but I came across the following ODE for $y(x)$:
$$\left(y^3y^{\prime\prime\prime}\right)^\prime+\frac{5}{8}xy^\prime-\frac{1}{2}y+\frac{a}{y}=0,$$
where the prime denote ...
- 157
0
votes
1
answer
512
views
Finding a composite solution to an ODE (boundary layer problem)
Given $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, where $a(t)>0$, $u(0)=1$, $u(1)=1$, and assuming that the boundary layer is at $t=1$, and the boundary layer variable is $T=\frac{1-t}{...
- 9,708