All Questions
Tagged with fa.functional-analysis mp.mathematical-physics
221 questions
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The Quantum Operations On The Bipartite Systems
Given two distinct and noninteracting quantum mechanical
systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces
$\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space
of the ...
- 1
0
votes
1
answer
258
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Exponential derivative operator and continuous functions
I would like to know how to write down the following expression
$$f(y)=\frac{1}{y^{n} e^{\frac{d}{dy}}g(y)}$$
in the form of $e^{-\frac{d}{dy}}y^{-n}(\frac{1}{g(y)})$ where $n$ is an integer and $f,g: ...
- 159
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2
answers
2k
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fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
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1
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79
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The operator of exponential derivative applied in quotients
I have an other question for a function different to the example given before in the link below:
Exponential derivative operator and continuous functions
We define for instance a function as:
$$H(y)=\...
- 159
0
votes
1
answer
267
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Invariance of sets under Schrödinger equations
We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$
$$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$
$$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$
...
- 115
0
votes
1
answer
912
views
Bounding derivative of a function
Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, |f|>f_{max}$. Can we have a upper bound on the following,
$\Big|\frac{a'(t)}{a(t)}\...
- 151
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votes
0
answers
77
views
Nice formula for powers of modified Bessel function
Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series
$$1+aK_v+(aK_v)^2+(aK_v)^3...$$
I know there are formula for product of two such functions. I would ...
- 275
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0
answers
66
views
Equality between operators, on dense subspace, from a quadratic form point of view
Let $L \ge 1$ and consider a finite box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$. The set of functions:
$$\psi_{p}(x) = \frac{1}{L^{d/2}}e^{i\langle p,x\rangle} \quad p\in \frac{2\pi}{L}\mathbb{Z}^{...
- 1,305
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192
views
Reference request: an introduction to nuclear spaces
I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
- 721
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138
views
Question about a step in the proof of the min-max principle
I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
- 1,305
0
votes
0
answers
415
views
Spectral theorem for commuting operators
Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
- 3,515
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241
views
About the proof of Lebesgue decomposition theorem for Hilbert spaces
Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
- 1,305
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210
views
Reed-Simon Vol. IV: Question regarding convergence of eigenvalues
I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
- 91
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164
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Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
- 101
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0
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230
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Definition of tensor product of dense subspaces of Hilbert spaces
Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...
- 1,305
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0
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168
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Creation and Annihilation operators in QFT - Part II
Following some suggestions on my previous posts, I'm trying to reformulate my question in a more specific way. This is a continuation of my original post. Since the mentioned post, I think I've ...
- 3,515
0
votes
0
answers
85
views
Sine-Gordon transformation and functional integrals
In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, ...
- 3,515
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145
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“Chapman-Kolmogorov”-convolution vs. smoothness
Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
- 1,461
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237
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Spectrum of a Hamiltonian on the real line
Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
- 21.8k
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1
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142
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A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 17
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155
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General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411