# Tagged Questions

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### Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
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Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ... 1answer 72 views ### Is there a useful relationship between pointwise and L^2 distance? It would be really convenient to get a bound on the point-wise closeness of functions by knowing their L^2 distance. Clearly, if two functions are close in the L^2 sense, you cannot get a general ... 1answer 36 views ### spectral measure and integral query I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that \int_{{\sigma}(T)} {\lambda}\,dE=T If ({\lambda}_{n}) is the ... 2answers 95 views ### Find the fallacy in using the Cauchy–Schwarz inequality Let \int_{a}^{b}\frac{f(x)}{x}dx=k, wherein f(x),a,b,k are positive. According to the Cauchy–Schwarz inequality: \int_{a}^{b}xf(x)dx=\int_{a}^{b}x^{2}\frac{f(x)}{x}dx\leq \left ( ... 2answers 92 views ### (From Lang SL_2) Orthonormal bases for L^2 (X \times Y) Lang SL_2 p. 13 :Let \{\phi_i\}, \{\psi_i\} be orthonormal bases for L^2(X) and L^2(Y) respectively. Let$$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$Then \{\theta_{ij}\} is an ... 1answer 116 views ### Computing an explicit solution to an integral equation via the Neumann Series. I am hoping that someone can provide guidance for solving the integral equation$$u=f+\lambda Au$$where 1/\lambda\notin\sigma(A), f\in L^2[0,2\pi], and A:L^2[0,2\pi]\to L^2[0,2\pi] is defined ... 0answers 171 views ### Prove Heisenberg uncertainty principle (measure and integration theory) Here is a question in measure and integration theory, Let f be a continuously differentiable complex function on \mathbf{R} s.t. the functions x \mapsto xf(x) and f' are in ... 1answer 93 views ### Is my proof correct? I want to show if V \subset H is dense, then L^2(0,T;V) \subset L^2(0,T;H) is dense too. I want to show that if V \subset H is a dense embedding then L^2(0,T;V) \subset L^2(0,T;H) is dense too. Everything is a Hilbert space. Let h \in L^2(0,T;H). Then h(t) \in H for each t. By ... 0answers 278 views ### Is this question solvable? 2 non-linear equations and the proof that the solution is unique (with asymmetric bounty option) As mentioned in the title I want to show the uniqueness of the solution to 2 non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ... 1answer 146 views ### Prove or disprove that the given expression is “always” positive I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that ... 1answer 91 views ### Is this function positive? I was wondering if:$$\int_0^1x(t)\int_0^tx(s)ds\ dt is positive for a general $x\in L_2[0,1]$ . Can you help me with this?
Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
### Weak and pointwise convergence in a $L^2$ space
Let $I$ be a measured space (typically an interval of $\Bbb R$ with the Lebesgue measure), and let $(f_n)_n$ a sequence of function of $L^2(I)$. Assume that the sequence $(f_n)$ converge pointwise ...