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Consider the 2D Dirac operator $$H = \begin{pmatrix} 0 & \partial_{\bar z} \\ \partial_z & 0 \end{pmatrix}$$ where $\partial_z = \partial_x - i \partial_y$ and $\partial_{\bar z} = \partial_x + i \par …

Consider the matrix-valued operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ I am wondering if one can explicitly compute the eigenfunctions of that object on the spac …

I am curious about the limiting behaviour of a certian sequence of functions $$f_n:=\left(\sum_{k=1}^{\infty} 2^{-k} e^{i2^{k}/n}\right)^n$$ -where $i$ is the imaginary unit-to get a conjecture about …

We consider the following function $$f:(\mathbb S^n)^N \rightarrow \mathbb R^{n+1} \text{ such that } f(x_1,...,x_N)= \sum_{i=1}^N x_i.$$ This function can be written in Cartesian coordinates as $f(x) …

Consider the equation $$f'(x)+ g(x)f(x)=0$$ This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$ Similarly, we can look at complex variables and consider the equation and Wi …

It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $ …

Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$ Then, one can consider the function $f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space …

I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\alpha …

Consider the linear constant coefficient differential operator $P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$ $$P= \begin{pmatrix} D_{z}+c & a \\ b & D_{z}+c \end{pmatrix}$$ where $D_z=-i \parti …

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction. Imagine one has two evolution equations $$\partial_t u = p(x,\partial_x,f)u$$ $$\partia …

I would like to ask a follow-up question on a previous question of mine here whose proof does not seem to carry over to this case in an obvious way: We define the function $$F_{\varepsilon}(x) = \sum_ …

Consider two positive-semi definite matrices $T_1, T_2$ of unit trace. Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two. We then study $f(\lambda) := \operatorname{tr}(T(\ …

Consider a self-adjoint matrix $M$ that has block form $$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{11} \end{pmatrix}.$$ I am wondering if there exists any criterion to decide if this matrix …

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues …

When studying the blow-up for focusing non-linear Schrödinger equation (NLS) one often compares the initial-state to a stationary solution. In the energy-critical case, this stationary solution is fo …