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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 …

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) …

I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations $$\partial_{z_2}f_1(z_1,z_2) + \partial_{z_1} f_2(z_1,z_2)=0 \text{ and }$$ $$\partial_{\bar z_1}f_ …

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 …

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 think this is one of the quasi exactly solvable potentials for the Schrödinger equation, see this paper. To say a bit more: For particular choices of $h$ certain eigenfunctions are explicit but not …

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_ …

This is either Helffer-Sjostrand https://www.tandfonline.com/doi/abs/10.1080/03605308408820335 or Barry Simon https://www.jstor.org/stable/pdf/2007072.pdf?refreqid=excelsior%3A258084917fff9e0c10088abb …

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 …

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. …

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind al …

The following result is well-known: Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have $$H \left( (1 - \lambda)(x_1,y_1) + …

It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values. We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \righ …

I am looking for a function with the following property: Let $v_1,v_2$ be two linearly independent vectors in $\mathbb{R}^2.$ I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$ I am trying …

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators …