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| cwf:space-analyticity_of_wave_function:start [2016/11/22 13:16] – created markus | cwf:space-analyticity_of_wave_function:start [2017/01/10 17:48] (current) – markus |
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| ====== Space-analyticity of wave function ====== | ====== Space-analyticity of wave function ====== |
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| The object of this topical section is to discuss conditions such that a wave function $\Psi(t,x)$ as a solution to the time-dependent Schrödinger equation with initial state $\Psi_0$ is real-analytic in several or all $n=3N$ space coordinates.[(Analyticity in several variables might not be entirely trivial, but [[wp>Osgood's lemma]] adds some confidence.)] We always assume to start with a real-analytic initial state $\Psi_0$. //Real-analytic// means that at every point $x_0$ one can write the function locally as a power series (Taylor expansion). Because there is a positive convergence radius at every $x_0$ in real space, the area of convergence also extends into the complex domain. Thus real-analytic automatically means there is an open domain $D$ in the complexified configuration space $\mathbb{C}^n$ that includes the full real configuration space $\mathbb{R}^n$, i.e., $\mathbb{R}^n \subset D \subset \mathbb{C}^n$, and on which the function is holomorphic. It is yet not guaranteed that this domain $D$ includes a strip of finite width around the real configuration space as assumed in some related results about analyticity of the wave function. | The object of this topical section is to discuss conditions such that a wave function $\Psi(t,x)$ as a solution to the time-dependent Schrödinger equation with initial state $\Psi_0$ is jointly real-analytic in all $n=3N$ space coordinates.[(Through Hartogs' theorem this is equivalent to separate analyticity in all coordinates, see [[https://www.encyclopediaofmath.org/index.php/Hartogs_theorem|Hartogs' theorem (Encyclopedia of Mathematics)]].)] We always assume to start with a real-analytic initial state $\Psi_0$. //Real-analytic// means that at every point $x_0$ one can write the function locally as a power series (Taylor expansion). Because there is a positive convergence radius at every $x_0$ in real space, the area of convergence also extends into the complex domain. Thus real-analytic automatically means there is an open domain $D$ in the complexified configuration space $\mathbb{C}^n$ that includes the full real configuration space $\mathbb{R}^n$, i.e., $\mathbb{R}^n \subset D \subset \mathbb{C}^n$, and on which the function is holomorphic. It is yet not guaranteed that this domain $D$ includes a strip of finite width around the real configuration space as assumed in some related results about analyticity of the wave function. |
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| The first positive result is [[free_evolution|space-analyticity under free evolution]] of an analytic initial state. | The first positive result is [[free_evolution|space-analyticity under free evolution]] of an analytic initial state. |
| | Then it should be studied, if analyticity is conserved also with non-analytic potentials like the [[Coulomb potential|singular Coulomb potential]]. |
