The discussion centers on the application of the Schrödinger equation within the context of the complex disk, specifically the region defined by ##\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}##. Participants highlight that the wave function ##\psi(z)## cannot be analytically defined in this space, as functions like sine or cosine lead to non-normalizable exponential behaviors when extended to the imaginary axis. The conversation also touches on the implications of using the real 2D disk ##\mathbb{D}_{\mathbb{R}}=\{(x,y)\in\mathbb{R}^{2}: x^2+y^2<1\}##, suggesting that energy eigenfunctions in this scenario can be expressed in terms of Bessel functions, specifically ##\psi(z) \propto J_l(k\sqrt{x^2 + y^2})##.
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Ok this is clear because from the complex point of view ##z## cannot be treated as a real position vector and, as conseguence, the physical interpretation of ##\psi(z)## is not so clear ... but if we consider the real 2D disk ## \mathbb{D}_{\mathbb{R}}=\{(x,y)\in\mathbb{R}^{2}: x^2+y^2<1\}## ? Is the situation similar to the 2D plane case?hilbert2 said:You can describe a point on 2D plane as a complex number, but the product of two such complex numbers doesn't have any clear interpretation in that case. The wave function ##\psi (z)## could not be required to be analytical in that situation, because a sine or cosine type function will behave like an increasing exponential function (not normalizable) if analytically continued to the imaginary axis.