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Asymptotics of the nodal lines of solutions of 2-dimensional Schrödinger equations. (English) Zbl 0627.35024
Results on nodal properties of $$L^ 2$$-solutions of 2-dimensional Schrödinger equations, recently obtained in [1] [see the author, T. Hoffmann-Ostenhof and J. Swetina, Duke Math. J. 53, 271-306 (1986; Zbl 0599.35036)] are sharpened. Let $$(-\Delta +V-E)\psi =0$$ in $$\Omega _ R=\{x\in {\mathbb{R}}^ 2|$$ $$r>R\}$$ where $$R>0$$, $$r\equiv | x|$$, $$\psi \in L^ 2(\Omega _ R)$$, $$E<0$$ and $$V(x)=V_ 1(r)+V_ 2(x)$$ with $$V_ 1$$, $$V_ 2$$ continuous and $$V_ 1,V_ 2\to 0$$ for $$r\to \infty$$. Then for \=R$$\geq R$$ large enough, $$v\in L^ 2(\Omega _{\bar R})$$ exists with $$v>0$$, $$v=v(r)$$ and $$(-\Delta +V_ 1(r)-E)v=0$$ in $$\Omega _{\bar R}$$. Further assume $$| V_ 1'| r^{1+\epsilon}\leq c$$, $$| V_ 2| r^{3/2+\delta}\leq C$$ in $$\Omega _ R$$ for some c, C, $$\epsilon$$, $$\delta >0$$ and let r, $$\omega$$ denote the usual polar coordinates. Under suitable assumptions on the regularity of $$V_ 2$$ with respect to $$\omega$$ uniformly for $$r\to \infty$$ it follows (see [1]) that $$\lim _{r\to \infty}\psi (r,\omega)/v(r)\equiv A(\omega)$$ exists, and A is real analytic. In the main theorem of this work it is shown that for large r the nodal set of $$\psi$$ $$\{x\in \Omega _ R|$$ $$\psi (x)=0\}$$ consists of non- intersecting nodal lines which look roughly speaking either like straight lines or like branches of parabolas. More explicitly: Suppose $$A(0)=0$$ with $$A(\omega)=\omega ^ M+d\omega ^{M+1}+O(\omega ^{M+2})$$ for $$| \omega |$$ small for some $$d\in {\mathbb{R}}$$, $$M\in {\mathbb{N}}$$. Let $$z_ i\in {\mathbb{R}}$$, $$1\leq i\leq M$$ denote the zeros of the Hermite polynomial $$H_ M(z)$$ of order M. Then for $$\epsilon >0$$ small and $$R_{\epsilon}$$ large $$\{x\in \Omega _ R|$$ $$r>R_{\epsilon}$$, $$| \omega | <\epsilon$$, $$\psi (x)=0\}$$ consists of M nodal lines, which, represented in cartesian coordinates by $$x_ 2=G_ i(x_ 1)$$, $$1\leq i\leq M$$, show the following asymptotic behaviour for $$x_ 1\to \infty:$$ For $$M\geq 2$$ and $$z_ i\neq 0$$, $$G_ i(x_ 1)=(z_ i/(| E| /4)^{1/4}+o(1))\sqrt{x_ 1}$$. For M odd $$H_ M(0)=0$$ and without loss let $$z_ 1=0$$, then $$G_ 1(x_ 1)=d/\sqrt{| E|}+o(1)$$. These asymptotic nodal patterns turn up already in a non- trivial way in the radial case $$(V_ 2=0)$$ as can be demonstrated easily.

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
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