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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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