zbMATH — the first resource for mathematics

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.

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
Full Text: DOI EuDML
[1] Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Swetina, J.: Continuity and nodal properties near infinity for solutions of 2-dimensional Schrödinger equations. Duke Math. J.53, 271-306 (1986) · Zbl 0599.35036 · doi:10.1215/S0012-7094-86-05318-4
[2] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0361.35003
[3] Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover 1968 · Zbl 0171.38503
[4] Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Swetina, J.: Pointwise bounds on the asymptotics of spherically averagedL 2-solutions of one-body Schrödinger equations. Ann. Inst. H. Poincaré42, 341-361 (1985) · Zbl 0595.35033
[5] Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Swetina, J.: Asymptotics and continuity properties near infinity of solutions of Schrödinger equations in exterior domains. Ann. Inst. H. Poincaré46, 247-280 (1987) · Zbl 0641.35017
[6] Bers, L.: Local behaviour of solutions of general elliptic equations. Commun. Pure Appl. Math.8, 473-496 (1955) · Zbl 0066.08101 · doi:10.1002/cpa.3160080404
[7] Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helvetici51, 43-55 (1976) · Zbl 0334.35022 · doi:10.1007/BF02568142
[8] Cafarelli, L.A., Friedmann, A.: Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations. J. Differ. Equations60, 420-433 (1985) · Zbl 0593.35047 · doi:10.1016/0022-0396(85)90133-0
[9] Albert, J.H.: Generic properties of eigenfunctions of elliptic partial differential operators. Trans. Am. Math. Soc.238, 341-354 (1978) · Zbl 0379.35023 · doi:10.1090/S0002-9947-1978-0471000-3
[10] Uhlenbeck, K.: Generic properties of eigenfunctions. Am. J. Math.98, 1059-1078 (1976) · Zbl 0355.58017 · doi:10.2307/2374041
[11] Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: On the asymptotics of nodes ofL 2-solutions of Schrödinger equations in dimensions ?3. Commun. Math. Phys. (in press) · Zbl 0658.35021
[12] Mitrinovic, D.S.: Analytic inequalities, Die Grundlehren d. math. Wiss., Bd 165. Berlin-Heidelberg-New York: Springer 1970
[13] Reed, M., Simon, B.: Methods of modern mathematical physics I, Functional analysis. New York: Academic Press 1972 · Zbl 0242.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.