Nodal and other properties of the second eigenfunction of the Laplacian in the plane
Author:
Richard L. Liboff
Journal:
Quart. Appl. Math. 63 (2005), 673-679
MSC (2000):
Primary 65Nxx, 35Jxx
DOI:
https://doi.org/10.1090/S0033-569X-05-00975-3
Published electronically:
August 3, 2005
MathSciNet review:
2187925
Full-text PDF Free Access
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Abstract: Rules are established for the intersection of nodals at a boundary in the plane relevant to the second eigenfunction of the Laplacian. Employing these results together with regularity theorems related to odd reflection of solutions of the Helmholtz equation, as well as a variation of C.S. Lin’s analysis, the following theorem is revisited: The nodal curve of the second eigenstate of the Laplacian for bounded convex domains in the plane, with Dirichlet boundary conditions, is a simple curve that intersects the boundary in two distinct points. Application is made to the regular convex polygons with $C_{n}, n \geq 2$, symmetry and to convex billiards with smooth boundaries.
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1 R. L. Liboff, Many faces of the Helmholtz equation, Phys. Essays 12 (1999) 1-8.
2 S. W. McDonald and A. N. Kaufmanm, Wave chaos in the stadium: Statistical properties of short-wave solutions to the Helmholtz equation, Phys. Rev. 37 (1988) 3067-3078.
3 M. A. Pinsky, The eigenvalues of an equilateral triangle, Siam J. Math. Anal. 11 (1980) 819-849.
4 G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comm. Math. Helv. 69 (1994) 142-154.
5 R. Courant and D. Hilbert, Methods of mathematical physics Vol. 1, Interscience, New York, 1966.
6 P. R. Garabedian, Partial differential equations, 2nd ed., Chelsea, New York, 1986.
7 R. L. Liboff, (a) The polygon quantum billiard problem, J. Math Phys. 35 (1994) 596-607, (b) Circular-sector quantum billiard and allied configurations J. Math Phys. 35 (1994) 2218-2228, (c) Nodal-surface conjectures for the convex quantum billiard, J. Math Phys. 35 3881-3886 (1994). ; ;
8 V. Amar, M. Pauri and A. Scotti, Schrödinger equation for convex plane polygons: A tiling method for the derivation of eigenvalues and eigenfunctions, J. Math Phys. 32 (1991) 2442-2449.
9 P. J. Richens and M. V. Berry, Pseudointegrable systems in classical and quantum mechanics, Physica 2D (1981) 495-412.
10 L. Payne, On two conjectures in the fixed membrane problem, J. Appl. Math. Phys. 24 (1973) 720-729.
11 L. Halbeisen and N. Hungerbühker, On periodic billiard trajectories in obtuse triangles, Siam Review 42 (1999) 657-670.
12 R. L. Liboff, Conical quantum billiard revisited, Quart. Appl. Math. LIX (2001) 343-351.
13 R. L. Liboff, Mapping onto the plane of a class of concave billiards. Quart. Appl. Math. LXII (2004) 323-335.
14 H. Lewy, On the reflection laws of second order differential equations in two independent variables. Bull. American Math. Soc. 65 (1959) 37-58.
15 C. S. Lin, On the second eigenstate of the Laplacian in ${\bf \mathbb {R}}^2$, Comm. Math Phys. 111 (1987) 111-161.
16 A. Melas. On the nodal line of the second eigenfunction of the Laplacian in ${\mathbb {R}}^2$, J. Differential Geom. 35 (1992) 255-263.
17 S.-Y. Cheng, Eigenfunctions and nodal sets, Comment. Math Helvetici 51 (1976) 43-55.
18 H. S. M. Coxeter, Regular polytopes 3rd. ed., Dover, New York, 1973.
19 R. L. Liboff, Complex hyperspherical equations, nodal-partitioning, and first-excited-state theorems in $\mathbb {R}^n$, Int. J. Theor. Phys. 41 (2002) 1957-1970.
20 R. L. Liboff, Convex polyhedra in $\mathbb {R}^n$, Quart. Appl. Math. LX (2002) 75-85.
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Additional Information
Richard L. Liboff
Affiliation:
Schools of Electrical Engineering and Applied Physics, and Center for Applied Math, Cornell University, Ithaca, New York 14853-5401
Email:
richaard@ece.cornell.edu
Received by editor(s):
January 11, 2005
Received by editor(s) in revised form:
February 24, 2005
Published electronically:
August 3, 2005
Additional Notes:
Fruitful discussions on these topics with my colleagues Alfred Schatz, Bradley Minch, Mason Porter and Sidney Leibovich are gratefully acknowledged. I am particularly indebted to Lawrence Payne for sharing his expertise with me in this study.
Article copyright:
© Copyright 2005
Brown University
