# Complex hill's equation and the complex periodic korteweg‐de vries equations

Complex hill's equation and the complex periodic korteweg‐de vries equations Introduction In this paper, we continue the discussion of several authors of the spectral class of differential operators L = d2/dx2 q, with a potential q of period one, the spectral class being defined by a common periodic and antiperiodic spectrum. When the potential is real, the operator is selfadjoint and the periodic and antiperiodic eigenvalues form an increasing sequence furthermore, the multiplicity of eigenfunctions and eigenvalues is equal. When 2 g 1 < 00 eigenvalues are simple, the spectral class has g real dimensions. This case was treated by McKean and Moerbeke [29], Novikov [37], Lax [23] and [24], Dubrovin, Matveev and Novikov [ll], and others. An infinite simple spectrum gives rise to an infinite real-dimensional spectral class and was treated by McKean and Trubowitz [31]. In the following, we treat complex potentials and non-selfadjoint operators. For such operators one must distinguish between two types of spectral multiplicities. The geometric multiplicity is the number of eigenfunctions and is either one or two. The spectrum is determined by the equation A*(A) - 1 = 0, A being half the trace of the monodromy matrix of the first-order system associated with L . The algebraic multiplicity is the order http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications on Pure & Applied Mathematics Wiley

# Complex hill's equation and the complex periodic korteweg‐de vries equations

, Volume 39 (1) – Jan 1, 1986
49 pages

/lp/wiley/complex-hill-s-equation-and-the-complex-periodic-korteweg-de-vries-M3xHvU7HKB

# References (40)

Publisher
Wiley
ISSN
0010-3640
eISSN
1097-0312
DOI
10.1002/cpa.3160390102
Publisher site
See Article on Publisher Site

### Abstract

Introduction In this paper, we continue the discussion of several authors of the spectral class of differential operators L = d2/dx2 q, with a potential q of period one, the spectral class being defined by a common periodic and antiperiodic spectrum. When the potential is real, the operator is selfadjoint and the periodic and antiperiodic eigenvalues form an increasing sequence furthermore, the multiplicity of eigenfunctions and eigenvalues is equal. When 2 g 1 < 00 eigenvalues are simple, the spectral class has g real dimensions. This case was treated by McKean and Moerbeke [29], Novikov [37], Lax [23] and [24], Dubrovin, Matveev and Novikov [ll], and others. An infinite simple spectrum gives rise to an infinite real-dimensional spectral class and was treated by McKean and Trubowitz [31]. In the following, we treat complex potentials and non-selfadjoint operators. For such operators one must distinguish between two types of spectral multiplicities. The geometric multiplicity is the number of eigenfunctions and is either one or two. The spectrum is determined by the equation A*(A) - 1 = 0, A being half the trace of the monodromy matrix of the first-order system associated with L . The algebraic multiplicity is the order

### Journal

Communications on Pure & Applied MathematicsWiley

Published: Jan 1, 1986