Third order operator with periodic coefficients on the real line

@article{Badanin2011ThirdOO,
  title={Third order operator with periodic coefficients on the real line},
  author={Andrei Badanin and Evgeny L. Korotyaev},
  journal={arXiv: Mathematical Physics},
  year={2011}
}
We consider the third order operator with periodic coefficients on the real line. This operator is used in the integration of the non-linear evolution Boussinesq equation. For the minimal smoothness of the coefficients we prove that: 1) the operator is self-adjoint and it is decomposable into the direct integral, 2) the spectrum is absolutely continuous, fills the whole real axis, and has multiplicity one or three, 3) the Lyapunov function, analytic on a three-sheeted Riemann surface, is… 
6 Citations
Third-order operators with three-point conditions associated with Boussinesq's equation
We consider a non-self-adjoint third-order operator on the interval with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator
Resonances of third order differential operators
  • E. Korotyaev
  • Mathematics
    Journal of Mathematical Analysis and Applications
  • 2019
Dependence of solutions and eigenvalues of third order linear measure differential equations on measures
This paper deals with a complex third order linear measure differential equation $${\rm{id}}{(y')^ \bullet} + 2{\rm{i}}q(x)y'{\rm{d}}x + y({\rm{id}}q(x) + {\rm{d}}p(x)) = \lambda

References

SHOWING 1-10 OF 48 REFERENCES
Even order periodic operators on the real line
We consider $2p\ge 4$ order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by
Third order operator with small periodic coefficients
We consider the third order operator with small 1-periodic coefficients on the real line. The spectrum of the operator is absolutely continuous and covers all real line. Under the minimal conditions
Spectral asymptotics for periodic fourth-order operators
We consider the operator d 4 dt4 +V on the real line with a real periodic potential V . The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a
Spectral estimates for matrix-valued periodic Dirac operators
TLDR
The asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy (in terms of the Fourier coefficients of the potential) are determined.
Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line
We consider the Schrodinger operator on the real line with an N × N matrix-valued periodic potential, N > 1. The spectrum of this operator is absolutely continuous and consists of intervals separated
Spectral estimates for a periodic fourth-order operator
We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic coefficients on the real line. The spectrum of $H$ is absolutely continuous and consists of intervals separated by gaps. We
Conformal spectral theory for the monodromy matrix
For any N x N monodromy matrix we define the Lyapunov function which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the
The periodic Euler-Bernoulli equation
We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem [a(x)u(x)] = λρ(x)u(x), -∞ < x < ∞, where the functions a and p are periodic
...
...