1d periodic differential operator of order 4

Vadim Tkachenko Ben-Gurion University of the Negev, Beer-Sheva, Israel

We consider a differential operator

L = \frac{d^{4}}{dx^{4}} + \frac{d}{dx}p(x)\frac{d}{dx} + q(x), x ∈ R^{1}

with 1-periodic functions p(x) and q(x). We prove that the characteristic equation for its Floquet multipliers is inverse and hence its spectrum in L^{2}(R^{1}) may be described using some hyper-elliptic Riemann surface.

We prove the following uniqueness theorem:

Let U(λ) be the monodromy matrix of operator L and let its characteristic determinant det(U(λ) − ρI) be the same as of operator L_{0} = \frac{d^{4}}{dx^{4}} . Then p(x) ≡ q(x) ≡ 0.