On the discrete spectrum of the one-dimensional Schr�dinger operator

Abstract

Suppose the potential V of the one-dimensional Schrodinger operator is a solution of the higher Kortewegde Wies equation (1) $$\frac{{\partial \tilde L}}{{\partial t}} = [\tilde L,M_{2n + 1} + c_1 M_{2n - 1} + \cdots + c_n M_1 ],$$ that vanishes at infinity along with its derivatives, where $$\tilde L = - \frac{{d^2 }}{{dx^2 }} + V(x,t)$$ and the Mt form Lax pairs with $$\tilde L$$ . The complex parameters Ci in Eq. (1) are such that the equation 2 $$\lambda ^{2n} + c_1 \lambda ^{2(n - 1)} + \cdots + c_{n - 1} \lambda ^2 + c_n = 0$$ has only nonreal roots. If λ0 is a root of Eq. (2),then λ 2 0 belongs to the discrete spectrum of the Schrodinger operator with potential V.