Completeness of Derivatives of Squared Schroedinger Eigenfunctions and Explicit Solutions of the Linearized KdV Equation

摘要

Explicit solutions to the Cauchy problem for the linearized Kortweg-deVries (KdV) equation are constructed when the initial data is integrable. The method is analogous to the Fourier decomposition for a constant coefficient equation and uses the connection between the one-dimensional Schroedinger equation and the KdV equation, as discovered by Gardner, Greene, Kruskal, and Miura. An expansion theorem expressing any integrable function in terms of derivatives of squared Schroedinger (generalized) eigenfunctions is proved. These functions evolve according to the linearized KdV equation, hence the expansion of the initial data leads to a generalized solution of the linearized KdV equation. Under suitable restrictions on the initial data, the solution constructed is classical. The proof of the expansion theorem may be interpreted as the skew-adjoint analogue of the more familiar process of simultaneously diagonalizing two self-adjoint operators. (Author)