The Boundary Value Problem of Scattering Theory
Владимир Александрович Марченко · Operator theory · 1986
This chapter is devoted to a detailed study of the boundary value problem generated on the half line 0 ≤ x < ∞ by the differential equation (3.1.1) %MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBbXgBd9 % gzLbvyNv2CaeHbafKCPfgBGuLBPn2BKvginnfaiuaacaWFTaGaa8xE % aiaa-jcacaWFRaGaa8xCamaabmaabaGaa8hEaaGaayjkaiaawMcaai % aa-LhacaWF9aGaa83UdmaaCaaaleqabaGaa8Nmaaaakiaa-Lhaaaa!4CF1! $$-y+q\left(x\right)y={\lambda^2}y$$ and the boundary condition (3.1.2) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaabm % aabaGaaGimaaGaayjkaiaawMcaaiabg2da9iaaicdaaaa!3AF3! $$y\left(0\right)=0$$ for the important special case where the function q(x) is real and satisfies the condition (3.1.3) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaca % WG4bWaaqWaaeaacaWGXbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaa % caGLhWUaayjcSdGaamizaiaadIhaaSqaaiaaicdaaeaacqGHEisPa0 % Gaey4kIipakiabgYda8iabg6HiLcaa!4686! $$\int\limits_0^\infty{x\left|{q\left(x\right)}\right|dx}<\infty$$ which is assumed to hold throughout the chapter. From condition (3.1.3) it is clear that (3.1.1) reduces to the simpler equation -y″ = λ2 y when x → ∞. This permits us a complete investigation of the properties of the solution to equation (3.1.1), and this is our goal in the present section.