Schrödinger equation with finitely many $\delta$-interactions: closed form, integral and series representations for solutions

Abstract

A closed form solution for the one-dimensional Schrödinger equation with a finite number of $\delta$-interactions \[ \mathbf{L}_{q,\mathfrak{I}_{N}}y:=-y^{\prime\prime}+\left( q(x)+\sum _{k=1}^{N}α_{k}δ(x-x_{k})\right) y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] is presented in terms of the solution of the unperturbed equation \[ \mathbf{L}_{q}y:=-y^{\prime\prime}+q(x)y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] and a corresponding transmutation operator ${\mathbf{T}}_{{\mathfrak{I}}_N}^f$ is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator ${\mathbf{T}}_{{\mathfrak{I}}_N}^f$ transmutes the second derivative into the Schrödinger operator ${\mathbf{L}}_{q,{\mathfrak{I}}_N}$ on a Sobolev space $H^2$. A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived.

37 pages, 1 figure