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

Abstract: A closed form solution for the one-dimensional Schr\"{o}dinger 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}\alpha{k}\delta(x-x_{k})\right) y=\lambda y,\quad0<x<b,\;\lambda \in\mathbb{C}% ] is presented in terms of the solution of the unperturbed equation [ \mathbf{L}{q}y:=-y^{\prime\prime}+q(x)y=\lambda y,\quad0<x<b,\;\lambda \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 Schr\"{o}dinger 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.