  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.
