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.
Categories: math.CA math-ph math.MP