# Schr\"odinger equation with finitely many $\delta$-interactions: closed form, integral and series representations for solutions

## TL;DR

This paper derives explicit solutions and representations for the one-dimensional Schrödinger equation with finitely many delta interactions, using transmutation operators, spectral series, and Bessel function expansions.

## Contribution

It introduces a closed form solution, integral and series representations, and a practical construction method for the Schrödinger equation with multiple delta interactions.

## Key findings

- Explicit closed form solutions for the equation with delta interactions.
- Representation of solutions as Neumann series of Bessel functions.
- Development of a spectral parameter power series method for transmutation operators.

## 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.