# Lamb's reconstruction of potentials and spatially localized scattering in nonrelativistic quantum mechanics

## TL;DR

This paper presents a simple condition for reconstructing certain Hamiltonians with real potentials from their eigenfunctions, with implications for quantum state preparation and localized scattering states.

## Contribution

It introduces a new reconstructibility condition for Hamiltonians from eigenfunctions, relevant to quantum state preparation and engineering localized scattering states.

## Key findings

- Reconstruction condition for Hamiltonians with real potentials.
- Identification of localized scattering states in specific potentials.
- Potential applications in quantum state engineering.

## Abstract

We formulate a simple condition for reconstructibility of a certain class of Hamiltonians with real potentials from the knowledge of their complex-valued eigenfunctions. This may be relevant to the question of preparability of quantum states raised by W. Lamb in his 1969 paper on operational interpretation of quantum mechanics. Of particular interest to engineering applications are: (a) an exotic case of an upside-down harmonic-oscillator-type potential (a variation of the inverted "Mexican hat" potential) whose square-integrable complex eigenfunction describes a localized scattering state similar to the "bound state in the continuum" of von Neumann and Wigner, and (b) a spatially confined scattering state of a particle moving in an infinite well with the properly shaped potential bottom.