Reply to Comment by Brown and Carrington on "Phase-Space Approach to Solving the Time-Independent Schr\"{o}dinger Equation"

TL;DR

This paper responds to comments on a phase-space method for solving the Schrödinger equation, clarifying the applicability of contraction ideas and the use of biorthogonal bases, and confirming their relevance in prior work.

Contribution

The authors clarify that the contraction idea applies to any DVR basis and demonstrate that the biorthogonal basis is implicitly used in previous methods.

Findings

Contraction idea is valid for any DVR basis, not just periodic.

Biorthogonal basis is implicitly used in prior work.

Clarification of methodological assumptions in phase-space approaches.

Abstract

The Comment of Brown and Carrington Jr. (BC) has two main points: 1) that the contraction idea of Shimshovitz and Tannor (ST) can be used for any DVR basis, not necessarily periodic, and 2) that the biorthogonal basis introduced by ST is unnecessary. We fully agree with the first point and in fact have several works that were in press at the time of the Comment that illustrate this. On the second point we show that BC implicitly use the biorthogonal basis of ST.