[Paper Review] Sum rules for Confining Potentials

C V Sukumar|arXiv (Cornell University)|Nov 6, 2006

Quantum Mechanics and Applications3 references4 citations

TL;DR

This paper develops sum rules for eigenvalues of confining potentials in quantum mechanics by leveraging Green's functions and separating contributions from even and odd states. It shows that divergences in individual sum rules for even and odd states can be canceled by subtraction, yielding a convergent sum rule involving both spectra—verified numerically for power-law and quartic potentials, including a sum rule for Airy function zeros.

ABSTRACT

Using the Green's function associated with the one-dimensional Schroedinger equation it is possible to establish a hierarchy of sum rules involving the eigenvalues of confining potentials which have only a boundstate spectrum. For some potentials the sum rules could lead to divergences. It is shown that when this happens it is possible to examine the separate sum rules satisfied by the even and odd eigenstates of a symmetric confining potential and by subtraction cancel the divergences exactly and produce a new sum rule which is free of divergences. The procedure is illustrated by considering symmetric power law potentials and the use of several examples. One of the examples considered shows that the zeros of the Airy function and its derivative obey a sum rule and this sum rule is verified. It is also shown how the procedure may be generalised to establish sum rules for arbitrary symmetric confining potentials.

Motivation & Objective

• To establish sum rules for eigenvalues of confining potentials using Green's functions.
• To address the issue of divergent sum rules in symmetric confining potentials when summing over all bound states.
• To show that divergences in separate even and odd state sum rules can be canceled by subtraction to yield a finite, convergent sum rule.
• To verify the derived sum rules numerically and analytically for power-law and solvable potentials, including the quartic oscillator and particle in a box.
• To demonstrate that the inverses of eigenvalues for certain potentials yield simple expressions involving gamma functions, revealing deep mathematical structure.

Proposed method

• Construct Green's functions for one-dimensional Schrödinger operators with confining potentials using eigenfunction expansions and Wronskian-based solutions.
• Derive sum rules by equating two representations of the Green's function: one via eigenfunction completeness and one via fundamental solutions.
• Separate the sum rules into contributions from even and odd states by imposing appropriate boundary conditions on the Green's function.
• Identify divergences in individual even and odd state sum rules for power-law potentials and subtract them to obtain a finite, convergent sum rule.
• Use WKB approximation for high-lying eigenvalues and numerical evaluation for low-lying states to verify sum rules.
• Apply scaling arguments and asymptotic limits (e.g., N→∞) to connect to the particle-in-a-box model and confirm analytical results.

Experimental results

Research questions

• RQ1 Can sum rules for eigenvalues of confining potentials be derived using Green's functions, even when individual sums diverge?
• RQ2 Is it possible to cancel divergences in separate sum rules for even and odd states of symmetric potentials by subtraction?
• RQ3 Do the resulting finite sum rules for combined even and odd spectra yield closed-form expressions involving special functions like the gamma function?
• RQ4 Can the derived sum rules be verified numerically and analytically for solvable potentials such as the quartic oscillator and particle in a box?
• RQ5 Do the zeros of the Airy function and its derivative satisfy a sum rule, and can this be derived and confirmed from the formalism?

Key findings

• For symmetric confining potentials, divergences in sum rules for even and odd states can be exactly canceled by subtraction, yielding a finite sum rule.
• The sum rule for the difference between odd and even state eigenvalue inverses converges and is given by a closed-form expression involving gamma functions.
• For the quartic potential (N=4), the sum rule evaluates to approximately 0.76330, matching numerical estimates within 0.0002.
• The sum rule for the particle in a box (N→∞) yields S = π²/12 ≈ 0.822467, consistent with analytical evaluation of ζ(2) series.
• The zeros of the Airy function and its derivative satisfy a sum rule that is derived and verified using the proposed method.
• Higher-order sum rules derived from the Green's function formalism reveal intricate algebraic relationships between eigenvalues of the Schrödinger equation.