Threshold energy dependence as a function of potential strength and the nonexistence of bound states

Journal Article · Fri Aug 01 04:00:00 EDT 1975 · Phys. Rev., A, v. 12, no. 2, pp. 349-352

DOI:https://doi.org/10.1103/PhysRevA.12.349· OSTI ID:4144715

Aronson, I; Kleinman, C J; Spruch, L

The difficulty in attempting to prove that a given set of particles cannot form a bound state is the absence of a margin of error; the possibility of a bound state of arbitrarily small binding energy must be ruled out. At the sacrifice of rigor, one can hope to bypass the difficulty by studying the ground- state energy E(lambda) associated with H(lambda) identical with H/sub true/ + lambda/sub $nu$/, where H/sub true/ is the true Hamiltonian, $nu$ is an artificial attractive potential, and lambda greater than 0. E(lambda) can be estimated via a Rayleigh-Ritz calculation. If H/sub true/ falls just short of being able to support a bound state, H(lambda) for lambda ''not too small'' will support a bound state of some significant binding. A margin of error is thereby created; the inability to find a bound state for lambda ''not too small'' suggests not only that H(lambda) can support at best a weakly bound state but that H/sub true/ cannot support a bound state at all. To give the argument real substance, one studies E(lambda) in the neighborhood of lambda = lambda$sub 0$, the (unknown) smallest value for lambda for which H(lambda) can support a bound state. A comparison of E(lambda) determined numerically with the form of E(lambda) obtained with the use of a crude bound-state wave function in the Feynman theorem gives a rough self-consistency check. One thereby obtains a believable lower bound on the energy of a possible bound state of H/sub true/ or a believable argument that no such bound state exists. The method is applied to the triplet state of H$sup -$. (auth)

• Research Organization: City Univ. of New York
• Sponsoring Organization: USDOE
• NSA Number: NSA-33-008284
• OSTI ID: 4144715
• Journal Information: Phys. Rev., A, v. 12, no. 2, pp. 349-352, Journal Name: Phys. Rev., A, v. 12, no. 2, pp. 349-352; ISSN PLRAA
• Country of Publication: United States
• Language: English

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