Discrete States for Singular Potential Problems

Scarf, F. L.

doi:10.1103/physrev.109.2170

Cited by 27 publications

(10 citation statements)

References 8 publications

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“…The singular attractive inverse square potential has been extensively studied in the coordinate representation (see for instance [24,25,28,29,30]). In Ref.…”

Section: Mechanicsmentioning

confidence: 99%

Regularization of the singular inverse square potential in quantum mechanics with a minimal length

2007

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We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

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“…The singular attractive inverse square potential has been extensively studied in the coordinate representation (see for instance [24,25,28,29,30]). In Ref.…”

Section: Mechanicsmentioning

confidence: 99%

Regularization of the singular inverse square potential in quantum mechanics with a minimal length

2007

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“…( 4) are restored from solutions of eqs. ( 8) by transformation (7). In what follows, we use u 1 (x; W ), u 2 (x; W ), and υ 1 (x; W ) defined by…”

Section: Exact Solutions and Asymptoticsmentioning

confidence: 99%

“…On the physical level of rigor, the Schrödinger equation with potential (1) was studied for a long time in connection with different physical problems, see for example [3,7] and books [10,8]. In particular, this potential enters the stationary radial Schrödinger equation Figure 1: Potential V (x) = g 1 x −1 + g 2 x −2 , with g 1 = g 2 = 1 (dashed), g 1 = −g 2 = 1 (solid) and g 1 = −g 2 = −1 (thick).…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint extensions and spectral analysis in the generalized Kratzer problem

Baldiotti¹,

Гитман²,

Tyutin³

et al. 2011

Phys. Scr.

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We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic motion of a particle in the potential fieldFor g2 > 0 and g1 < 0, the potential is known as the Kratzer potential VK (x) and is usually used to describe molecular energy and structure, interactions between different molecules, and interactions between nonbonded atoms.We construct all self-adjoint Schrödinger operators with the potential V (x) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectral problems, we follow the Krein's method of guiding functionals. This work is a continuation of our previous works devoted to Coulomb, Calogero, and Aharonov-Bohm potentials.

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“…A discussion of the underlying singular periodic problem (1.5) on R, including the associated Floquet (Bloch) theory, was presented by Scarf [58]. These investigations focus on aspects of ordinary differential equations as opposed to operator theory even though Dirichlet problems associated with singular endpoints were formally discussed (in this context see also [57]). An operator theoretic approach for (1.5) and (1.6) over a finite interval bounded by singularities and a variety of associated self-adjoint boundary conditions including coupled boundary conditions leading to energy bands (Floquet-Bloch theory) in the periodic problem on R, was discussed in in [22] and [26].…”

Section: Introductionmentioning

confidence: 99%