Bound states of the potential<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:math>

Mehta, C. H.; Patil, S. H.

doi:10.1103/physreva.17.43

Cited by 39 publications

(24 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same assertion is also valid for modified Coulomb interactions having the form M ε (x) = − 1 |x| + ε ; this potentials also diverge in D (R). Such regularizations were considered in [1,2,[4][5][6]. It should be noted that the equivalence of penetrability in the limit and the convergence in the space of distributions for potentials Q ε + ε −1 V (ε −1 ·) is consistent with Kurasov's results [12,14].…”

Section: 3supporting

confidence: 57%

1D Schrödinger operators with Coulomb-like potentials

Golovaty

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

We study the convergence of 1D Schrödinger operators Hε with the potentials which are regularizations of a class of pseudo-potentials having in particular the formThe limit behaviour of Hε in the norm resolvent topology, as ε → 0, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for δ -like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of L ∞ (R)-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.2000 Mathematics Subject Classification. Primary 34L40, 34B09; Secondary 81Q10. Key words and phrases. 1D Schrödinger operator, Coulomb potential, one-dimensional hydrogen atom, δ -potential, scattering problem, penetrability of potential, point interaction.

show abstract

Section: 3supporting

confidence: 57%

1D Schrödinger operators with Coulomb-like potentials

Golovaty

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…The bound states are obtained in terms of three potential parameters: the coupling Z > 0, the cut-off parameter β > 0, and the power parameter q ≥ 1. The cases q = 1 and 2 are of special physical significance [4,5,6,7,8,9,10,11,12]. The potential V 1 represents the potential due to a smeared charge and is useful in describing mesonic atoms.…”

Section: Introductionmentioning

confidence: 99%

Soft-core Coulomb potentials and Heun’s differential equation

Hall

Saad

Sen

2010

Journal of Mathematical Physics

View full text Add to dashboard Cite

Schrödinger's equation with the attractive potential V (r) = −Z/(r q + β q ) 1 q , Z > 0, β > 0, q ≥ 1, is shown, for general values of the parameters Z and β, to be reducible to the confluent Heun equation in the case q = 1, and to the generalized Heun equation in case q = 2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases this factor becomes a polynomial. The Asymptotic Iteration Method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detail solutions for both cases are provided.

show abstract

“…Let us consider two comparison potentials: the cut-off Coulomb potential V a [21,22], which has a known analytical solution [14], and the sech-squared potential V b [23][24][25][26] …”

Section: An Examplementioning

confidence: 99%

Sharp comparison theorems for the Klein–Gordon equation in d dimensions

Hall

Zorin

2016

Int. J. Mod. Phys. E

View full text Add to dashboard Cite

We establish sharp (or 'refined') comparison theorems for the Klein-Gordon equation. We show that the condition Va ≤ V b , which leads to Ea ≤ E b , can be replaced by the weaker assumption Ua ≤ U b which still implies the spectral ordering Ea ≤ E b . In the simplest case, for d = 1,Vi(t)dt, i = a or b, and for d > 1, Ui(r) = r 0 Vi(t)t d−1 dt, i = a or b. We also consider sharp comparison theorems in the presence of a scalar potential S (a 'variable mass') in addition to the vector term V (the time component of a 4-vector). The theorems are illustrated by a variety of explicit detailed examples.

show abstract

Bound states of the potentialV(r)=−Z(r+β)

Abstract: We have studied the bound states of the potential V(r) = -Z/(r + P). The S-wave energy level E(P), .as a function of the parameter p, has a logarithmic singularity at the origin, The imaginary part of E(p) for P~0 goes as Z + ' P ', where I is the angular momentum.

Cited by 39 publications

References 6 publications

1D Schrödinger operators with Coulomb-like potentials

1D Schrödinger operators with Coulomb-like potentials

Soft-core Coulomb potentials and Heun’s differential equation

Sharp comparison theorems for the Klein–Gordon equation in d dimensions

Contact Info

Product

Resources

About