Exact Solutions of the Time-Independent Axially Symmetric Schrödinger Equation

Kudryavtsev, A. G.

doi:10.1134/s0021364020020095

Cited by 10 publications

(12 citation statements)

References 6 publications

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“…This transformation corresponds to the generalisation of Moutard transformation. The generalized Moutard transformation was applied for the Schrödinger equation in cylindrical coordinates in the paper [15]. Here we write out formulae of generalized Moutard transformation for the sake of completeness and subsequent use.…”

Section: The Nonlocal Darboux Transformationmentioning

confidence: 99%

“…Now the potential (29) can be taken as the initial potential for new transformations. In the case of the two -dimensional Schrödinger equation it was shown that the twofold application of the Moutard transformation can be effective for obtaining nonsingular potentials in cartesian coordinates [17] and in cylindrical coordinates [15]. To avoid cumbersome formulas let us consider the simple case C 1 = 1.…”

Section: Application Of the Nonlocal Darboux Transformationmentioning

confidence: 99%

“…The main idear of this papers was inspired by the approach to nonlocal symmetries in the symmetry group analysis of differential equations [12], [13] (for the variant based on the theory of coverings, see [14] and references therein) . In the paper [15] the generalized Moutard transformation was considered and applied for the Schrödinger equation in cylindrical coordinates. In the present paper we consider the nonlocal Darboux transformation for the stationary Schrödinger equation in cylindrical coordinates (1) using the approach of papers [10], [11].…”

Section: Introductionmentioning

confidence: 99%

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On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations

Kudryavtsev

2021

Preprint

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The nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations is considered. Formulae for the nonlocal Darboux transformation are obtained and its relation to the generalized Moutard transformation is established. New examples of two -dimensional potencials and exact solutions for the stationary axially symmetric Schrödinger and Helmholtz equations are obtained as an application of the nonlocal Darboux transformation.

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Section: The Nonlocal Darboux Transformationmentioning

confidence: 99%

Section: Application Of the Nonlocal Darboux Transformationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations

Kudryavtsev

2021

Preprint

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“…that parametrically depend on the collective coordinate X via ξ±. These integrals can be computed analytically [12,20,21] and an example is detailed in the appendix of Ref. [22].…”

Section: Separation As Collective Coordinatementioning

confidence: 99%

Collective Coordinate Methods and Their Applicability to $$\phi ^4$$ Models

Weigel

2019

Nonlinear Systems and Complexity

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Collective coordinate methods are frequently applied to study dynamical properties of solitons. These methods simplify the field equations -typically partial differential equations -to ordinary differential equations for selected excitations. More importantly though, collective coordinates provide a practical means to focus on particular modes of otherwise complicated dynamical processes. We review the application of collective coordinate methods in the analysis of the kink-antikink interaction within the ϕ 4 soliton model and illuminate discrepancies between these methods and the exact results from the field equations. Kink-antikink scatteringKink-antikink scattering can be simulated as a solution to the field equations with suitable initial conditions. Here we discuss this treatment and its results briefly.1 Here we restrain to the classical description, i.e. canonical commutation relations are not imposed. Then this factor has no dynamical relevance.

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“…Historically the K − d → π − Λp reaction has been studied to investigate the Λp scattering and also the ΣN interaction with kaons at rest [30,31] and in-flight [32][33][34][35]. There are many theoretical calculations of this reaction with kaons at rest so far [28,[36][37][38][39][40][41][42][43][44][45][46]. Some of these works have mainly concerned a possible bound state below the ΣN threshold by considering Σ-Λ conversion in the intermediate states.…”

Section: Introductionmentioning

confidence: 99%

$K^- d\rightarrow πΛN$ reaction for studying charge symmetry breaking in the $ΛN$ interaction

Yutaro¹,

Ishikawa²

2022

Preprint

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We discuss charge symmetry breaking in the ΛN interaction. In order to investigate the Λp and Λn interactions at low energies, we propose to utilize the K − d → π − Λp and K − d → π 0 Λn reactions. They are symmetric under the exchange of both the pion and nucleon isospin partners in the final states. This advantage allows us to study charge symmetry breaking in the ΛN interaction. We calculate the differential cross sections of these reactions with stopped kaons theoretically and discuss the possibility to extract the scattering length and effective range of the ΛN scattering. With stopped kaons, the ΛN interaction takes place dominantly in the spin-triplet state thanks to the deuteron spin and s-wave dominance of the scattering amplitudes at the low energy. We find that the ratio of the differential cross sections as a function of the ΛN invariant mass between the two reactions is useful for revealing how charge symmetry breaking, or isospin symmetry breaking appears in the low-energy ΛN scattering.

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Exact Solutions of the Time-Independent Axially Symmetric Schrödinger Equation

Cited by 10 publications

References 6 publications

On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations

On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations

Collective Coordinate Methods and Their Applicability to $$\phi ^4$$ Models

$K^- d\rightarrow πΛN$ reaction for studying charge symmetry breaking in the $ΛN$ interaction

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