Transition from discrete to continuous spectrum: separable potential in one dimension

1990 ◽  
Vol 68 (4-5) ◽  
pp. 394-402 ◽  
Author(s):  
J. G. Muga ◽  
R. F. Snider
Keyword(s):  
Perturbation Theory ◽  
Phase Shift ◽  
Continuous Spectrum ◽  
Wave Functions ◽  
Phase Factor ◽  
One Dimension ◽  
Separable Potential ◽  
Level Shift ◽  
Momentum Representation ◽  
Convergence Parameter

An analysis is made of the transition from the discrete to the continuous spectrum for a separable potential in one dimension. The role played by the length of the box and the convergence parameter, ε, in the different limiting operations is discussed. Relations are found between scattering and perturbation theory matrices and wave functions in momentum representation. In particular, the known expression relating the level shift to the phase shift is recovered. The scattering and Brillouin–Wigner perturbation wave functions are in general not simply related by a phase factor.

Scattering by a separable potential in one dimension

1990 ◽  
Vol 68 (4-5) ◽  
pp. 403-410 ◽  
Author(s):  
J. G. Muga ◽  
R. F. Snider
Keyword(s):  
Phase Shift ◽  
Partial Wave Analysis ◽  
Three Dimensions ◽  
One Dimension ◽  
Separable Potential ◽  
Wave Analysis ◽  
S Matrix ◽  
Low Energies ◽  
Reflectance Coefficient ◽  
Excess Density

The scattering by a separable potential in one dimension is described by means of explicit expressions for the reflectance coefficient, phase shift, and excess density. Resonances, bound and virtual states are analyzed in the light of the position of the poles of the resolvent, and a physical consequence of the spurious poles of the S matrix is indicated. Deviations from the usual partial wave analysis in three dimensions, especially a modification of Levinson's theorem, are also studied. It is shown that at very low energies the slope of the phase shift is not always a valid indicator of the excess density in the interaction region.

scholarly journals Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Atoms ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 53
Author(s):  
Jack C. Straton
Keyword(s):  
Quantum Theory ◽  
Arbitrary Function ◽  
Plane Waves ◽  
Wave Functions ◽  
One Dimension ◽  
Two Dimensional ◽  
Transition Amplitudes ◽  
Multidimensional Integrals ◽  
Coulomb Potentials

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

scholarly journals Hermitizing the HAL QCD potential in the derivative expansion

2020 ◽  
Vol 2020 (2) ◽  
Author(s):  
Sinya Aoki ◽  
Takumi Iritani ◽  
Koichi Yazaki
Keyword(s):  
Phase Shift ◽  
Wave Functions ◽  
Leading Order ◽  
Many Body ◽  
Derivative Expansion ◽  
Local Part ◽  
Scattering Phase ◽  
Many Body Systems ◽  
Local Term ◽  
Better Than

Abstract A formalism is given to hermitize the HAL QCD potential, which needs to be non-Hermitian except for the leading-order (LO) local term in the derivative expansion as the Nambu– Bethe– Salpeter (NBS) wave functions for different energies are not orthogonal to each other. It is shown that the non-Hermitian potential can be hermitized order by order to all orders in the derivative expansion. In particular, the next-to-leading order (NLO) potential can be exactly hermitized without approximation. The formalism is then applied to a simple case of $\Xi \Xi (^{1}S_{0}) $ scattering, for which the HAL QCD calculation is available to the NLO. The NLO term gives relatively small corrections to the scattering phase shift and the LO analysis seems justified in this case. We also observe that the local part of the hermitized NLO potential works better than that of the non-Hermitian NLO potential. The Hermitian version of the HAL QCD potential is desirable for comparing it with phenomenological interactions and also for using it as a two-body interaction in many-body systems.

scholarly journals Entanglement perturbation theory for the elementary excitation in one dimension

2009 ◽  
Vol 373 (26) ◽  
pp. 2277-2280 ◽  
Author(s):  
Sung Gong Chung ◽  
Lihua Wang
Keyword(s):  
Perturbation Theory ◽  
Elementary Excitation ◽  
One Dimension

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

scholarly journals Universality of excited three-body bound states in one dimension

2021 ◽  
Author(s):  
Lucas Happ ◽  
Matthias Zimmermann ◽  
Maxim A Efremov
Keyword(s):  
Excited States ◽  
Bound States ◽  
Space Dimension ◽  
Binding Energies ◽  
Wave Functions ◽  
One Dimension ◽  
Strong Indication ◽  
Body System ◽  
Weakly Bound ◽  
Three Body

Abstract We study a heavy-heavy-light three-body system confined to one space dimension in the regime where an excited state in the heavy-light subsystems becomes weakly bound. The associated two-body system is characterized by (i) the structure of the weakly-bound excited heavy-light state and (ii) the presence of deeply-bound heavy-light states. The consequences of these aspects for the behavior of the three-body system are analyzed. We find a strong indication for universal behavior of both three-body binding energies and wave functions for different weakly-bound excited states in the heavy-light subsystems.

Generalized Møller—Plesset perturbation theory applied to general MCSCF reference wave functions

1991 ◽  
Vol 183 (5) ◽  
pp. 443-448 ◽  
Author(s):  
Robert B. Murphy ◽  
Richard P. Messmer
Keyword(s):  
Perturbation Theory ◽  
Wave Functions ◽  
Reference Wave ◽  
Moller Plesset ◽  
Plesset Perturbation Theory

scholarly journals Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

2021 ◽  
Vol 121 (2) ◽  
pp. 171-194
Author(s):  
Son N.T. Tu
Keyword(s):  
Rate Of Convergence ◽  
Classical Mechanics ◽  
One Dimension ◽  
Separable Potential ◽  
Periodic Homogenization ◽  
Periodic Functions ◽  
Effective Equation ◽  
Optimal Rate Of Convergence ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1.

Momentum Representation of the Coulomb Scattering Wave Functions

1951 ◽  
Vol 83 (3) ◽  
pp. 667-668 ◽  
Author(s):  
E. Guth ◽  
C. J. Mullin
Keyword(s):  
Wave Functions ◽  
Coulomb Scattering ◽  
Momentum Representation ◽  
Scattering Wave
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