scholarly journals Exact WKB methods in SU(2) Nf = 1

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Alba Grassi ◽  
Qianyu Hao ◽  
Andrew Neitzke
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fredholm Determinant ◽  
Quantization Condition ◽  
Direct Computation ◽  
Borel Summation ◽  
Local Solutions ◽  
Bps States ◽  
The Relationship ◽  
Good Agreement

Abstract We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) $$ \mathcal{N} $$ N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

ARBITRARY l-WAVE SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR THE SCREEN COULOMB POTENTIAL

2013 ◽  
Vol 22 (06) ◽  
pp. 1350036 ◽  
Author(s):  
SHISHAN DONG ◽  
GUO-HUA SUN ◽  
SHI-HAI DONG
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
State Energy ◽  
Energy Levels ◽  
Bound State ◽  
Screen Coulomb Potential ◽  
Normalization Constant ◽  
Angular Momentum State ◽  
Good Agreement

Using improved approximate schemes for centrifugal term and the singular factor 1/r appearing in potential itself, we solve the Schrödinger equation with the screen Coulomb potential for arbitrary angular momentum state l. The bound state energy levels are obtained. A closed form of normalization constant of the wave functions is also found. The numerical results show that our results are in good agreement with those obtained by other methods. The key issue is how to treat two singular points in this quantum system.

On the blowup of solutions of a Schrödinger equation with an inhomogeneous damping coefficient

2014 ◽  
Vol 16 (03) ◽  
pp. 1350036 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Damping Coefficient ◽  
Blowup Of Solutions ◽  
Blowing Up ◽  
The Cauchy Problem ◽  
Local Solutions ◽  
Blowup Result ◽  
Suitable Initial Data ◽  
Blowing Up Solutions

We consider the Cauchy problem for the nonlinear self-focusing Schrödinger equation in ℝNwith an inhomogeneous smooth damping coefficient and we prove, for suitable initial data, and in the spirit of the seminal work of R. Glassey, a blowup result for the corresponding local solutions. We also give some lower bound estimates for the blowing-up solutions.

SUPER-RECURRENCE PHENOMENON RELEVANT TO THE PHASE IN THE NONLINEAR SCHRÖDINGER EQUATION

1995 ◽  
Vol 09 (17) ◽  
pp. 1045-1052 ◽  
Author(s):  
Y. YANG ◽  
Y. TAN ◽  
W.Y. ZHANG ◽  
C.Y. ZHENG
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Periodic Boundary Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Initial Condition ◽  
Nonlinear Schrödinger ◽  
Recurrence Period ◽  
Spatially Periodic ◽  
Good Agreement

The nonlinear Schrödinger equation with spatially periodic boundary conditions is numerically solved by means of the spectrum method. It is found that with the initial condition carefully chosen, the phase recurrence just appears when the amplitudes have the nineteenth recurrences to the initial condition. This phenomenon is called as the phase super-recurrence. Using a simple perturbation model, the amplitude recurrence period Ta and the phase change ∆φa in the period Ta are estimated, and a good agreement between this estimation and the numerical results of the nonlinear Schrödinger equation is shown.

The Schrödinger Equation

2020 ◽  
pp. 265-288
Author(s):  
M. Suhail Zubairy
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Quantum Tunneling ◽  
Schrodinger Equation ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Particle Dynamics ◽  
Quantization Condition ◽  
De Broglie Waves ◽  
Macroscopic Objects

In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.

scholarly journals Solutions of the Schrödinger equation for pseudo-Coulomb potential plus a new improved ring-shaped potential in the cosmic string space–time

2016 ◽  
Vol 94 (5) ◽  
pp. 517-521 ◽  
Author(s):  
Akpan N. Ikot ◽  
Tamunoimi M. Abbey ◽  
Ephraim O. Chukwuocha ◽  
Michael C. Onyeaju
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Cosmic String ◽  
Schrodinger Equation ◽  
State Energy ◽  
Bound State ◽  
Space Time ◽  
Minkowski Space Time ◽  
Uvarov Method ◽  
Good Agreement

In this paper, we obtain the bound state energy eigenvalues and the corresponding eigenfunctions of the Schrödinger equation for the pseudo-Coulomb potential plus a new improved ring-shaped potential within the framework of cosmic string space–time using the generalized parametric Nikiforov–Uvarov method. Our results are in good agreement with other works in the cosmic string space–time and reduced to those in the Minkowski space–time when α = 1.

The second Exton potential for the Schrödinger equation

2019 ◽  
Vol 34 (24) ◽  
pp. 1950195
Author(s):  
Artur M. Ishkhanyan ◽  
Jacek Karwowski
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Fractional Power ◽  
Fundamental Solutions ◽  
Transcendental Equation ◽  
Quantization Condition ◽  
Hermite Functions ◽  
Energy Quantization ◽  
Half Axis

Analytical solutions of the Schrödinger equation with a singular, fractional-power potential, referred to as the second Exton potential, are derived and analyzed. The potential is defined on the positive half-axis and supports an infinite number of bound states. It is conditionally integrable and belongs to a biconfluent Heun family. The fundamental solutions are expressed as irreducible linear combinations of two Hermite functions of a scaled and shifted argument. The energy quantization condition results from the boundary condition imposed at the origin. For the exact eigenvalues, which are solutions of a transcendental equation involving two Hermite functions, highly accurate approximation by simple closed-form expressions is derived. The potential is a good candidate for the description of quark–antiquark interaction.

scholarly journals Spatiotemporal Complexity Mediated by Higher-Order Peregrine-Like Extreme Events

2021 ◽  
Vol 9 ◽  
Author(s):  
Saliya Coulibaly ◽  
Camus G. L. Tiofack ◽  
Marcel G. Clerc
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Analytical Description ◽  
Rogue Waves ◽  
Dissipative Systems ◽  
Largest Lyapunov Exponent ◽  
Dissipative Effects ◽  
Non Linear ◽  
The Stability ◽  
Good Agreement

The Peregrine soliton is the famous coherent solution of the non-linear Schrödinger equation, which presents many of the characteristics of rogue waves. Usually studied in conservative systems, when dissipative effects of injection and loss of energy are included, these intrigued waves can disappear. If they are preserved, their role in the dynamics is unknown. Here, we consider this solution in the framework of dissipative systems. Using the paradigmatic model of the driven and damped non-linear Schrödinger equation, the profile of a stationary Peregrine-type solution has been found. Hence, the Peregrine soliton waves are persistent in systems outside of the equilibrium. In the weak dissipative limit, analytical description has a good agreement with the numerical simulations. The stability has been studied numerically. The large bursts that emerge from the instability are analyzed by means of the local largest Lyapunov exponent. The observed spatiotemporal complexity is ruled by the unstable second-order Peregrine-type soliton.

scholarly journals The Schrödinger Equation with Deng-Fan-Eckart Potential (DFEP): Nikiforov-Uvarov-Functional Analysis (NUFA) Method

2021 ◽  
Vol 3 (5) ◽  
pp. 58-62
Author(s):  
E. B. Ettah
Keyword(s):  
Functional Analysis ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Spectra ◽  
Analysis Method ◽  
Radial Part ◽  
Molecular Physics ◽  
Centrifugal Barrier ◽  
Eckart Potential ◽  
Good Agreement

In this study, the radial part of the Schrödinger equation with the Deng-Fan-Eckart potential (DFEP) is solved analytically by employing the improved Greene and Aldrich approximation to bypass the centrifugal barrier and using the Nikiforov-Uvarov-Functional Analysis method (NUFA). The energy expression and wave function are obtained respectively. The numerical energy spectra for some diatomic molecules have been studied and compared with the findings of earlier studies and it has been found to be in good agreement. The NUFA method used in this study is easy and very less cumbersome compared to other methods that currently exist and it is recommended that researchers in this area adopt this method. The findings of this study will find direct applications in molecular physics.

scholarly journals Charmonium Properties Using the Discrete Variable Representation (DVR) Method

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Bhaghyesh A.
Keyword(s):  
Numerical Methods ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hamiltonian Matrix ◽  
Discrete Variable ◽  
Discrete Variable Representation ◽  
Eigenvalues And Eigenfunctions ◽  
Decay Widths ◽  
Variable Representation ◽  
Good Agreement

The Schrödinger equation is solved numerically for charmonium using the discrete variable representation (DVR) method. The Hamiltonian matrix is constructed and diagonalized to obtain the eigenvalues and eigenfunctions. Using these eigenvalues and eigenfunctions, spectra and various decay widths are calculated. The obtained results are in good agreement with other numerical methods and with experiments.

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