^{1}

^{2}

^{1}

^{2}

^{3}.

In this paper, we studied the approximate scattering state solutions of the Dirac equation with the hyperbolical potential with pseudospin and spin symmetries. By applying an improved Greene-Aldrich approximation scheme within the formalism of functional analytical method, we obtained the spin-orbit quantum numbers dependent scattering phase shifts for the spin and pseudospin symmetries. The normalization constants, lower and upper radial spinor for the two symmetries, and the relativistic energy spectra were presented. Our results reveal that both the symmetry constants (

Scattering theory is very central to the study of several fields such as atomic, nuclear, high energy or condensed matter physics. It allows for descriptions and interpretations of many collisions processes such as excitation and ionization by particle or radiation impact [

As a result, several authors in quantum mechanics have strictly followed different approaches to study the scattering state solutions of the relativistic and nonrelativistic wave equations for central and noncentral potential models [

However, literature revealed that the investigations on the spin and pseudosymmetries [

This is owing to the fact that the symmetries in Hadron and nuclear spectroscopy [

In view of the above works, we are motivated to investigate the scattering state solutions of Dirac equation with hyperbolical potential suggested by Schiöberg in 1986 and apply an improved Greene-Aldrich approximation scheme within the formalism of functional analytical method. In the present study the new Schiöberg potential is ignored and we focus on the effects of the symmetry constants and the positive potential parameters on the relativistic energy and the scattering phase shifts of the hyperbolical potential (Schiöberg potential).

This paper is organized as follows: Section

By considering the Dirac wave equation and its corresponding spinors, the two-coupled first-order differential equations for the upper and lower components of the spinor may be obtained as [

By following the pseudospin symmetry conditions and considering the hyperbolical potential

Under the pseudospin symmetry condition (

In a similar way, we consider the spin symmetry conditions and take

To obtain the approximate solutions in the presence of the spin symmetry and pseudospin symmetry, we use the Greene-Aldrich approximation [

Defining a variable

In order to solve (

By considering the boundary condition that

To obtain the phase shifts

Here, the analytical properties of partial-wave

Using the previously defined transformation variable and approximation, (

Similarly, we also assume the following upper wave function for the spin symmetry

To avoid repetition, we follow the same procedures in previous subsection and write the upper component of spin symmetry radial wave functions for any arbitrary

Following the same steps in Section

Following the same fashion in Section

To study the nonrelativistic limit, we apply the following appropriate mapping to (

The pseudospin symmetry bound states energy spectra displayed in Table

Pseudospin symmetry bound state energies at the poles of ^{−1}) for the hyperbolical potential as a function of positive potential parameter

| | | | |
---|---|---|---|---|

1 | −1 | 0.10 | 2.279163696, 1.029316001 | 2.1233814120, −3.993501276 |

1 | −1 | 0.15 | 1.861759552, 1.031095657 | 1.3219168610, −3.993392561 |

1 | −1 | 0.20 | 1.595805327, 1.034022382 | 0.8412753547, −3.993236371 |

1 | −1 | 0.25 | 1.410036018, 1.038851771 | 0.5132318062, −3.993028490 |

1 | −2 | 0.10 | 2.499986930, 1.049190512 | 2.3968826230, −3.989115288 |

1 | −2 | 0.15 | 2.038697268, 1.051647743 | 1.4930060260, −3.988979900 |

1 | −2 | 0.20 | 1.734781325, 1.055647948 | 0.9587751244, −3.988786147 |

1 | −2 | 0.25 | 1.518635403, 1.062147486 | 0.5989218610, −3.988529604 |

1 | −3 | 0.10 | 2.699088639, 1.074153615 | 2.7421356920, −3.983691020 |

1 | −3 | 0.15 | 2.214785842, 1.077522035 | 1.7216780460, −3.983519794 |

1 | −3 | 0.20 | 1.879801953, 1.082984677 | 1.1206711830, −3.983275390 |

1 | −3 | 0.25 | 1.634694558, 1.091816104 | 0.7192390968, −3.982952907 |

1 | −4 | 0.10 | 2.860475911, 1.104471112 | 3.1199566410, −3.977227902 |

1 | −4 | 0.15 | 2.372200392, 1.109000893 | 1.9871570450, −3.977014236 |

1 | −4 | 0.20 | 2.015688603, 1.116352376 | 1.3149710440, −3.976709786 |

1 | −4 | 0.25 | 1.745900039, 1.128273463 | 0.8667745016, −3.976309013 |

2 | −1 | 0.10 | 2.615061529, 1.055125495 | 3.1191738270, −3.987973646 |

2 | −1 | 0.15 | 2.170952096, 1.059520099 | 2.1743591190, −3.987691971 |

2 | −1 | 0.20 | 1.854820744, 1.066898930 | 1.5763871650, −3.987286660 |

2 | −1 | 0.25 | 1.610697844, 1.079577280 | 1.1529573900, −3.986746084 |

2 | −2 | 0.10 | 2.751633202, 1.082483207 | 3.3080559170, −3.982048026 |

2 | −2 | 0.15 | 2.286444614, 1.087766383 | 2.2967933160, −3.981747832 |

2 | −2 | 0.20 | 1.948294269, 1.096614037 | 1.6626435590, −3.981316476 |

2 | −2 | 0.25 | 1.684969993, 1.111784132 | 1.2171612360, −3.980742235 |

2 | −3 | 0.10 | 2.880415615, 1.115474816 | 3.5565298640, −3.975069259 |

2 | −3 | 0.15 | 2.407308925, 1.122116505 | 2.4653804670, −3.974727589 |

2 | −3 | 0.20 | 2.050458310, 1.133236516 | 1.7841645900, −3.974237859 |

2 | −3 | 0.25 | 1.766983989, 1.152366772 | 1.3088621540, −3.973588060 |

2 | −4 | 0.10 | 2.985001626, 1.154437579 | 3.8391364550, −3.967043330 |

2 | −4 | 0.15 | 2.516623028, 1.162869448 | 2.6672425580, −3.966648795 |

2 | −4 | 0.20 | 2.146938664, 1.177051056 | 1.9336438780, −3.966084415 |

2 | −4 | 0.25 | 1.844677707, 1.201768936 | 1.4235514910, −3.965337543 |

Spin symmetry energies at the poles of ^{−1}) for the hyperbolical potential as a function of positive potential parameter

| | | | |
---|---|---|---|---|

0 | −2 | 0.10 | 2.852765813, −0.9971975973 | 4.894486374, 4.004871783 |

0 | −2 | 0.15 | 2.286257855, −0.9971671352 | 4.484669284, 4.005013210 |

0 | −2 | 0.20 | 1.976243190, −0.9971235606 | 4.291486196, 4.005228383 |

0 | −2 | 0.25 | 1.777926670, −0.9970658947 | 4.186281947, 4.005540044 |

0 | −3 | 0.10 | 3.349240971, −0.9941222745 | 5.374173995, 4.010207082 |

0 | −3 | 0.15 | 2.597878123, −0.9940696246 | 4.789639030, 4.010461236 |

0 | −3 | 0.20 | 2.190050696, −0.9939945667 | 4.496311909, 4.010844649 |

0 | −3 | 0.25 | 1.933251119, −0.9938956907 | 4.328836785, 4.011393011 |

0 | −4 | 0.10 | 3.899001828, −0.9899271098 | 5.855422017, 4.017491955 |

0 | −4 | 0.15 | 2.970664174, −0.9898454341 | 5.119514795, 4.017895505 |

0 | −4 | 0.20 | 2.458599114, −0.9897291933 | 4.729861208, 4.018501874 |

0 | −4 | 0.25 | 2.135123157, −0.9895764138 | 4.498048414, 4.019363991 |

0 | −5 | 0.10 | 4.445008402, −0.9846098450 | 6.309268779, 4.026738666 |

0 | −5 | 0.15 | 3.365034712, −0.9844924012 | 5.451065869, 4.027328768 |

0 | −5 | 0.20 | 2.754614425, −0.9843254210 | 4.975122387, 4.028213547 |

0 | −5 | 0.25 | 2.364503739, −0.9841062477 | 4.681992554, 4.029467529 |

1 | −2 | 0.10 | 4.307228207, −0.9929477534 | 5.855267990, 4.012335498 |

1 | −2 | 0.15 | 3.494799666, −0.9928150205 | 5.188967231, 4.012888014 |

1 | −2 | 0.20 | 3.001609225, −0.9926233942 | 4.804823066, 4.013746766 |

1 | −2 | 0.25 | 2.661245323, −0.9923666479 | 4.556984308, 4.015030469 |

1 | −3 | 0.10 | 4.595853112, −0.9881911614 | 6.178641705, 4.020607401 |

1 | −3 | 0.15 | 3.680442629, −0.9880244067 | 5.414291047, 4.021342479 |

1 | −3 | 0.20 | 3.132109286, −0.9877847131 | 4.970695195, 4.022473524 |

1 | −3 | 0.25 | 2.758390431, −0.9874654265 | 4.683014161, 4.024139623 |

1 | −4 | 0.10 | 4.953432668, −0.9823076582 | 6.534290370, 4.030858520 |

1 | −4 | 0.15 | 3.924112012, −0.9820955447 | 5.675159767, 4.031828444 |

1 | −4 | 0.20 | 3.308851693, −0.9817915310 | 5.167984043, 4.033311444 |

1 | −4 | 0.25 | 2.892600410, −0.9813881332 | 4.835489514, 4.035475765 |

1 | −5 | 0.10 | 5.338358623, −0.9752962991 | 6.885521151, 4.043099865 |

1 | −5 | 0.15 | 4.202075842, −0.9750304881 | 5.945991145, 4.044348875 |

1 | −5 | 0.20 | 3.517208069, −0.9746502453 | 5.378418925, 4.046250921 |

1 | −5 | 0.25 | 3.054278778, −0.9741470070 | 5.001075188, 4.049010392 |

In Tables

Nonrelativistic energies at the poles of ^{−1}) for the hyperbolical potential as a function of positive potential parameter

| | | States | | | | |
---|---|---|---|---|---|---|---|

0 | 1 | 0.10 | 2p | 2.61890 | 3.90580 | 5.00395 | 5.88694 |

0.15 | 1.68043 | 2.57796 | 3.43332 | 4.21023 | |||

0.20 | 1.20892 | 1.86672 | 2.52064 | 3.14766 | |||

| |||||||

1 | 1 | 0.10 | 3p | 4.73556 | 6.04579 | 6.91727 | 7.48500 |

0.15 | 3.46030 | 4.62316 | 5.50084 | 6.15070 | |||

0.20 | 2.68324 | 3.67163 | 4.46580 | 5.09331 | |||

| |||||||

0 | 2 | 0.10 | 3d | 3.62747 | 5.29513 | 6.47684 | 7.25824 |

0.15 | 2.27024 | 3.56732 | 4.69715 | 5.59908 | |||

0.20 | 1.57921 | 2.54881 | 3.48311 | 4.31406 | |||

| |||||||

2 | 1 | 0.10 | 4p | 6.00303 | 7.11562 | 7.71968 | 8.02132 |

0.15 | 4.66775 | 5.80666 | 6.52479 | 6.95757 | |||

0.20 | 3.75708 | 4.81251 | 5.53175 | 6.00386 | |||

| |||||||

1 | 2 | 0.10 | 4d | 5.33170 | 6.73691 | 7.54672 | 7.97921 |

0.15 | 3.85825 | 5.19508 | 6.13602 | 6.75665 | |||

0.20 | 2.95305 | 4.10518 | 5.00371 | 5.66610 | |||

| |||||||

0 | 3 | 0.10 | 4f | 4.69061 | 6.43208 | 7.43782 | 7.98144 |

0.15 | 3.00365 | 4.60199 | 5.79916 | 6.61027 | |||

0.20 | 2.07438 | 3.35838 | 4.47793 | 5.35424 |

The pseudospin symmetry and spin symmetry phase shifts are displayed in Tables

Pseudospin scattering phase shifts for hyperbolical potential with positive potential parameter

| | | |
---|---|---|---|

0 | −1, 1 | 1.521873210037270, 1.570796326794897 | −8.076217263995554, −6.037944299866905 |

−2, 2 | 0.446819990038913, 1.521873210037270 | −9.291029083349983, −8.076217263995554 | |

−3, 3 | −1.097154578790937, 0.446819990038913 | −9.904139528549850, −9.291029083349983 | |

−4, 4 | −2.972292099735542, −1.097154578790937 | −10.013382196552033, −9.904139528549850 | |

−5, 5 | −5.103985153357378, −2.972292099735542 | −9.621579256682395, −10.013382196552033 | |

| |||

1 | −1, 1 | 3.092669536832167, 3.141592653589793 | −6.505420937200658, −4.467147973072009 |

−2, 2 | 2.017616316833810, 3.092669536832167 | −7.720232756555086, −6.505420937200658 | |

−3, 3 | 0.473641748003960, 2.017616316833810 | −8.333343201754955, −7.720232756555086 | |

−4, 4 | −1.401495772940645, 0.473641748003960 | −8.442585869757139, −8.333343201754955 | |

−5, 5 | −3.533188826562482, −1.401495772940645 | −8.050782929887498, −8.442585869757139 | |

| |||

2 | −1, 1 | 4.663465863627064, 4.712388980384690 | −4.934624610405762, −2.896351646277112 |

−2, 2 | 3.588412643628707, 4.663465863627064 | −6.149436429760190, −4.934624610405762 | |

−3, 3 | 2.044438074798856, 3.588412643628707 | −6.762546874960058, −6.149436429760190 | |

−4, 4 | 0.169300553854252, 2.044438074798856 | −6.871789542962241, −6.762546874960058 | |

−5, 5 | −1.962392499767585, 0.169300553854252 | −6.479986603092602, −6.871789542962241 | |

| |||

3 | −1, 1 | 6.234262190421960, 6.283185307179586 | −3.363828283610865, −1.325555319482215 |

−2, 2 | 5.159208970423603, 6.234262190421960 | −4.578640102965293, −3.363828283610865 | |

−3, 3 | 3.615234401593753, 5.159208970423603 | −5.191750548165161, −4.578640102965293 | |

−4, 4 | 1.740096880649148, 3.615234401593753 | −5.300993216167345, −5.191750548165161 | |

−5, 5 | −0.391596172972688, 1.740096880649148 | −4.909190276297705, −5.300993216167345 |

Spin scattering phase shifts for hyperbolical potential with positive potential parameter

| | | |
---|---|---|---|

0 | −1, 1 | −15.357449458632775, −15.177770279380065 | −34.356717558868027, −34.211165018163769 |

−2, 2 | −15.177770279380065, −14.803384517679635 | −34.211165018163769, −33.916113793943268 | |

−3, 3 | −14.803384517679635, −14.204750986373117 | −33.916113793943268, −33.463715091492688 | |

−4, 4 | −14.204750986373117, −13.337466036355023 | −33.463715091492688, −32.842249720235664 | |

−5, 5 | −13.337466036355023, −12.138509328495319 | −32.842249720235664, −32.036048006960613 | |

| |||

1 | −1, 1 | −13.786653131837877, −13.606973952585166 | −32.785921232073129, −32.640368691368877 |

−2, 2 | −13.606973952585166, −13.232588190884737 | −32.640368691368877, −32.345317467148369 | |

−3, 3 | −13.232588190884737, −12.633954659578222 | −32.345317467148369, −31.892918764697786 | |

−4, 4 | −12.633954659578222, −11.766669709560125 | −31.892918764697786, −31.271453393440765 | |

−5, 5 | −11.766669709560125, −10.567713001700424 | −31.271453393440765, −30.465251680165718 | |

| |||

2 | −1, 1 | −12.215856805042982, −12.036177625790272 | −31.215124905278238, −31.069572364573975 |

−2, 2 | −12.036177625790272, −11.661791864089839 | −31.069572364573975, −30.774521140353471 | |

−3, 3 | −11.661791864089839, −11.063158332783328 | −30.774521140353471, −30.322122437902888 | |

−4, 4 | −11.063158332783328, −10.195873382765226 | −30.322122437902888, −29.700657066645867 | |

−5, 5 | −10.195873382765226, −8.9969166749055290 | −29.700657066645867, −28.894455353370819 | |

| |||

3 | −1, 1 | −10.645060478248087, −10.465381298995377 | −29.644328578483339, −29.498776037779084 |

−2, 2 | −10.465381298995377, −10.090995537294944 | −29.498776037779084, −29.203724813558573 | |

−3, 3 | −10.090995537294944, −9.4923620059884290 | −29.203724813558573, −28.751326111107996 | |

−4, 4 | −9.4923620059884290, −8.6250770559703320 | −28.751326111107996, −28.129860739850976 | |

−5, 5 | −8.6250770559703320, −7.4261203481106310 | −28.129860739850976, −27.323659026575921 |

A plot of pseudospin scattering phase shifts for the hyperbolical potential as a function of spin-orbit number

A plot of pseudospin scattering phase shifts for the hyperbolical potential as a function of spin-orbit number

A plot of spin scattering phase shifts for the hyperbolical potential as a function of spin-orbit number

A plot of spin scattering phase shifts for the hyperbolical potential as a function of spin-orbit number

Figure

In conclusion, we have studied the approximate scattering state solution of Dirac equation with the hyperbolical potential using a short-range approximation within the framework of functional analytical method. We have obtained the spin and pseudospin symmetry bound state energies and their corresponding nonrelativistic energies, spin and pseudospin symmetry phase shifts, normalization constants, pseudospin symmetry lower component, and spin symmetry upper component of radial spinor wave functions for any arbitrary

We also studied the behaviour of phase shifts with spin-orbit quantum numbers under spin and pseudospin symmetries and we have successfully showed that relativistic scattering phase shifts largely depend on the symmetry constants (

K. J. Oyewumi is on Sabbatical Leave from Theoretical Physics Section, Department of Physics, University of Ilorin, Ilorin, Nigeria. O. J. Oluwadare declares that this paper partially represents the results of their Ph.D. research thesis supported by the Tertiary Education Trust Funds (TETFunds) through the Federal University Oye-Ekiti, Ekiti State, Nigeria.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank Professor C. S. Jia for supplying some of his papers.