<inline-formula> <mml:math id="mm10000" display="block"> <mml:semantics> <mml:mi mathvariant="script">PT</mml:mi> </mml:semantics> </mml:math> </inline-formula>-Symmetric Potentials from the Confluent Heun Equation
We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantic...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-01-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/1/68 |
| _version_ | 1827698516270514176 |
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| author | Géza Lévai |
| author_facet | Géza Lévai |
| author_sort | Géza Lévai |
| collection | DOAJ |
| description | We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula> symmetry. We point out that for the implementation of <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula> symmetry, the symmetrical canonical form of the Heun equation is more suitable than its non-symmetrical canonical form. The potentials identified in this construction depend on twelve parameters, of which three contribute to scaling and shifting the energy and the coordinate. Five parameters control the <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> function that detemines the variable transformation taking the Heun equation into the one-dimensional Schrödinger equation, while four parameters play the role of the coupling coefficients of four independently tunable potential terms. The potentials obtained this way contain Natanzon-class potentials as special cases. Comparison with the results of an earlier study based on potentials obtained from the non-symmetrical canonical form of the confluent Heun equation is also presented. While the explicit general solutions of the confluent Heun equation are not available, the results are instructive in identifying which potentials can be obtained from this equation and under which conditions they exhibit <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula> symmetry, either unbroken or broken. |
| first_indexed | 2024-03-10T13:31:24Z |
| format | Article |
| id | doaj.art-d201ed33cf5f49d9904ae7174797fd89 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T13:31:24Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-d201ed33cf5f49d9904ae7174797fd892023-11-21T07:58:45ZengMDPI AGEntropy1099-43002021-01-012316810.3390/e23010068<inline-formula> <mml:math id="mm10000" display="block"> <mml:semantics> <mml:mi mathvariant="script">PT</mml:mi> </mml:semantics> </mml:math> </inline-formula>-Symmetric Potentials from the Confluent Heun EquationGéza Lévai0Institute for Nuclear Research (Atomki), P. O. Box 51, H-4001 Debrecen, HungaryWe derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula> symmetry. We point out that for the implementation of <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula> symmetry, the symmetrical canonical form of the Heun equation is more suitable than its non-symmetrical canonical form. The potentials identified in this construction depend on twelve parameters, of which three contribute to scaling and shifting the energy and the coordinate. Five parameters control the <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> function that detemines the variable transformation taking the Heun equation into the one-dimensional Schrödinger equation, while four parameters play the role of the coupling coefficients of four independently tunable potential terms. The potentials obtained this way contain Natanzon-class potentials as special cases. Comparison with the results of an earlier study based on potentials obtained from the non-symmetrical canonical form of the confluent Heun equation is also presented. While the explicit general solutions of the confluent Heun equation are not available, the results are instructive in identifying which potentials can be obtained from this equation and under which conditions they exhibit <inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula> symmetry, either unbroken or broken.https://www.mdpi.com/1099-4300/23/1/68confluent Heun differential equationsolvable potentialsPT symmetry |
| spellingShingle | Géza Lévai <inline-formula> <mml:math id="mm10000" display="block"> <mml:semantics> <mml:mi mathvariant="script">PT</mml:mi> </mml:semantics> </mml:math> </inline-formula>-Symmetric Potentials from the Confluent Heun Equation Entropy confluent Heun differential equation solvable potentials PT symmetry |
| title | <inline-formula>
<mml:math id="mm10000" display="block">
<mml:semantics>
<mml:mi mathvariant="script">PT</mml:mi>
</mml:semantics>
</mml:math>
</inline-formula>-Symmetric Potentials from the Confluent Heun Equation |
| title_full | <inline-formula>
<mml:math id="mm10000" display="block">
<mml:semantics>
<mml:mi mathvariant="script">PT</mml:mi>
</mml:semantics>
</mml:math>
</inline-formula>-Symmetric Potentials from the Confluent Heun Equation |
| title_fullStr | <inline-formula>
<mml:math id="mm10000" display="block">
<mml:semantics>
<mml:mi mathvariant="script">PT</mml:mi>
</mml:semantics>
</mml:math>
</inline-formula>-Symmetric Potentials from the Confluent Heun Equation |
| title_full_unstemmed | <inline-formula>
<mml:math id="mm10000" display="block">
<mml:semantics>
<mml:mi mathvariant="script">PT</mml:mi>
</mml:semantics>
</mml:math>
</inline-formula>-Symmetric Potentials from the Confluent Heun Equation |
| title_short | <inline-formula>
<mml:math id="mm10000" display="block">
<mml:semantics>
<mml:mi mathvariant="script">PT</mml:mi>
</mml:semantics>
</mml:math>
</inline-formula>-Symmetric Potentials from the Confluent Heun Equation |
| title_sort | inline formula mml math id mm10000 display block mml semantics mml mi mathvariant script pt mml mi mml semantics mml math inline formula symmetric potentials from the confluent heun equation |
| topic | confluent Heun differential equation solvable potentials PT symmetry |
| url | https://www.mdpi.com/1099-4300/23/1/68 |
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