Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space
In this paper, we present a “non-linear” factorization of a family of non-normal operators arising from Gribov’s theory of the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel=&qu...
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2024-12-01
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| author | Abdelkader Intissar |
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| description | In this paper, we present a “non-linear” factorization of a family of non-normal operators arising from Gribov’s theory of the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><msup><mi>λ</mi><mo>′</mo></msup><msup><mi>A</mi><msup><mo>*</mo><mn>2</mn></msup></msup><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>μ</mi><msup><mi>A</mi><mo>*</mo></msup><mi>A</mi><mo>+</mo><mi>i</mi><mi>λ</mi><msup><mi>A</mi><mo>*</mo></msup></mrow></mstyle></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>+</mo><msup><mi>A</mi><mo>*</mo></msup><mo>)</mo><mi>A</mi><mo>,</mo></mrow></semantics></math></inline-formula> where the quartic Pomeron coupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>λ</mi><mo>′</mo></msup></semantics></math></inline-formula>, the Pomeron intercept <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and the triple Pomeron coupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> are real parameters, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>A</mi><mo>*</mo></msup></semantics></math></inline-formula> and <i>A</i> are, respectively, the usual creation and annihilation operators of the one-dimensional harmonic oscillator obeying the canonical commutation relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>A</mi><mo>,</mo><msup><mi>A</mi><mo>*</mo></msup><mo>]</mo><mo>=</mo><mi>I</mi></mrow></semantics></math></inline-formula>. In Bargmann representation, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>A</mi><mo>⟷</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac></mrow></mstyle></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>A</mi><mo>*</mo></msup><mo>⟷</mo><mi>z</mi></mrow></mstyle></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula>. It follows that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub></mstyle></semantics></math></inline-formula> can be written in the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac><mo>,</mo></mrow></mstyle></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>λ</mi><mo>′</mo></msup><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>λ</mi><mi>z</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>i</mi><mi>λ</mi><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>μ</mi><mi>z</mi></mrow><mo>.</mo></mrow></semantics></math></inline-formula> This operator is an operator of the Heun type where the Heun operator is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac></mstyle><mo>+</mo><mi>v</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a cubic complex polynomial, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> are polynomials of degree at most 2 and 1, respectively, which are given. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub></semantics></math></inline-formula> takes the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac></mstyle><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi><mo>(</mo><mi>λ</mi><mo>−</mo><msup><mi>λ</mi><mo>′</mo></msup><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mi>b</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>y</mi><mo>(</mo><mi>λ</mi><mi>y</mi><mo>+</mo><mi>μ</mi><mo>)</mo></mrow><mo>.</mo></mrow></mstyle></semantics></math></inline-formula> We introduce the change of variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>λ</mi><mrow><mn>2</mn><msup><mi>λ</mi><mo>′</mo></msup></mrow></mfrac></mstyle><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>c</mi><mi>o</mi><mi>s</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></mrow></semantics></math></inline-formula> to obtain the main result of transforming <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub></mstyle></semantics></math></inline-formula> into a product of two first-order operators: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">˜</mo></mover><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><msup><mi>λ</mi><mo>′</mo></msup><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>θ</mi></mrow></mfrac><mo>+</mo><mi>α</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>θ</mi></mrow></mfrac><mo>+</mo><mi>α</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></mstyle></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> being explicitly determined. |
| format | Article |
| id | doaj-art-f048390dea304249839b39045d1a5e84 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-f048390dea304249839b39045d1a5e842025-01-10T13:18:01ZengMDPI AGMathematics2227-73902024-12-011313110.3390/math13010031Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann SpaceAbdelkader Intissar0Le Prador, 129, Rue Commandant Rolland, 13008 Marseille, FranceIn this paper, we present a “non-linear” factorization of a family of non-normal operators arising from Gribov’s theory of the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><msup><mi>λ</mi><mo>′</mo></msup><msup><mi>A</mi><msup><mo>*</mo><mn>2</mn></msup></msup><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>μ</mi><msup><mi>A</mi><mo>*</mo></msup><mi>A</mi><mo>+</mo><mi>i</mi><mi>λ</mi><msup><mi>A</mi><mo>*</mo></msup></mrow></mstyle></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>+</mo><msup><mi>A</mi><mo>*</mo></msup><mo>)</mo><mi>A</mi><mo>,</mo></mrow></semantics></math></inline-formula> where the quartic Pomeron coupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>λ</mi><mo>′</mo></msup></semantics></math></inline-formula>, the Pomeron intercept <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and the triple Pomeron coupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> are real parameters, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>A</mi><mo>*</mo></msup></semantics></math></inline-formula> and <i>A</i> are, respectively, the usual creation and annihilation operators of the one-dimensional harmonic oscillator obeying the canonical commutation relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>A</mi><mo>,</mo><msup><mi>A</mi><mo>*</mo></msup><mo>]</mo><mo>=</mo><mi>I</mi></mrow></semantics></math></inline-formula>. In Bargmann representation, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>A</mi><mo>⟷</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac></mrow></mstyle></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>A</mi><mo>*</mo></msup><mo>⟷</mo><mi>z</mi></mrow></mstyle></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula>. It follows that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub></mstyle></semantics></math></inline-formula> can be written in the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac><mo>,</mo></mrow></mstyle></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>λ</mi><mo>′</mo></msup><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>λ</mi><mi>z</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>i</mi><mi>λ</mi><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>μ</mi><mi>z</mi></mrow><mo>.</mo></mrow></semantics></math></inline-formula> This operator is an operator of the Heun type where the Heun operator is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac></mstyle><mo>+</mo><mi>v</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a cubic complex polynomial, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> are polynomials of degree at most 2 and 1, respectively, which are given. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub></semantics></math></inline-formula> takes the following form: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>z</mi></mrow></mfrac></mstyle><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi><mo>(</mo><mi>λ</mi><mo>−</mo><msup><mi>λ</mi><mo>′</mo></msup><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mi>b</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>y</mi><mo>(</mo><mi>λ</mi><mi>y</mi><mo>+</mo><mi>μ</mi><mo>)</mo></mrow><mo>.</mo></mrow></mstyle></semantics></math></inline-formula> We introduce the change of variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>λ</mi><mrow><mn>2</mn><msup><mi>λ</mi><mo>′</mo></msup></mrow></mfrac></mstyle><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>c</mi><mi>o</mi><mi>s</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></mrow></semantics></math></inline-formula> to obtain the main result of transforming <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><msub><mi>H</mi><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub></mstyle></semantics></math></inline-formula> into a product of two first-order operators: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">˜</mo></mover><mrow><msup><mi>λ</mi><mo>′</mo></msup><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi></mrow></msub><mo>=</mo><msup><mi>λ</mi><mo>′</mo></msup><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>θ</mi></mrow></mfrac><mo>+</mo><mi>α</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>θ</mi></mrow></mfrac><mo>+</mo><mi>α</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></mstyle></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> being explicitly determined.https://www.mdpi.com/2227-7390/13/1/31non-normal operatorsfactorization methodharmonic oscillatorHamiltonian of Reggeon theoryBargmann space |
| spellingShingle | Abdelkader Intissar Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space Mathematics non-normal operators factorization method harmonic oscillator Hamiltonian of Reggeon theory Bargmann space |
| title | Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space |
| title_full | Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space |
| title_fullStr | Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space |
| title_full_unstemmed | Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space |
| title_short | Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space |
| title_sort | factorization of the non normal hamiltonian of reggeon field theory in bargmann space |
| topic | non-normal operators factorization method harmonic oscillator Hamiltonian of Reggeon theory Bargmann space |
| url | https://www.mdpi.com/2227-7390/13/1/31 |
| work_keys_str_mv | AT abdelkaderintissar factorizationofthenonnormalhamiltonianofreggeonfieldtheoryinbargmannspace |