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SUMMARY:Low-order momentum correlation functions of the symmetric Tamm-Dan
 coff $q$-bosons\, derived via under-trace operations
DTSTART;VALUE=DATE-TIME:20260922T144500Z
DTEND;VALUE=DATE-TIME:20260922T145000Z
DTSTAMP;VALUE=DATE-TIME:20260914T040531Z
UID:indico-contribution-556@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Yuriy Mishchenko (Bogolyubov Institute for Theoretic
 al Physics of NAS of Ukraine)\n$\\quad$Using the set of independent Symmet
 ric Tamm-Dancoff (STD) type $q$-deformed quantum oscillators [1] with the 
 standard linear-in-$N$ Hamiltonian\, we construct respective gas of $q$-bo
 sons. Such a deformed $q$-Bose gas model admits [2] exact analytic express
 ions for the $r$-particle momentum correlation function\, and correspondin
 g intercepts\, of any $r$th order. This is the third example of deformed o
 scillators' based Bose gas model\, along with the $qp$-Bose gas model [3] 
 and the $\\mu$-deformed one [4] (based on Jannussis's oscillator)\, where 
 correlation average $\\langle (a^\\dagger_{\\bf k})^r (a_{\\bf k})^r \\ran
 gle$ was calculated explicitly. In view of the formal resemblance between 
 the $qp$-bracket and the STD deformation structure function\, with $N$-mul
 tiplier ignored\, we apply the idea of the derivation from [3] for $qp$-bo
 sons\, to the current case of STD $q$-bosons.\n\n$\\quad$Indeed\, using ju
 st under-trace algebraic operations\, for the STD $q$-bosons in a fixed $\
 \bf k$-mode we find\n$\\qquad\\qquad \\langle a^\\dagger a\\rangle_\\mathr
 m{STD}\n=  \\frac1{(e^{\\beta\\omega} -\\\, q)^2}\\\,\n\\biggl\\{\\\n \\Bi
 gl\\langle\n  \\frac{\\bigl[a\, [a\,a^\\dagger]_q \\bigr]_q\\\, a^\\dagger
 }{\\varphi_\\mathrm{STD}(N+1)}\n \\Bigr\\rangle\n    + \\frac{ [1]^\\mathr
 m{STD}_q!\\\, e^{\\beta\\omega} }{\\mathrm{Tr}\\\, e^{-\\beta\\omega N}}\n
  \\biggr\\}$\nversus $\\langle a^\\dagger a\\rangle_{qp} =  \\frac{[1]_{qp
 }!}{e^{\\beta\\omega}\\\,-\\\,q}\n\\\, \\bigl\\langle[a\,a^\\dagger]_q\\bi
 gr\\rangle$ for $qp$-bosons\, see [3]. What concerns the quadratic correla
 tions\, using the same approach of under-trace transformations we derive\n
 $\\qquad \\bigl\\langle a^{\\dagger2}a^2 \\bigr\\rangle_\\mathrm{STD}\n =\
 n\\frac1{(e^{\\beta\\omega} -\\\, q^2)^3 (e^{\\beta\\omega} -\\\, 1)^3}\\\
 n\\biggl\\{\\\n\\Bigl\\langle \\ \n  \\sum\\limits_{i=0}^4 A_i(q)\\\; \\te
 xtrm{g}_q(N\\!+\\!1\\!+\\!i)\\\,\\textrm{g}_q(N\\!+\\!i)\n         \\cdot 
 q^{-2N} \\Bigr\\rangle$\n$\\qquad\\qquad\\qquad\\qquad\n-\\  (e^{\\beta\\o
 mega}-1)\\\, \\textrm{g}_q(0) P_2\\bigl(q\;e^{\\beta\\omega}\\bigr)\n\\big
 gr\\}\\\n  +\\\n\\varphi_\\mathrm{STD}(2)! \\frac{e^{\\beta\\omega}-\\\,1}
 {(e^{\\beta\\omega}\\\,-\\\,q^2)^3}$\nwhere underlying role belongs to the
  double commutator as well\, through $\\textrm{g}_q(N)$-function\, $[a\,[a
 \,a^\\dagger]_q]_q = - \\textrm{g}_q({N+1})\\\, q^{1-N} \\cdot a$\,  and $
 P_2(q\;e^{\\beta\\omega})$ is some quadratic polynomial in $e^{\\beta\\ome
 ga}$\, with coefficients depending on $q$.\n\n 1. W.S. Chung\, A.M. Gavril
 ik\, I.I. Kachurik\,\n    A.P. Rebesh: J. Phys. A: Math. Theor. **47**\, 3
 05304 (2014).\n 2. A.M. Gavrilik\, Yu.A. Mishchenko\, arXiv preprint\,\n  
   arXiv:1410.0538  (2014).\n 3. L.V. Adamska\, A.M. Gavrilik\, J. Phys. A:
  Math. Gen.\n    **37**\, 4787-4796 (2004).\n 4. A.M. Gavrilik\, Yu.A. Mis
 hchenko\, Phys. Lett. A\n    **376**\, 2484-2489 (2012).\n\nhttps://indico
 .bitp.kiev.ua/event/18/contributions/556/
LOCATION:Bogolyubov Institute for Theoretical Physics Conference Hall
URL:https://indico.bitp.kiev.ua/event/18/contributions/556/
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