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We have the following variant of harmonic oscillators.

$$ \left\{ \begin{array}{**lr**} T = a + a^\dagger\\ a | n \rangle = \sqrt{[n]} |n-1 \rangle \\ a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\rangle\\ [n] = \frac{1}{2}(\alpha\frac{1-q_1^n}{1-q_1} + \beta\frac{1-q_2^n}{1-q_2}) \end{array} \right. $$

where $0<q_1,q_2<1$.

My question is how to calculate the quantity $$ \langle n \mid T^k \mid m\rangle $$

For the special case $\alpha = \beta =1, q_1 = q_2 = q$, this system becomes a q-deformed harmonic oscillator, which can be solved analytically.

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  • $\begingroup$ Are you expecting a closed form for all $n,m,k$? For any given $n,m,k$, it's an elementary computation of course that you can teach a computer to do. Maybe that way you can see a pattern ... $\endgroup$
    – Michael Engelhardt
    Commented Jul 2, 2023 at 13:39
  • $\begingroup$ @MichaelEngelhardt I expect an analytical expression for $\langle n| T^k | m \rangle$. Just as we can get an analytical expression for $\langle n| x^k | m \rangle$ in a usual harmonic oscillator model. $\endgroup$
    – Lili Si
    Commented Jul 2, 2023 at 15:30

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