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The quantum DELL system

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Abstract

We propose quantum Hamiltonians of the double-elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system which was previously found in the string theory literature. The eigenfunctions for the N-body model are instanton partition functions of 6d SU(N) gauge theory with adjoint matter compactified on a torus with a codimension-two defect. As a by-product, we discover new family of symmetric orthogonal polynomials which provide an elliptic generalization to Macdonald polynomials.

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Notes

  1. See section 4 of [44] for review and references.

  2. We thank Francesco Benini for discussions about this topic.

  3. N degrees of freedom with removed center of mass.

  4. We thank the authors of [4] for continuing discussions on this.

  5. Since theta-functions often appear in ratios in our calculations, we shall omit prefactor \(2x^{1/4}\) in the definition of \(\theta _1(x|p)\).

  6. Extra compact directions of the 6d theory will be implemented in the character formula.

  7. We refer the reader to section 4.2 of [12] for detailed explanations.

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Acknowledgements

This manuscript took some time to complete primarily due to the technical computational difficulties of expressions involving double-periodic functions. We would like to thank numerous people with whom we discussed these and other matters, as well as various institutions which we visited in the last several years, but our special acknowledgements go to Babak Haghigat, Wenbin Yan, Can Kozcaz and Francesco Benini who participated in earlier stages of the project.

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Authors and Affiliations

  1. Department of Mathematics, University of California Berkeley, Evans Hall 970, Berkeley, CA, 94720, USA

    Peter Koroteev

  2. Department of Mathematics, Uppsala University, Box 480 751 06, Uppsala, Sweden

    Shamil Shakirov

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  1. Peter Koroteev
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Correspondence to Peter Koroteev.

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Appendix: Spectrum of the three-body DELL model

Appendix: Spectrum of the three-body DELL model

Similarly to the U(2) example in (4.19,4.20) for U(3) 6d theory, the eigenfunction (4.12) for \(\mathbf j =(1,0,0)\) reads (we highlight elliptic parameters p and w in the expansion in bold)

$$\begin{aligned} {\mathscr {Z}}(\mathbf{x})= & {} x_{1}+x_{2}+x_{3}\\&+ \mathbf{{p}}\frac{q (-1+t)}{(q-1) (q t-1)^3 (q t+1)^2 t}\\&\cdot (q^4 t^6 x_{1}^3 x_{2}+q^4 t^6 x_{1}^3 x_{3}+2 q^4 t^6 x_{1}^2 x_{2}^2+3 q^4 t^6 x_{1}^2 x_{2} x_{3}\\&+2 q^4 t^6 x_{1}^2x_{3}^2+q^4 t^6 x_{1} x_{2}^3+3 q^4 t^6 x_{1} x_{2}^2 x_{3}\\&+3 q^4 t^6 x_{1} x_{2} x_{3}^2+q^4 t^6 x_{1} x_{3}^3+q^4 t^6 x_{2}^3x_{3}+2 q^4 t^6 x_{2}^2 x_{3}^2\\&+q^4 t^6 x_{2} x_{3}^3-2 q^4 t^5 x_{1}^2 x_{2} x_{3}\\&-2 q^4 t^5 x_{1} x_{2}^2 x_{3}-2 q^4 t^5 x_{1} x_{2} x_{3}^2+2 q^3 t^6 x_{1}^2 x_{2} x_{3}\\&+2 q^3 t^6 x_{1} x_{2}^2 x_{3}+2 q^3 t^6 x_{1} x_{2} x_{3}^2-q^4 t^4 x_{1}^2 x_{2}^2\\&-2 q^4 t^4 x_{1}^2 x_{2} x_{3}-q^4 t^4 x_{1}^2 x_{3}^2-2 q^4 t^4 x_{1} x_{2}^2 x_{3}\\&-2 q^4 t^4 x_{1} x_{2} x_{3}^2-q^4 t^4 x_{2}^2 x_{3}^2-q^3 t^5 x_{1}^3 x_{2}-q^3 t^5 x_{1}^3 x_{3}\\&-q^3 t^5 x_{1}^2 x_{2}^2-q^3 t^5 x_{1}^2 x_{2} x_{3}-q^3 t^5 x_{1}^2 x_{3}^2-q^3 t^5 x_{1} x_{2}^3-q^3 t^5 x_{1} x_{2}^2 x_{3}\\&-q^3 t^5 x_{1} x_{2} x_{3}^2-q^3 t^5 x_{1} x_{3}^3-q^3 t^5 x_{2}^3 x_{3}-q^3 t^5 x_{2}^2 x_{3}^2-q^3 t^5 x_{2} x_{3}^3-q^4 t^3 x_{1}^2 x_{2}^2\\&-q^4 t^3 x_{1}^2 x_{3}^2-q^4 t^3 x_{2}^2 x_{3}^2+2 q^3 t^4 x_{1}^2 x_{2}^2+2 q^3 t^4 x_{1}^2 x_{3}^2+2 q^3 t^4 x_{2}^2 x_{3}^2\\&-q^2 t^5 x_{1}^2 x_{2}^2-q^2 t^5 x_{1}^2 x_{3}^2-q^2 t^5 x_{2}^2 x_{3}^2+q^4 t^2 x_{1}^2 x_{2} x_{3}+q^4 t^2 x_{1} x_{2}^2 x_{3}\\&+q^4 t^2 x_{1} x_{2} x_{3}^2-q^3 t^3 x_{1}^3 x_{2}-q^3 t^3 x_{1}^3 x_{3}\\&-q^3 t^3 x_{1}^2 x_{2}^2-5 q^3 t^3 x_{1}^2 x_{2} x_{3}-q^3 t^3 x_{1}^2 x_{3}^2\\&-q^3 t^3 x_{1} x_{2}^3-5 q^3 t^3 x_{1} x_{2}^2 x_{3}-5 q^3 t^3 x_{1} x_{2} x_{3}^2\\&-q^3 t^3 x_{1} x_{3}^3-q^3 t^3 x_{2}^3 x_{3}-q^3 t^3 x_{2}^2 x_{3}^2\\&-q^3 t^3 x_{2} x_{3}^3-q^2 t^4 x_{1}^3 x_{2}-q^2 t^4 x_{1}^3 x_{3}-3 q^2 t^4 x_{1}^2 x_{2}^2\\&-3 q^2 t^4 x_{1}^2 x_{3}^2-q^2 t^4 x_{1} x_{2}^3-q^2 t^4 x_{1} x_{3}^3-q^2 t^4 x_{2}^3 x_{3}-3 q^2 t^4 x_{2}^2 x_{3}^2\\&-q^2 t^4 x_{2} x_{3}^3-2 q t^5 x_{1}^2 x_{2} x_{3}-2 q t^5 x_{1} x_{2}^2 x_{3}-2 q t^5 x_{1} x_{2} x_{3}^2\\&+2 q^3 t^2 x_{1}^2 x_{2} x_{3}+2 q^3 t^2 x_{1} x_{2}^2 x_{3}+2 q^3 t^2 x_{1} x_{2} x_{3}^2 \end{aligned}$$
$$\begin{aligned}&-2 q t^4 x_{1}^2 x_{2} x_{3}-2 q t^4 x_{1} x_{2}^2 x_{3}-2 q t^4 x_{1} x_{2} x_{3}^2+2 q^3 t x_{1}^2 x_{2} x_{3}+2 q^3 t x_{1} x_{2}^2 x_{3}\\&+2 q^3 t x_{1} x_{2} x_{3}^2+q^2 t^2 x_{1}^3 x_{2}+q^2 t^2 x_{1}^3 x_{3}+3 q^2 t^2 x_{1}^2 x_{2}^2\\&+3 q^2 t^2 x_{1}^2 x_{3}^2+q^2 t^2 x_{1} x_{2}^3+q^2 t^2 x_{1} x_{3}^3+q^2 t^2 x_{2}^3 x_{3}\\&+3 q^2 t^2 x_{2}^2 x_{3}^2+q^2 t^2 x_{2} x_{3}^3+q t^3 x_{1}^3 x_{2}+q t^3 x_{1}^3 x_{3}\\&+q t^3 x_{1}^2 x_{2}^2+5 q t^3 x_{1}^2 x_{2} x_{3}+q t^3 x_{1}^2 x_{3}^2+q t^3 x_{1} x_{2}^3\\&+5 q t^3 x_{1} x_{2}^2 x_{3}+5 q t^3 x_{1} x_{2} x_{3}^2+q t^3 x_{1} x_{3}^3+q t^3 x_{2}^3 x_{3}\\&+q t^3 x_{2}^2 x_{3}^2+q t^3 x_{2} x_{3}^3-t^4 x_{1}^2 x_{2} x_{3}-t^4 x_{1} x_{2}^2 x_{3}-t^4 x_{1} x_{2} x_{3}^2\\&+q^2 t x_{1}^2 x_{2}^2+q^2 t x_{1}^2 x_{3}^2+q^2 t x_{2}^2 x_{3}^2-2 q t^2 x_{1}^2 x_{2}^2\\&-2 q t^2 x_{1}^2 x_{3}^2-2 q t^2 x_{2}^2 x_{3}^2+t^3 x_{1}^2 x_{2}^2+t^3 x_{1}^2 x_{3}^2+t^3 x_{2}^2 x_{3}^2+q t x_{1}^3 x_{2}\\&+q t x_{1}^3 x_{3}+q t x_{1}^2 x_{2}^2+q t x_{1}^2 x_{2} x_{3}+q t x_{1}^2 x_{3}^2+q t x_{1} x_{2}^3+q t x_{1} x_{2}^2 x_{3}\\&+q t x_{1} x_{2} x_{3}^2+q t x_{1} x_{3}^3+q t x_{2}^3 x_{3}+q t x_{2}^2 x_{3}^2+q t x_{2} x_{3}^3+t^2 x_{1}^2 x_{2}^2\\&+2 t^2 x_{1}^2 x_{2} x_{3}+t^2 x_{1}^2 x_{3}^2+2 t^2 x_{1} x_{2}^2 x_{3}\\&+2 t^2 x_{1} x_{2} x_{3}^2+t^2 x_{2}^2 x_{3}^2-2 q x_{1}^2 x_{2} x_{3}-2 q x_{1} x_{2}^2 x_{3}\\&-2 q x_{1} x_{2} x_{3}^2+2 t x_{1}^2 x_{2} x_{3}+2 t x_{1} x_{2}^2 x_{3}+2 t x_{1} x_{2} x_{3}^2-x_{1}^3 x_{2}\\&-x_{1}^3 x_{3}-2 x_{1}^2 x_{2}^2-3 x_{1}^2 x_{2} x_{3}-2 x_{1}^2 x_{3}^2-x_{1} x_{2}^3\\&-3 x_{1} x_{2}^2 x_{3}-3 x_{1} x_{2} x_{3}^2-x_{1} x_{3}^3-x_{2}^3 x_{3}-2 x_{2}^2 x_{3}^2-x_{2} x_{3}^3) \\&+\mathbf w p \dfrac{ (-1+t) (t^3 q-1) (-t+q)}{q (q-1) (q t-1)^3 (q t+1)^2 t^4}\\&\cdot (q^6 t^8 x_{1}^3 x_{2}+q^6 t^8 x_{1}^3 x_{3}+q^6 t^8 x_{1}^2 x_{2}^2\\&+2 q^6 t^8 x_{1}^2 x_{2} x_{3}+q^6 t^8 x_{1}^2 x_{3}^2+q^6 t^8 x_{1} x_{2}^3\\&+2 q^6 t^8 x_{1} x_{2}^2 x_{3}+2 q^6 t^8 x_{1} x_{2} x_{3}^2\\&+q^6 t^8 x_{1} x_{3}^3+q^6 t^8 x_{2}^3 x_{3}+q^6 t^8 x_{2}^2 x_{3}^2\\&+q^6 t^8 x_{2} x_{3}^3-q^6 t^7 x_{1}^2 x_{2}^2-2 q^6 t^7 x_{1}^2 x_{2} x_{3}\\&-q^6 t^7 x_{1}^2 x_{3}^2-2 q^6 t^7 x_{1} x_{2}^2 x_{3}-2 q^6 t^7 x_{1} x_{2} x_{3}^2\\&-q^6 t^7 x_{2}^2 x_{3}^2+q^5 t^8 x_{1}^2 x_{2}^2+2 q^5 t^8 x_{1}^2 x_{2} x_{3}\\&+q^5 t^8 x_{1}^2 x_{3}^2+2 q^5 t^8 x_{1} x_{2}^2 x_{3}+2 q^5 t^8 x_{1} x_{2} x_{3}^2+q^5 t^8 x_{2}^2 x_{3}^2\\&-q^6 t^6 x_{1}^2 x_{2}^2-q^6 t^6 x_{1}^2 x_{2} x_{3}-q^6 t^6 x_{1}^2 x_{3}^2-q^6 t^6 x_{1} x_{2}^2 x_{3}\\&-q^6 t^6 x_{1} x_{2} x_{3}^2-q^6 t^6 x_{2}^2 x_{3}^2-q^5 t^7 x_{1}^3 x_{2}\\&-q^5 t^7 x_{1}^3 x_{3}+q^5 t^7 x_{1}^2 x_{2}^2-q^5 t^7 x_{1}^2 x_{2} x_{3}\\&+q^5 t^7 x_{1}^2 x_{3}^2-q^5 t^7 x_{1} x_{2}^3-q^5 t^7 x_{1} x_{2}^2 x_{3}\\&-q^5 t^7 x_{1} x_{2} x_{3}^2-q^5 t^7 x_{1} x_{3}^3-q^5 t^7 x_{2}^3 x_{3}+q^5 t^7 x_{2}^2 x_{3}^2\\&-q^5 t^7 x_{2} x_{3}^3+q^4 t^8 x_{1}^2 x_{2} x_{3}+q^4 t^8 x_{1} x_{2}^2 x_{3}+q^4 t^8 x_{1} x_{2} x_{3}^2\\&+q^6 t^5 x_{1}^2 x_{2}^2+2 q^6 t^5 x_{1}^2 x_{2} x_{3}+q^6 t^5 x_{1}^2 x_{3}^2\\&+2 q^6 t^5 x_{1} x_{2}^2 x_{3}+2 q^6 t^5 x_{1} x_{2} x_{3}^2+q^6 t^5 x_{2}^2 x_{3}^2\\&+q^5 t^6 x_{1}^3 x_{2}+q^5 t^6 x_{1}^3 x_{3}+3 q^5 t^6 x_{1}^2 x_{2}^2+q^5 t^6 x_{1}^2 x_{2} x_{3}\\&+3 q^5 t^6 x_{1}^2 x_{3}^2+q^5 t^6 x_{1} x_{2}^3+q^5 t^6 x_{1} x_{2}^2 x_{3}+q^5 t^6 x_{1} x_{2} x_{3}^2 \end{aligned}$$
$$\begin{aligned}&+q^5 t^6 x_{1} x_{3}^3+q^5 t^6 x_{2}^3 x_{3}+3 q^5 t^6 x_{2}^2 x_{3}^2+q^5 t^6 x_{2} x_{3}^3-2 q^4 t^7 x_{1}^2 x_{2}^2\\&-2 q^4 t^7 x_{1}^2 x_{3}^2-2 q^4 t^7 x_{2}^2 x_{3}^2-q^6 t^4 x_{1}^2 x_{2} x_{3}-q^6 t^4 x_{1} x_{2}^2 x_{3}\\&-q^6 t^4 x_{1} x_{2} x_{3}^2-3 q^5 t^5 x_{1}^2 x_{2}^2-4 q^5 t^5 x_{1}^2x_{2} x_{3}-3 q^5 t^5 x_{1}^2 x_{3}^2\\&-4 q^5 t^5 x_{1} x_{2}^2x_{3}-4 q^5 t^5 x_{1} x_{2} x_{3}^2-3 q^5 t^5 x_{2}^2 x_{3}^2\\&-q^4 t^6 x_{1}^3 x_{2}-q^4 t^6 x_{1}^3 x_{3}+q^4 t^6 x_{1}^2 x_{2}^2+4 q^4 t^6 x_{1}^2 x_{2} x_{3}\\&+q^4 t^6 x_{1}^2 x_{3}^2-q^4 t^6 x_{1} x_{2}^3+4 q^4 t^6 x_{1} x_{2}^2 x_{3}+4 q^4 t^6 x_{1} x_{2} x_{3}^2-q^4 t^6 x_{1} x_{3}^3\\&-q^4 t^6 x_{2}^3 x_{3}+q^4 t^6 x_{2}^2 x_{3}^2-q^4 t^6 x_{2} x_{3}^3-2 q^3 t^7 x_{1}^2 x_{2}x_{3}\\&-2 q^3 t^7 x_{1} x_{2}^2 x_{3}-2 q^3 t^7 x_{1} x_{2} x_{3}^2-2 q^5 t^4 x_{1}^2 x_{2}^2\\&+q^5 t^4 x_{1}^2 x_{2} x_{3}-2 q^5 t^4 x_{1}^2 x_{3}^2+q^5 t^4 x_{1} x_{2}^2 x_{3}+q^5 t^4 x_{1} x_{2} x_{3}^2\\&-2 q^5 t^4 x_{2}^2 x_{3}^2-2 q^4 t^5 x_{1}^3 x_{2}-2 q^4 t^5 x_{1}^3 x_{3}\\&+q^4 t^5 x_{1}^2 x_{2}^2-7 q^4 t^5 x_{1}^2 x_{2} x_{3}+q^4 t^5 x_{1}^2 x_{3}^2-2 q^4 t^5 x_{1} x_{2}^3\\&-7 q^4 t^5 x_{1} x_{2}^2 x_{3}-7 q^4 t^5 x_{1} x_{2} x_{3}^2-2 q^4 t^5 x_{1}x_{3}^3\\&-2 q^4 t^5 x_{2}^3 x_{3}+q^4 t^5 x_{2}^2 x_{3}^2-2 q^4 t^5 x_{2} x_{3}^3\\&-3 q^3 t^6 x_{1}^2 x_{2}^2-3 q^3 t^6 x_{1}^2 x_{3}^2-3 q^3 t^6 x_{2}^2 x_{3}^2\\&+q^5 t^3 x_{1}^2 x_{2} x_{3}+q^5 t^3 x_{1} x_{2}^2 x_{3}+q^5 t^3 x_{1} x_{2} x_{3}^2\\&+3 q^4 t^4 x_{1}^2 x_{2}^2-q^4 t^4 x_{1}^2 x_{2} x_{3}+3 q^4 t^4 x_{1}^2 x_{3}^2-q^4 t^4 x_{1} x_{2}^2 x_{3}\\&-q^4 t^4 x_{1} x_{2} x_{3}^2+3 q^4 t^4 x_{2}^2 x_{3}^2+q^3 t^5 x_{1}^3 x_{2}+q^3 t^5 x_{1}^3 x_{3}\\&-4 q^3 t^5 x_{1}^2 x_{2}^2+4 q^3 t^5 x_{1}^2 x_{2} x_{3}-4 q^3 t^5 x_{1}^2 x_{3}^2\\&+q^3 t^5 x_{1} x_{2}^3+4 q^3 t^5 x_{1} x_{2}^2 x_{3}+4 q^3 t^5 x_{1} x_{2} x_{3}^2\\&+q^3 t^5 x_{1} x_{3}^3+q^3 t^5 x_{2}^3 x_{3}-4 q^3 t^5 x_{2}^2 x_{3}^2+q^3 t^5 x_{2} x_{3}^3-4 q^2 t^6 x_{1}^2 x_{2} x_{3}\\&-4 q^2 t^6 x_{1} x_{2}^2 x_{3}-4 q^2 t^6 x_{1} x_{2} x_{3}^2-3 q^4 t^3 x_{1}^2 x_{2}^2\\&-q^4 t^3 x_{1}^2 x_{2} x_{3}-3 q^4 t^3 x_{1}^2 x_{3}^2-q^4 t^3 x_{1} x_{2}^2 x_{3}\\&-q^4 t^3 x_{1} x_{2} x_{3}^2-3 q^4 t^3 x_{2}^2 x_{3}^2+3 q^2 t^5 x_{1}^2 x_{2}^2\\&+q^2 t^5 x_{1}^2 x_{2} x_{3}+3 q^2 t^5 x_{1}^2 x_{3}^2+q^2 t^5 x_{1} x_{2}^2 x_{3}+q^2 t^5 x_{1} x_{2} x_{3}^2\\&+3 q^2 t^5 x_{2}^2 x_{3}^2+4 q^4 t^2 x_{1}^2 x_{2} x_{3}+4 q^4 t^2 x_{1} x_{2}^2 x_{3}+4 q^4 t^2 x_{1} x_{2} x_{3}^2\\&-q^3 t^3 x_{1}^3 x_{2}-q^3 t^3 x_{1}^3 x_{3}+4 q^3 t^3 x_{1}^2 x_{2}^2-4 q^3 t^3 x_{1}^2 x_{2} x_{3}+4 q^3 t^3 x_{1}^2 x_{3}^2\\&-q^3 t^3 x_{1} x_{2}^3-4 q^3 t^3 x_{1} x_{2}^2 x_{3}-4 q^3 t^3 x_{1} x_{2} x_{3}^2-q^3 t^3 x_{1} x_{3}^3\\&-q^3 t^3 x_{2}^3 x_{3}+4 q^3 t^3 x_{2}^2 x_{3}^2-q^3 t^3 x_{2} x_{3}^3-3 q^2 t^4 x_{1}^2 x_{2}^2\\&+q^2 t^4 x_{1}^2 x_{2} x_{3}-3 q^2 t^4 x_{1}^2 x_{3}^2+q^2 t^4 x_{1} x_{2}^2 x_{3}+q^2 t^4 x_{1} x_{2} x_{3}^2\\&-3 q^2 t^4 x_{2}^2 x_{3}^2-q t^5 x_{1}^2 x_{2} x_{3}-q t^5 x_{1} x_{2}^2 x_{3}-q t^5 x_{1} x_{2} x_{3}^2\\&+3 q^3 t^2 x_{1}^2 x_{2}^2+3 q^3 t^2 x_{1}^2 x_{3}^2+3 q^3 t^2 x_{2}^2 x_{3}^2+2 q^2 t^3 x_{1}^3 x_{2}\\&+2 q^2 t^3 x_{1}^3 x_{3}-q^2 t^3 x_{1}^2 x_{2}^2+7 q^2 t^3 x_{1}^2 x_{2} x_{3}-q^2 t^3 x_{1}^2 x_{3}^2+2 q^2 t^3 x_{1} x_{2}^3\\&+7 q^2 t^3 x_{1} x_{2}^2 x_{3}+7 q^2 t^3 x_{1} x_{2} x_{3}^2+2 q^2 t^3 x_{1} x_{3}^3+2 q^2 t^3 x_{2}^3 x_{3}-q^2 t^3 x_{2}^2 x_{3}^2\\&+2 q^2 t^3 x_{2} x_{3}^3+2 q t^4 x_{1}^2 x_{2}^2-q t^4 x_{1}^2 x_{2} x_{3}+2 q t^4 x_{1}^2 x_{3}^2-q t^4 x_{1} x_{2}^2 x_{3}\\&-q t^4 x_{1} x_{2} x_{3}^2+2 q t^4 x_{2}^2 x_{3}^2+2 q^3 t x_{1}^2 x_{2} x_{3}+2 q^3 t x_{1} x_{2}^2 x_{3}+2 q^3 t x_{1} x_{2} x_{3}^2\\&+q^2 t^2 x_{1}^3 x_{2}+q^2 t^2 x_{1}^3 x_{3}-q^2 t^2 x_{1}^2 x_{2}^2-4 q^2 t^2 x_{1}^2 x_{2} x_{3} \end{aligned}$$
$$\begin{aligned}&-q^2 t^2 x_{1}^2 x_{3}^2+q^2 t^2 x_{1} x_{2}^3-4 q^2 t^2 x_{1} x_{2}^2 x_{3}-4 q^2 t^2 x_{1} x_{2} x_{3}^2\\&+q^2 t^2 x_{1} x_{3}^3+q^2 t^2 x_{2}^3 x_{3}-q^2 t^2 x_{2}^2 x_{3}^2+q^2 t^2 x_{2} x_{3}^3\\&+3 q t^3 x_{1}^2 x_{2}^2+4 q t^3 x_{1}^2 x_{2} x_{3}+3 q t^3 x_{1}^2 x_{3}^2\\&+4 q t^3 x_{1} x_{2}^2 x_{3}+4 q t^3 x_{1} x_{2} x_{3}^2+3 q t^3 x_{2}^2 x_{3}^2+t^4 x_{1}^2 x_{2} x_{3}\\&+t^4 x_{1} x_{2}^2 x_{3}+t^4 x_{1} x_{2} x_{3}^2+2 q^2 t x_{1}^2 x_{2}^2\\&+2 q^2 t x_{1}^2 x_{3}^2+2 q^2 t x_{2}^2 x_{3}^2-q t^2 x_{1}^3 x_{2}\\&-q t^2 x_{1}^3 x_{3}-3 q t^2 x_{1}^2 x_{2}^2-q t^2 x_{1}^2 x_{2} x_{3}\\&-3 q t^2 x_{1}^2 x_{3}^2-q t^2 x_{1} x_{2}^3-q t^2 x_{1} x_{2}^2 x_{3}\\&-q t^2 x_{1} x_{2} x_{3}^2-q t^2 x_{1} x_{3}^3-q t^2 x_{2}^3 x_{3}\\&-3 q t^2 x_{2}^2 x_{3}^2-q t^2 x_{2} x_{3}^3-t^3 x_{1}^2 x_{2}^2-2 t^3 x_{1}^2 x_{2} x_{3}-t^3 x_{1}^2 x_{3}^2-2 t^3 x_{1} x_{2}^2 x_{3}\\&-2 t^3 x_{1} x_{2} x_{3}^2-t^3 x_{2}^2 x_{3}^2-q^2 x_{1}^2 x_{2} x_{3}\\&-q^2 x_{1} x_{2}^2 x_{3}-q^2 x_{1} x_{2} x_{3}^2+q t x_{1}^3 x_{2}\\&+q t x_{1}^3 x_{3}-q t x_{1}^2 x_{2}^2+q t x_{1}^2 x_{2} x_{3}-q t x_{1}^2 x_{3}^2\\&+q t x_{1} x_{2}^3+q t x_{1} x_{2}^2 x_{3}+q t x_{1} x_{2} x_{3}^2\\&+q t x_{1} x_{3}^3+q t x_{2}^3 x_{3}-q t x_{2}^2 x_{3}^2+q t x_{2} x_{3}^3\\&+t^2 x_{1}^2 x_{2}^2+t^2 x_{1}^2 x_{2} x_{3}+t^2 x_{1}^2 x_{3}^2\\&+t^2 x_{1} x_{2}^2 x_{3}+t^2 x_{1} x_{2} x_{3}^2+t^2 x_{2}^2 x_{3}^2\\&-q x_{1}^2 x_{2}^2-2 q x_{1}^2 x_{2} x_{3}-q x_{1}^2 x_{3}^2-2 q x_{1} x_{2}^2 x_{3}\\&-2 q x_{1} x_{2} x_{3}^2-q x_{2}^2 x_{3}^2+t x_{1}^2 x_{2}^2+2 t x_{1}^2 x_{2} x_{3}\\&+t x_{1}^2 x_{3}^2+2 t x_{1} x_{2}^2 x_{3}+2 t x_{1} x_{2} x_{3}^2\\&+t x_{2}^2 x_{3}^2-x_{1}^3 x_{2}-x_{1}^3 x_{3}-x_{1}^2 x_{2}^2-2 x_{1}^2 x_{2} x_{3}\\&-x_{1}^2 x_{3}^2-x_{1} x_{2}^3-2 x_{1} x_{2}^2 x_{3}-2 x_{1} x_{2} x_{3}^2\\&-x_{1} x_{3}^3-x_{2}^3 x_{3}-x_{2}^2 x_{3}^2-x_{2} x_{3}^3) + \cdots \end{aligned}$$

Notice that the series purely in w is absent because we took the smallest possible weight \(\mathbf j \).

The corresponding eigenvalues are

$$\begin{aligned} \lambda _1= & {} -q t-1-t^{-1}\\&- \frac{\mathbf{p}}{(q t+1) (q t-1)^2 t^2} (-1+t) (t+1) (-t+q)\\&\quad (t^3 q^3-q^2 t^4-t^3 q^2+t^3 q+q^2 t-t^2 q-q t+t^2+t-1) \\&+ \frac{\mathbf{w}}{t^3 q} (t^6 q^3+t^5 q^3+t^5 q^2+2 q^2 t^4+t^3 q^2+t^4 q\\&\quad +2 t^3 q+2 t^2 q+q t+t^2+t+1) \\&+\frac{\mathbf{w p}}{q^2 (q t+1) (q t-1)^2 t^5} (-t+q) (-1+t)(t+1) \\&\quad (t^9 q^7-t^{10} q^6-t^8 q^7+2 t^8 q^6-q^5 t^9+2 t^7 q^6-3 t^8 q^5-t^9 q^4\\&\quad +t^7 q^5-t^8 q^4+2 t^5 q^6+2 t^6 q^5-4 t^7 q^4+2 t^5 q^5-5 t^6 q^4\\&\quad +2 t^4 q^5-t^6 q^3+3 t^7 q^2+t^3 q^5-3 t^5 q^3+3 t^6 q^2+2 t^3 q^4\\&\quad -4 t^4 q^3+2 t^5 q^2+q t^6+t^2 q^4-2 t^3 q^3+2 q^2 t^4+3 t^5 q-t^2 q^3\\&\quad -t^3 q^2+q^3 t-4 t^2 q^2+2 t^3 q-2 q^2 t+2 t^2 q-q t-t^2-q+t) + \ldots \end{aligned}$$

and

$$\begin{aligned} \lambda _2= & {} q t+q+t^{-1}\\&- \frac{\mathbf{p}}{(q t+1) (q t-1)^2 t^2} (-1+t) (t+1) (-t+q)\\&\quad (t^4 q^3-t^3 q^3-t^2 q^3+t^3 q^2+t^2 q^2-t^3 q-q^2 t+q t+q-t) \\&- \frac{\mathbf{w}}{t^3 q} (t^6 q^3+t^5 q^3+t^4 q^3\\&\quad +t^5 q^2+2 q^2t^4+2 t^3 q^2+t^2 q^2+t^3 q+2 t^2 q+q t+t+1) \\&- \frac{\mathbf{w p}}{q^2 (q t+1) (q t-1)^2 t^5} (-1+t) (t+1) (-t+q) \\&\quad (t^9 q^7-t^{10} q^6-t^8 q^7-t^9 q^6+2 t^8 q^6-2 q^5 t^9+2 t^7 q^6-4 t^8 q^5\\&\quad +t^9 q^4-t^7 q^5-t^8 q^4+3 t^5 q^6+2 t^6 q^5-2 t^7 q^4+t^8 q^3+t^4 q^6\\&\quad +2 t^5 q^5-4 t^6 q^4+2 t^7 q^3+3 t^4 q^5-3 t^5 q^4+t^7 q^2+3 t^3 q^5\\&\quad -t^4 q^4+2 t^6 q^2-5 t^4 q^3+2 t^5 q^2-4 t^3 q^3+2 q^2 t^4+2 t^5 q\\&\quad -t^2 q^3+t^3 q^2-q^3 t-3 t^2 q^2+2 t^3 q-q^2 t+2 t^2 q-t^2-q+t) + \ldots \end{aligned}$$

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Koroteev, P., Shakirov, S. The quantum DELL system. Lett Math Phys 110, 969–999 (2020). https://doi.org/10.1007/s11005-019-01247-y

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