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On tau-functions for the Toda lattice hierarchy


We extend a recent result of Dubrovin et al. in On tau-functions for the KdV hierarchy, arXiv:1812.08488 to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions and use them to give explicit formulae for the generating series of k-point correlation functions of the solution. Applications to computing GUE correlators and Gromov–Witten invariants of the Riemann sphere are under consideration.

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  1. Bertola, M., Dubrovin, B., Yang, D.: Correlation functions of the KdV hierarchy and applications to intersection numbers over \(\overline{{\cal{M}}}_{g, n}\). Physica D 327, 30–57 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  2. Bertola, M., Dubrovin, B., Yang, D.: Simple Lie algebras and topological ODEs. IMRN 2016, 1368–1410 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Bertola, M., Dubrovin, B., Yang, D.: Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multi-point correlation functions. arXiv:1610.07534v2

  4. Bessis, D., Itzykson, C., Zuber, J.B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109–157 (1980)

    MathSciNet  Article  Google Scholar 

  5. Carlet, G.: The extended bigraded Toda hierarchy. J. Phys. A Math. Gen. 39, 9411–9435 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  6. Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J. 4, 313–332 (2004)

    MathSciNet  Article  Google Scholar 

  7. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rep. 254, 1–133 (1995)

    ADS  Article  Google Scholar 

  8. Dickey, L.A.: Soliton Equations and Hamiltonian Systems, 2nd edn. World Scientific, Singapore (2003)

    Book  Google Scholar 

  9. Dubrovin, B.: Geometry of 2D topological field theories. In: Francaviglia, M., Greco, S. (eds.) Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Springer Lecture Notes in Mathematics, vol. 1620, pp. 120–348. Springer, Berlin, Heidelberg (1996)

    Chapter  Google Scholar 

  10. Dubrovin, B., Yang, D.: Generating series for GUE correlators. Lett. Math. Phys. 107, 1971–2012 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  11. Dubrovin, B., Yang, D.: On Gromov–Witten invariants of \({\mathbb{P}}^1\). Math. Res. Lett. 26, 729–748 (2019)

  12. Dubrovin, B., Yang, D., Zagier, D.: Gromov–Witten invariants of the Riemann sphere. Pure Appl. Math. Q. (to appear)

  13. Dubrovin, B., Yang, D., Zagier, D.: On tau-functions for the KdV hierarchy. arXiv:1812.08488

  14. Dubrovin, B., Zhang, Y.: Virasoro symmetries of the extended Toda hierarchy. Commun. Math. Phys. 250, 161–193 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  15. Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants. arXiv:math/0108160

  16. Eguchi, T., Yang, S.-K.: The topological \(CP^1\) model and the large-\(N\) matrix integral. Mod. Phys. Lett. A 9, 2893–2902 (1994)

    ADS  Article  Google Scholar 

  17. Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  18. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons, Translated from Russian by Reyman, A.G. Springer, Berlin (1987)

    Chapter  Google Scholar 

  19. Flaschka, H.: On the Toda lattice. II. Inverse-scattering solution. Prog. Theor. Phys. 51, 703–716 (1974)

    ADS  MathSciNet  Article  Google Scholar 

  20. Grünbaum, F.A., Yakimov, M.: Discrete bispectral Darboux transformations from Jacobi operators. Pac. J. Math. 204, 395–431 (2002)

    MathSciNet  Article  Google Scholar 

  21. Kazakov, V., Kostov, I., Nekrasov, N.: D-particles, matrix integrals and KP hierarchy. Nucl. Phys. B 557, 413–442 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  22. Manakov, S.V., Complete integrability and stochastization of discrete dynamical systems. J. Exp. Theor. Phys. 67, 543–555 (in Russian) (English translation. In: Sov. Phys. JETP 40(2), 269–274 (1974))

  23. Marchal, O.: WKB solutions of difference equations and reconstruction by the topological recursion. Nonlinearity 31, 226–262 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  24. Mehta, M.L.: Random Matrices, 2nd edn. Academic Press, New York (1991)

    MATH  Google Scholar 

  25. Milanov, T.E.: Hirota quadratic equations for the extended Toda hierarchy. Duke Math. J. 138, 161–178 (2007)

    MathSciNet  Article  Google Scholar 

  26. Okounkov, A., Pandharipande, R.: Gromov–Witten theory, Hurwitz theory, and completed cycles. Ann. Math. 163, 517–560 (2006)

    MathSciNet  Article  Google Scholar 

  27. Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Okamoto, K. (ed.) Group Representations and Systems of Differential Equations (Tokyo, 1982), Advanced Studies in Pure Mathematics, vol. 4, pp. 1–95. North-Holland, Amsterdam (1984)

    Google Scholar 

  28. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)

    MATH  Google Scholar 

  29. Zhang, Y.: On the \(CP^1\) topological sigma model and the Toda lattice hierarchy. J. Geom. Phys. 40, 215–232 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  30. Zhou, J.: Emergent geometry and mirror symmetry of a point. arXiv:1507.01679

  31. Zhou, J.: Hermitian one-matrix model and KP hierarchy. arXiv:1809.07951

  32. Zhou, J.: Genus expansions of Hermitian one-matrix models: fat graphs vs. thin graphs. arXiv:1809.10870

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The author is grateful to Youjin Zhang, Boris Dubrovin, Don Zagier for their advising and to Jian Zhou and Si-Qi Liu for helpful discussions. He thanks the referee for valuable suggestions; in particular, Appendix A comes out from the suggestions. He also wishes to thank Boris Dubrovin for introducing GUE to him and for helpful suggestions and discussions on this article. The work is partially supported by a starting research grant from University of Science and Technology of China.

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Correspondence to Di Yang.

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Dedicated to the memory of Boris Anatol’evich Dubrovin, with gratitude and admiration.

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Appendix A: Pair of abstract pre-wave functions

Appendix A: Pair of abstract pre-wave functions

Here, we construct a ring that is suitable for defining abstract pre-wave functions. Recall that \({{\mathcal {A}}}\) is the ring of polynomials of \(v_k,w_k\), \(k \in {{\mathbb {Z}}}\). Instead of the \({{\mathbb {Z}}}\)-coefficients, we will use in this appendix the \({{\mathbb {Q}}}\)-coefficients, i.e., \({{\mathcal {A}}}={{\mathbb {Q}}}\bigl [ \{v_k,w_k \,|\, k\in {{\mathbb {Z}}}\}\bigr ]\), is now under consideration. For each monic monomial \(\alpha \in {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\), we associate a symbol \(m_\alpha \). Denote by \({\mathcal {B}}\) the polynomial ring

$$\begin{aligned} {\mathcal {B}}\, := \, {{\mathbb {Q}}}\bigl [\{ \,m_\alpha \,|\,\alpha \text{ is } \text{ a } \text{ monic } \text{ monomial } \text{ in } {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\} \bigr ]. \end{aligned}$$

Define the action of \(\Lambda ^k\) on \({\mathcal {B}}\) with \(k\in {{\mathbb {Z}}}\) by

$$\begin{aligned} \Lambda ^k \bigl (m_{\alpha _1} \ldots m_{\alpha _l}) \; = \; m_{ \Lambda ^k(\alpha _1)} \ldots m_{ \Lambda ^k (\alpha _l)} \end{aligned}$$

for \(\alpha _1,\ldots ,\alpha _l\) being monic monomials in \({{\mathcal {A}}}\backslash {{\mathbb {Q}}}\), as well as by linearly extending it to other elements of \({\mathcal {B}}\). For a monic monomial \(\alpha =v_{i_1} \ldots v_{i_r} w_{j_1} \ldots w_{j_s}\in {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\) with \(i_1\le \cdots \le i_r\), \(j_1\le \cdots \le j_s\) and \(r+s\ge 1\), let \(k_\alpha :=-i_1\) (if \(r\ge 1\)), \(k_\alpha :=-j_1\) (if \(r=0\)); the monomial \(\Lambda ^{k_\alpha } (\alpha )\in {{\mathcal {A}}}\) is then called the (unique) reduced monomial (associated to \(\alpha \)). Denote by \({\mathcal {C}}\) the polynomial ring generated by \(m_{\beta }\), \(v_k\), \(w_k\) with \({{\mathbb {Q}}}\)-coefficients, where \(\beta \) are reduced monic monomials, and \(k\in {{\mathbb {Z}}}\). Let us also define an action of \(\Lambda ^k\) on \({\mathcal {C}}\), \(k\in {{\mathbb {Z}}}\). To this end, we introduce some notations: For \(\beta \) a reduced monic monomial of \({{\mathcal {A}}}\), denote

$$\begin{aligned} n_{\Lambda ^k(\beta )} \, := \, \left\{ \begin{array}{cc} m_\beta + \sum _{i=0}^{k-1} \Lambda ^i(\beta ), &{} ~ k\ge 0,\\ m_\beta - \sum _{i=k}^{-1} \Lambda ^i(\beta ), &{} ~ k\le -1. \end{array}\right. \end{aligned}$$

Then, for a monomial \(\alpha \cdot m_{\beta _1} \ldots m_{\beta _s}\) of \({\mathcal {C}}\) with \(\alpha \) being a monomial in \({{\mathcal {A}}}\), define

$$\begin{aligned} \Lambda ^k (\alpha \cdot m_{\beta _1} \ldots m_{\beta _s}) = \Lambda ^k (\alpha ) \cdot \prod _{j=1}^s n_{\Lambda ^k(\beta _j)}, \quad k\in {{\mathbb {Z}}}. \end{aligned}$$

Define the action of \(\Lambda ^k\) on other elements in \({\mathcal {C}}\) by requiring it as a linear operator. Denote by \(p:{\mathcal {B}}\rightarrow {\mathcal {C}}\) the linear map which maps \(m_{\alpha _1} \ldots m_{\alpha _l}\in {\mathcal {B}}\) to \(n_{\alpha _1} \ldots n_{\alpha _l}\in {\mathcal {C}}\), for \(\alpha _i\), \(i=1,\ldots ,l\) being monic monomials in \({{\mathcal {A}}}\backslash {{\mathbb {Q}}}\). Denote by \({\mathcal {B}}^0\) the image of p. Clearly, \({{\mathcal {A}}}\subset {\mathcal {B}}^0\). Indeed, the element \((\Lambda -1) \bigl (\sum _{i=1}^l \lambda _i \, m_{\alpha _i}\bigr ) \in {\mathcal {B}}\) becomes \(\sum _{i=1}^l \lambda _i \alpha _i \in {{\mathcal {A}}}\) under the map p. Here, \(\alpha _1,\ldots ,\alpha _l\) are distinct monic monomials in \({{\mathcal {A}}}\backslash {{\mathbb {Q}}}\). Finally, we define an operator \({\mathbb {S}}: {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\rightarrow {\mathcal {B}}^0\) by

$$\begin{aligned} {\mathbb {S}} \bigl (\lambda _1 \alpha _1 \; + \; \cdots \; + \; \lambda _l \alpha _l\bigr ) \; = \; \lambda _1 n_{\alpha _1} \; + \; \cdots \; + \; \lambda _l n_{\alpha _l} \, \end{aligned}$$

for \(\alpha _1,\ldots ,\alpha _l\) being distinct monic monomials and \(\lambda _1,\ldots ,\lambda _l\in {{\mathbb {Q}}}\).

Motivated by (62) and (63), define two families of elements \(y_i,z_i \in {{\mathcal {A}}}\), \(i\ge 1\) by

$$\begin{aligned}&y_{k+1} \ = - \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k y_i^{m_i}}{\prod _{i=1}^k m_i!} -v_0\delta _{k,0} \\&\qquad \qquad \quad - w_0 \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0 \sum _{i=1}^{k-1} im_i=k-1 \end{array}} \frac{\prod _{i=1}^{k-1} (-1)^{m_i} \bigl (\Lambda ^{-1}(y_i)\bigr )^{m_i}}{\prod _{i=1}^{k-1} m_i!}, \\&z_{k+1} \; = \; \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k (-1)^{m_i} z_i^{m_i}}{\prod _{i=1}^k m_i!} \; + \; v_1\delta _{k,0} \\&\quad \qquad \qquad + w_2 \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0\\ \sum _{i=1}^{k-1} im_i=k-1 \end{array} }\frac{\prod _{i=1}^{k-1} \bigl (\Lambda (z_i)\bigr )^{m_i}}{\prod _{i=1}^{k-1} m_i!}. \end{aligned}$$

Equivalently, the generating series \(y(\lambda ):=\sum _{i\ge 1} y_i/\lambda ^i\), \(z(\lambda ):=\sum _{i\ge 1} z_i/\lambda ^i\) satisfy

$$\begin{aligned}&\lambda \, e^{y(\lambda )} \; + \; v_0 \,-\, \lambda \; + \; w_0 \, \lambda ^{-1} \Lambda ^{-1}\bigl (e^{-y(\lambda )}\bigr ) \; = \; 0, \\&\lambda \, \Lambda ^{-1} \bigl (e^{-z(\lambda )}\bigr ) \; + \; v_0 \,-\, \lambda \; + \; w_1 \, \lambda ^{-1} e^{z(\lambda )} \; = \; 0. \end{aligned}$$


$$\begin{aligned} \psi _A\, := \, e^{{\mathbb {S}} (y(\lambda ))} \otimes \lambda ^n \otimes 1, \quad \psi _B \, := \, e^{{\mathbb {S}} (z(\lambda ))} \otimes \lambda ^{-n} \otimes e^{-\sigma } , \end{aligned}$$

where \(e^{-\sigma }\) is a formal element satisfying \(e^{(1-\Lambda ^{-1})(-\sigma )} = w_0\), and \(\lambda ^n\), \(\lambda ^{-n}\) are formal elements satisfying \(\Lambda ^k (1\otimes \lambda ^n)=\lambda ^k \otimes \lambda ^n\), \(\Lambda ^k (1\otimes \lambda ^{-n})=\lambda ^{-k} \otimes \lambda ^{-n}\), \(k\in {{\mathbb {Z}}}\). We have

$$\begin{aligned}&L \bigl (\psi _A\bigr ) \; = \; \lambda \, \psi _A,\quad L \bigl (\psi _B\bigr ) \; = \; \lambda \, \psi _B,\nonumber \\&\psi _A(\lambda ) \; = \; \bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr )\otimes \lambda ^n \;\in \; {\mathcal {C}} \left[ \left[ \lambda ^{-1}\right] \right] \otimes \lambda ^n, \end{aligned}$$
$$\begin{aligned}&\psi _B(\lambda ) \; = \; \bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr ) \otimes \lambda ^{-n} \otimes e^{-\sigma } \;\in \; {\mathcal {C}} \left[ \left[ \lambda ^{-1}\right] \right] \otimes \lambda ^{-n} \otimes e^{-\sigma }, \end{aligned}$$

where \(L=\Lambda +v_0 + w_0 \, \Lambda ^{-1}\). We call \(\psi _A\) and \(\psi _B\) the abstract pre-wave functions of type A and of type B, respectively, associated with \(v_0,w_0\).


$$\begin{aligned} d_{{\mathrm{pre}}}(\lambda )\, := \, \psi _A(\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) \,-\, \psi _B(\lambda ) \, \Lambda ^{-1}\bigl (\psi _A(\lambda )\bigr ) \end{aligned}$$


$$\begin{aligned} \Psi (\lambda ) \, := \, \begin{pmatrix} \psi _A (\lambda ) &{}\quad \psi _B(\lambda ) \\ \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) &{}\quad \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) \end{pmatrix}. \end{aligned}$$

Then, we have the following identity:

$$\begin{aligned} R(\lambda ) \; = \; \Psi (\lambda ) \, \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \, \Psi ^{-1}(\lambda ) \;=:\; M(\lambda ). \end{aligned}$$

The proof is similar to that of Proposition 3. (The main fact used in the proof is that from the definition, the coefficients of entries of \(R(\lambda )\) are uniquely determined in an algebraic way.) We omit its details here. However, let us explain the equality (131) by an equivalent form. From definition, we have

$$\begin{aligned} M(\lambda ) \; = \; \frac{1}{d_{{\mathrm{pre}}}(\lambda )} \begin{pmatrix} \psi _A (\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) &{}\quad -\psi _A (\lambda ) \, \psi _B(\lambda ) \\ \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) &{}\quad -\Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \, \psi _B(\lambda ) \end{pmatrix}. \end{aligned}$$

Then, from a straightforward calculation by using the definitions, we find

$$\begin{aligned} M_{11}(\lambda )&\; = \; \frac{1}{1 \,-\, \frac{w_0}{\lambda ^2} \, e^{ \Lambda ^{-1} (z(\lambda ) - y(\lambda ))}}, \end{aligned}$$
$$\begin{aligned} M_{12}(\lambda )&\; = \; \frac{1}{\lambda ^{-1} \, e^{-\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{\lambda }{w_0} \, e^{-\Lambda ^{-1}(z(\lambda ))}}, \end{aligned}$$
$$\begin{aligned} M_{21}(\lambda )&\; = \; \frac{1}{\lambda \, e^{\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{w_0}{\lambda } \, e^{\Lambda ^{-1}(z(\lambda ))}}, \end{aligned}$$
$$\begin{aligned} M_{22}(\lambda )&\; = \; \frac{1}{1 \,-\, \frac{\lambda ^2}{w_0} \, e^{ \Lambda ^{-1} (y(\lambda )- z(\lambda ))}}. \end{aligned}$$

Hence, the equality (131) means new expressions for the entries of the basic matrix resolvent \(R(\lambda )\) explicitly in terms of \(y(\lambda ),z(\lambda )\). Substituting the following expansions

$$\begin{aligned} y(\lambda ) \; = \; -\frac{v_0}{\lambda } \,-\, \frac{\frac{1}{2} v_0^2+w_0}{\lambda ^2} \; + \; \cdots , \qquad z(\lambda ) \; = \; \frac{v_1}{\lambda } \; + \; \frac{\frac{1}{2} v_1^2+w_2}{\lambda ^2} \; + \; \cdots \nonumber \\ \end{aligned}$$

into (132)–(135), we find that the new expressions agree with (24). Combining with (56), (57), we obtain

$$\begin{aligned}&\frac{1}{\frac{\lambda ^2}{w_0} \, e^{ \Lambda ^{-1} (y(\lambda )- z(\lambda ))} \,-\, 1} \; = \; \sum _{p\ge 0} \Omega _{p,0} \, \lambda ^{-p-2} \;=:\; A, \end{aligned}$$
$$\begin{aligned}&\frac{1}{\lambda \, e^{\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{w_0}{\lambda } \, e^{\Lambda ^{-1}(z(\lambda ))}} \; = \; \lambda ^{-1}\; + \;\sum _{p\ge 0} \Lambda ^{-1} \bigl (S_p\bigr ) \, \lambda ^{-p-2} \; =: \; B. \end{aligned}$$

We therefore arrive at the following formulae:

$$\begin{aligned} e^{\Lambda ^{-1}(y(\lambda ))} \; = \; \frac{1}{\lambda }\, \frac{1+A}{B}, \qquad e^{\Lambda ^{-1}(z(\lambda ))} \; = \; \frac{\lambda }{w_0} \, \frac{A}{B}. \end{aligned}$$

Let us proceed to the generating series of multi-point correlation functions. Define

$$\begin{aligned} D_{{\mathrm{pre}}}(\lambda ,\mu ) \, := \, \frac{\psi _A(\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\mu )\bigr ) \,-\, \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \,\psi _B(\mu )}{\lambda -\mu }. \end{aligned}$$

Using (131), Proposition 1, and a similar argument to the proof of Theorem 2, we obtain

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}}&= \frac{(-1)^{k-1}}{\prod _{j=1}^k d_{{\mathrm{pre}}}(\lambda _j)} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D_{{\mathrm{pre}}}(\lambda _{\pi (j)},\lambda _{\pi (j+1)}) \nonumber \\&\quad -\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

For the reader’s convenience, we give the first few terms of the abstract pre-wave functions \(\psi _A(\lambda )\) and \(\psi _B(\lambda )\) as follows:

$$\begin{aligned} \psi _A&= \biggl (1 \,-\, \frac{m_{v_0}}{\lambda } \; + \; \frac{m_{v_0}^2-m_{v_0^2}-2m_{w_0}}{2 \lambda ^2} \nonumber \\&\quad - \frac{1}{6\lambda ^3} \Bigl (m_{v_0}^3 + 2m_{v_0^3} - 3m_{v_0} m_{v_0^2} + 6m_{v_0 w_0} + 6m_{v_0 w_1} \nonumber \\&\quad - 6 m_{v_0} m_{w_0} - 6v_{-1} w_0\Bigr ) \; + \; {\mathrm{O}}\Bigl (\frac{1}{\lambda ^4}\Bigr )\biggr ) \, \lambda ^n,\end{aligned}$$
$$\begin{aligned} \psi _B&= \biggl (1 \; + \;\frac{m_{v_0}+v_0}{\lambda } \; + \; \frac{m_{v_0}^2+m_{v_0^2} +2v_0 m_{v_0}+2m_{w_0}+2v_0^2+2w_0+2w_1}{2\lambda ^2} \nonumber \\&\quad + \frac{1}{6 \lambda ^3} \Bigl (m_{v_0}^3+6m_{v_0} m_{w_0} +3 m_{v_0} m_{v_0^2} +2m_{v_0^3} +6m_{v_0 w_1} + 6m_{v_0 w_0} \nonumber \\&\quad +3 v_0 m_{v_0}^2+6v_0^2m_{v_0} +6w_0 m_{v_0} +6w_1 m_{v_0} +3v_0 m_{v_0^2}+6v_0 m_{w_0} \nonumber \\&\quad +6 v_0^3+12v_0 w_0+12 v_0 w_1 + 6 v_1 w_1 \Bigr ) \; + \; {\mathrm{O}}\Bigl (\frac{1}{\lambda ^4}\Bigr )\biggr ) \, \lambda ^{-n} e^{-\sigma }. \end{aligned}$$

It turns out that the above abstract pre-wave functions form a pair. Namely, \(d_{{\mathrm{pre}}}(\lambda ) = \lambda \, e^{\Lambda ^{-1}(-\sigma )}\). Interestingly, for given arbitrary initial value (f(n), g(n)), based on this statement, one obtains a constructive method for a pair of wave functions associated with (f(n), g(n)) (cf. (28) in Sect. 1.3 for the definition of a pair). This is important considering Theorem 1. We hope to confirm the statement on the pair property of the abstract pre-wave functions in another publication.

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Yang, D. On tau-functions for the Toda lattice hierarchy. Lett Math Phys 110, 555–583 (2020).

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  • Toda lattice hierarchy
  • Tau-function
  • Pair of wave functions
  • Matrix resolvent
  • Generating series

Mathematics Subject Classification

  • 37K10
  • 53D45
  • 14N35
  • 05A15
  • 33E15