Group algebras acting on the space of solutions of a special double confluent Heun equation

Abstract

We study properties of the space $\boldsymbol{\Omega}$ of solutions of a special double confluent Heun equation closely related to the model of a overdamped Josephson junction. We describe operators acting on $\boldsymbol{\Omega}$ and relations in the algebra $\mathcal{A}$ generated by them over the real number field. The structure of $\mathcal{A}$ depends on parameters. We give conditions under which $\mathcal{A}$ is isomorphic to a group algebra and describe two corresponding group structures.

A. Necessary information about $$\mathfrak{pqrs}$$ polynomials

We consider sequences $\{p_k\}$, $\{q_k\}$, $\{r_k\}$, and $\{s_k\}$, $k=0,1,2,\dots$, of functions of the complex variable $z$ defined by the recursive scheme

$$\begin{aligned} \, &p_0=0,\qquad q_0=1,\qquad r_0=z^{-2},\qquad s_0=-\mu, \end{aligned}$$

(49)

$$ \begin{aligned} \, &p_k=(1-\ell)z\,p_{k-1}+q_{k-1}+z^2p_{k-1}', \\[1mm] &q_k=z^2 (-\lambda+(\ell+1)\mu z)p_{k-1}+\mu(1- z^2)q_{k-1}+z^2q_{k-1}', \\[1mm] &r_k=2(k-2)z\,r_{k-1}-s_{k-1}-z^2r_{k-1}', \\[1mm] &s_k=z^2(\lambda-(\ell+1)\mu z)r_{k-1}+ ((2k-\ell-3)z+\mu(z^2-1))s_{k-1}-z^2s_{k-1}'. \end{aligned}$$

(50)

Although the function $r_0$ is singular at zero, all four functions $p_1$, $q_1$, $r_1$, and $s_1$ are already polynomials for $k=1$. This easily implies that for any $k>1$, the result is also four polynomials and, moreover, depends not only on $z$ but also on the constant parameters $\lambda$ and $\mu$ (and also on $\ell$).

Here, we need only one set of four polynomials (representatives of the sequences defined above), namely, the functions with the index $k=\ell$. For them, we introduce the notation

$$\mathfrak{p}=p_{\ell},\qquad\mathfrak{q}=q_{\ell},\qquad\mathfrak{r}=r_{\ell},\qquad\mathfrak{s}=s_{\ell},$$

and they are polynomials in $z$ of the respective degrees $2\ell-2$, $2\ell$, $2\ell-2$, and $2\ell$. They also depend polynomially on the parameters $\lambda$ and $\mu$ (see [5]).

For the first few values of $\ell$, the $\mathfrak{pqrs}$ polynomials have the forms

$$\begin{aligned} \, \ell=1\colon\quad\mathfrak{p}={}&1,\qquad\mathfrak{q}=-\mu(z^2-1),\qquad \mathfrak{r}=\mu,\qquad\mathfrak{s}=\lambda-\mu^2(z^2-1), \\[1mm] \ell=2\colon\quad\mathfrak{p}={}&-\mu(z^2-1)-z,\qquad \mathfrak{q}=(-\lambda+\mu z)z^2+\mu^2(z^2-1)^2, \\[1mm] \mathfrak{r}={}&-\lambda+\mu^2(z^2-1), \\[1mm] \mathfrak{s}={}&\lambda(-z+\mu(2z^2-1))-\mu^2(z+\mu(z^2-1)^2), \\[1mm] \ell=3\colon\quad\mathfrak{p}={}&(1-\lambda)z^2+(\mu(z^2-1)+z)^2, \\[1mm] \mathfrak{q}={}&-\mu(2z^4+(\mu^2(z^2-1)^2+2(\mu z-\lambda)z^2)(z^2-1)), \\[1mm] \mathfrak{r}={}&\mu(\mu^2(z^2-1)^2-\lambda(2z^2-1)), \\[1mm] \mathfrak{s}={}&-\lambda^2z^2+\mu^2\bigl(2z^2-\mu(z^2-1)(2z+\mu(z^2-1)^2)\bigr)+{} \\[1mm] &{}+\lambda\bigl(2z^2+\mu(z^2-1)(-2z+\mu(3z^2-1))\bigr), \\[1mm] \ell=4\colon\quad\mathfrak{p}={}&2(\lambda-3)z^3+2(\lambda-3)\mu z^2(z^2-1)- 3\mu^2z(z^2-1)^2-\mu^3(z^2-1)^3, \\[1mm] \mathfrak{q}={}&(\lambda-3)\lambda z^4-2(\lambda-3)\mu z^5+3\mu^3z^3(z^2-1)^2+{} \\[1mm] &{}+\mu^4(z^2-1)^4-3\mu^2z^2\bigl((3z^2-2z^4+\lambda(z^2-1)^2\bigr), \\[1mm] \mathfrak{r}={}&\lambda^2z^2-3\mu^2z^2+\mu^4(z^2-1)^3- \lambda\bigl(3 z^2+\mu^2(1-4z^2+3z^4)\bigr), \\[1mm] \mathfrak{s}={}&-\bigl(\lambda^2 z^2 (-2 z+\mu (3 z^2-2))+{} \\[1mm] &{}+\mu^2(6z^3+3\mu^2z(z^2-1)^2+\mu^3(z^2-1)^4+\mu(6z^2-9z^4))+{} \\[1mm] &{}+\lambda(6z^3-\mu^3(z^2-1)^2(4 z^2-1)+ \mu(6z^2-9z^4)+\mu^2z(3-8z^2+3z^4))\bigr). \end{aligned}$$

The $\mathfrak{pqrs}$ polynomials satisfy the system of linear differential equations

$$\begin{aligned} \, &z^2\mathfrak{p}'=(\mu+(\ell-1)z)\mathfrak{p}-\mathfrak{q}+(-1)^\ell z^2\mathfrak{r}, \\[1mm] &\mathfrak{q}'=(\lambda-(\ell+1)\mu z)\mathfrak{p}+\mu\,\mathfrak{q}+(-1)^\ell\mathfrak{s}, \\[1mm] &z^2\mathfrak{r}'=(-1)^{\ell+1}(\lambda+\mu^2)\mathfrak{p}+z(2 (\ell-1)-\mu z)\mathfrak{r}-\mathfrak{s}, \\[1mm] &z^2\mathfrak{s}'=(-1)^{\ell+1}(\lambda+\mu^2)\mathfrak{q}+ z^2(\lambda-(\ell+1)\mu z)\mathfrak{r}+((\ell-1)z-\mu)\mathfrak{s}. \end{aligned} $$

(51)

It follows from them that the expression

$$\mathcal{D}_\ell=z^{2(1-l)}\bigl(\mathfrak{p}(z)\mathfrak{s}(z)-\mathfrak{q}(z)\mathfrak{r}(z)\bigr) $$

(52)

is independent of $z$. Indeed, calculating its derivative and eliminating derivatives of the $\mathfrak{pqrs}$ polynomials using the equations given above, we obtain an expression that vanishes as a result of identical algebraic transformations. In other words, $\mathcal{D}_\ell$ is a first integral of Eqs. (51). It is easy to see that $\mathcal{D}_\ell$ is a polynomial in $\lambda$ and $\mu$.

As a consequence of the $z$-independence, we can calculate the value of $\mathcal{D}_\ell$ for any value of this variable except zero. In particular,

$$\mathcal{D}_\ell=\mathfrak{p}(1)\mathfrak{s}(1)-\mathfrak{q}(1)\mathfrak{s}(1).$$

We now note that as shown in [5], the $\mathfrak{pqrs}$ polynomials satisfy the functional equations

$$\begin{aligned} \, &\mathfrak{p}(-z^{-1})=(-1)^{\ell+1}z^{-2(\ell-1)}\mathfrak{p}(z), \\[1mm] &\mathfrak{q}(-z^{-1})=z^{-2\ell}[(-1)^\ell\mu\,\mathfrak{p}(z)+z^2\mathfrak{r}(z)], \\[1mm] &\mathfrak{r}(-z^{-1})=z^{-2(\ell-1)}[\mu z^2\mathfrak{p}(z)+\mathfrak{q}(z)], \\[1mm] &\mathfrak{s}(-z^{-1})=-z^{-2\ell}[\mu(\mu z^2\mathfrak{p}(z)+\mathfrak{q}(z))+ (-1)^\ell z^2(\mu z^2\mathfrak{r}(z)+\mathfrak{s}(z))], \end{aligned}$$

(53)

$$ \begin{aligned} \, &\mathfrak{p}(-z)=(-1)^{\ell+1}(\lambda+\mu^2)^{-1}(\mu z^2\mathfrak{r}(z)+\mathfrak{s}(z)), \\[1mm] &\mathfrak{q}(-z)=\mu z^2\mathfrak{p}(z)+\mathfrak{q}(z)+(-1)^\ell (\lambda+\mu^2)^{-1}\mu z^2(\mu z^2\mathfrak{r}(z)+\mathfrak{s}(z)), \\[1mm] &\mathfrak{r}(-z)=\mathfrak{r}(z), \\[1mm] &\mathfrak{s}(-z)=(-1)^{\ell+1}(\lambda+\mu^2)\mathfrak{p}(z)-\mu z^2\mathfrak{r}(z), \end{aligned}$$

(54)

$$ \begin{aligned} \, &\mathfrak{p}(z^{-1})=(\lambda+\mu^2)^{-1}z^{-2(\ell-1)}[\mu z^2\mathfrak{r}(z)+\mathfrak{s}(z)], \\[1mm] &\mathfrak{q}(z^{-1})=(\lambda+\mu^2)^{-1}z^{-2\ell}[\lambda z^2\mathfrak{r}(z)-\mu\,\mathfrak{s}(z)], \\[1mm] &\mathfrak{r}(z^{-1})=z^{-2(\ell-1)}[\mu z^2\mathfrak{p}(z)+\mathfrak{q}(z)], \\[1mm] &\mathfrak{s}(z^{-1})=z^{-2\ell}[\lambda z^2\mathfrak{p}(z)-\mu\,\mathfrak{q}(z)]. \end{aligned}$$

(55)

Substituting $z=1$ in the last four equalities, we obtain

$$\mathfrak{q}(1)=-\mu\mathfrak{p}(1)+\mathfrak{r}(1),\qquad\mathfrak{s}(1)=(\lambda+\mu^2)\mathfrak{p}(1)-\mu\mathfrak{r}(1).$$

Using these equalities, we obtain

$$\mathcal{D}_\ell=(\lambda+\mu^2)\mathfrak{p}^2(1)-\mathfrak{r}^2(1).$$

Finally, using this formula and the explicit expressions for the $\mathfrak{pqrs}$ polynomials in the first four orders given above, we obtain the explicit representations

$$\begin{aligned} \, \ell=1\colon&\quad\mathcal{D}_\ell=\lambda, \\[1mm] \ell=2\colon&\quad\mathcal{D}_\ell=\lambda-\lambda^2+\mu^2, \\[1mm] \ell=3\colon&\quad\mathcal{D}_\ell=-4\lambda^2+ \lambda^3+4\mu^2-4\lambda(\mu^2-1), \\[1mm] \ell=4\colon&\quad\mathcal{D}_\ell=10\lambda^3-\lambda^4- 9\mu^2(\mu^2-4)-6\lambda(7\mu^2-6)+\lambda^2(10\mu^2-33). \end{aligned} $$

(56)

B. Group properties of transformations generated by $$\mathcal{L}$$ -operators for $$\lambda+\mu^2>0$$

We interpreted the constant $\omega$ introduced in formulas (3) as an arbitrary nonzero real number. Its role consists in a “synchronous scaling” of the operators $\mathcal{L}_{\mathrm{B}}$ and $\mathcal{L}_{\mathrm{C}}$, and $\omega$ can be fixed in any desired way, as needed. It follows from formulas (6)–(14) that the useful effect simplifying their form is obtained if $\omega$ is chosen from the condition $|(2\omega)^2(\lambda+\mu^2)|=1$. More precisely, depending on the values of $\lambda$ and $\mu$, we obtain either $(2\omega)^2 (\lambda+\mu^2)=1$ or $(2\omega)^2(\lambda+\mu^2)=-1$. We considered the first case in the main text, and we therefore focus on the second case here. It corresponds to the domain in the space of the parameters $\lambda$ and $\mu$ such that

$$\lambda+\mu^2>0.$$

For a given choice of $\omega$, the composition laws for the $\mathcal{L}$-operators in this domain become

$$\begin{aligned} \, &\mathcal{L}_{\mathrm{A}}\circ\mathcal{L}_{\mathrm{A}}=-\mathcal{D}_\ell\cdot\mathrm{Id}, \\[1mm] &\mathcal{L}_{\mathrm{B}}\circ\mathcal{L}_{\mathrm{B}}=-\mathcal{D}_\ell\cdot\mathcal{M}, \\[1mm] &\mathcal{L}_{\mathrm{C}}\circ\mathcal{L}_{\mathrm{C}}=-\mathrm{Id}, \\[1mm] &\mathcal{L}_{\mathrm{A}}\circ\mathcal{L}_{\mathrm{B}}=\mathcal{D}_\ell\cdot\mathcal{M}^{-1}\circ\mathcal{L}_{\mathrm{C}}, \\[1mm] &\mathcal{L}_{\mathrm{B}}\circ\mathcal{L}_{\mathrm{A}}=-\mathcal{D}_\ell\cdot\mathcal{L}_{\mathrm{C}}, \\[1mm] &\mathcal{L}_{\mathrm{B}}\circ\mathcal{L}_{\mathrm{C}}=\mathcal{M}\circ\mathcal{L}_{\mathrm{A}}, \\[1mm] &\mathcal{L}_{\mathrm{C}}\circ\mathcal{L}_{\mathrm{B}}=-\mathcal{L}_{\mathrm{A}}, \\[1mm] &\mathcal{L}_{\mathrm{C}}\circ\mathcal{L}_{\mathrm{A}}=\mathcal{L}_{\mathrm{B}}, \\[1mm] &\mathcal{L}_{\mathrm{A}}\circ\mathcal{L}_{\mathrm{C}}=-\mathcal{M}^{-1}\circ\mathcal{L}_{\mathrm{B}}. \end{aligned} $$

(57)

A second simplification is achieved by leveling the constant $\mathcal{D}_\ell$. In contrast to $\omega$, this is an invariant, a particular (polynomial) function of the parameters $\lambda$ and $\mu$ (see Appendix A). But we can perform a specific normalization of the operators $\mathcal{L}_{\mathrm{A}}$ and $\mathcal{L}_{\mathrm{B}}$, multiplying them by $\mathcal{D}_\ell^{-1/2}$ for $\mathcal{D}_\ell>0$ or by $(-\mathcal{D}_\ell)^{-1/2}$ for $\mathcal{D}_\ell<0$. Only the sign of $\mathcal{D}_\ell$ then remains. This allows simplifying rules (57) as follows.

If $\mathcal{D}_\ell>0$, then we define

$$\begin{alignedat}2 &\mathbf{a}:=\mathcal{D}_\ell^{-1/2}\cdot\mathcal{L}_{\mathrm{A}},&\qquad &{}^*\mathbf{a}:=-\mathcal{D}_\ell^{-1/2}\cdot\mathcal{L}_{\mathrm{A}}, \\[1mm] &\mathbf{b}:=\mathcal{D}_\ell^{-1/2}\cdot\mathcal{L}_{\mathrm{B}},&\qquad &{}^*\mathbf{b}:=-\mathcal{D}_\ell^{-1/2}\cdot\mathcal{L}_{\mathrm{B}}, \\[1mm] &\mathbf{c}:=\mathcal{L}_{\mathrm{C}},&\qquad&{}^*\mathbf{c}:=-\mathcal{L}_{\mathrm{C}}. \end{alignedat} $$

(58)

If $\mathcal{D}_\ell>0$, then we define

$$\begin{alignedat}2 &\mathbf{a}:=\mathcal{L}_{\mathrm{C}},&\qquad&{}^*\mathbf{a}:=-\mathcal{L}_{\mathrm{C}}, \\[1mm] &\mathbf{b}:=(-\mathcal{D}_\ell)^{-1/2}\cdot\mathcal{L}_{\mathrm{A}},&\qquad &{}^*\mathbf{b}:=-(-\mathcal{D}_\ell)^{-1/2}\cdot\mathcal{L}_{\mathrm{A}}, \\[1mm] &\mathbf{c}:=(-\mathcal{D}_\ell)^{-1/2}\cdot\mathcal{L}_{\mathrm{B}},&\qquad &{}^*\mathbf{c}:=-(-\mathcal{D}_\ell)^{-1/2}\cdot\mathcal{L}_{\mathrm{B}}. \end{alignedat} $$

(59)

Using these definitions, we can rewrite composition rules (57) as

$$\begin{array}{l} \text{if }\mathcal{D}_\ell>0\colon\\ \mathbf{a}\circ\mathbf{a}=-\mathrm{Id},\\ \mathbf{b}\circ\mathbf{b}=-\mathcal{M},\\ \mathbf{c}\circ\mathbf{c}=-\mathrm{Id},\\ \mathbf{a}\circ\mathbf{b}=\mathcal{M}^{-1}\circ\mathbf{c},\\ \mathbf{b}\circ\mathbf{a}={}^*\mathbf{c},\\ \mathbf{b}\circ\mathbf{c}=\mathcal{M}\circ\mathbf{a},\\ \mathbf{c}\circ\mathbf{b}={}^*\mathbf{a},\\ \mathbf{c}\circ\mathbf{a}=\mathbf{b},\\ \mathbf{a}\circ\mathbf{c}=\mathcal{M}^{-1}\circ{}^*\mathbf{b}, \end{array}\quad \begin{array}{|@{\;\;\;\;\;\;}l} \text{if }\mathcal{D}_\ell<0\colon\\ \mathbf{a}\circ\mathbf{a}=-\mathrm{Id},\\ \mathbf{b}\circ\mathbf{b}=\mathrm{Id},\\ \mathbf{c}\circ\mathbf{c}=\mathcal{M},\\ \mathbf{a}\circ\mathbf{b}=\mathbf{c},\\ \mathbf{b}\circ\mathbf{a}=\mathcal{M}^{-1}\circ{}^*\mathbf{c},\\ \mathbf{b}\circ\mathbf{c}=\mathcal{M}^{-1}\circ{}^*\mathbf{a},\\ \mathbf{c}\circ\mathbf{b}=\mathbf{a},\\ \mathbf{c}\circ\mathbf{a}=\mathcal{M}\circ\mathbf{b},\\ \mathbf{a}\circ\mathbf{c}={}^*\mathbf{b}. \end{array} $$

(60)

It follows from an analysis of tables (60) that there exist two groups, each containing the subgroup $\mathbb{Z}$ generated by the monodromy operator $\mathcal{M}$ as a normal divisor. These groups will be described in a subsequent paper. In conclusion, we note that the quotient group by this normal divisor is the quaternion group for $\mathcal{D}_\ell>0$ and the dihedral group for $\mathcal{D}_\ell<0$.