Determinantal formulas for zonal spherical functions on hyperboloids (Q5929684)

Let \(X\) be a hyperboloid \(x^2_1+\cdots +x^2_p-x^1_{p+1}-\cdots- x^2_{p+q}=1\) in \(\mathbb{R}^{p+q}\). It is a pseudo-Riemannian symmetric space of rank one: \(X=G/H\) with \(G=SO_0(p,q)\) and \(H=SO_0 (p-1,q)\). Introduce on \(X\) polar coordinates associated with the subgroup \(K=SO(p)\times SO(q)\) (maximal compact): \(x=\text{(cosh} t\cdot u,\sinh t\cdot v)\), where \(u,v\) range over the unit spheres in \(\mathbb{R}^p\) and \(\mathbb{R}^q\) respectively. Let \(\varphi\) be a \(K\)-invariant normalized eigenfunction of the Laplace-Beltrami operator \(\Delta\) on \(X\): \(\Delta\varphi =(z^2-\rho^2) \varphi\), where \(\rho=(p+ q-2)/2\). Then it depends on \(t\) only (and on the spectral parameter \(z)\). Therefore, it is an eigenfunction of the radial part of the Laplace-Beltrami operator: \(((d/dt)^2 +((p-1) \tanh t+(q-1)\coth T)) \varphi=(z^2- \rho^2)\varphi\) and can be expressed in terms of the Gauss hypergeometric function: \(\varphi(t,z)= F(( \rho+ z)/2\), \((\rho-z)/2\); \(q/2;- \sinh^2t)\). Denote \(m=(p-3)/2\), \(n=(q-3)/2\). The main result of the paper under review is a determinantal formula for \(\varphi\) in the case when \(m\) and \(n\) are nonnegative integers. Namely, let \(m> n\). Then \(\varphi\) is up to a factor the even part (in \(t)\) of the function \(\Phi (t,z)= e^{(z-\rho)t} b_{m,n}(t,z)/a_{m+1, n+1}(t)\), where \(a_{m,n}(t)= \det (\mathbf{1}+{\mathbf A}^{m,n}(t))\), \(b_{m,n}(t,z)= \det(\mathbf{1}+ {\mathbf B}^{m,n} (t,z))\) and \({\mathbf A}^{m,n}(t)\) and \({\mathbf B}^{m,n}(t,z)\) are some explicitly written square matrices of order \(m\). The entries of these matrices are labelled by \(j,k\in I^{m,n}\), some set of positive integers of cardinality \(m\), and are functions of the form \((j+k)^{-1}\alpha_j e^{-2jt}\) and \((j+k)^{-1} \alpha_j e^{-2jt} (z+j)/(z-j)\) respectively, here \(\alpha_j\) are some numbers. It turns out that \(a_{m,n}(t)\) can be evaluated in a very simple form: it amounts to \((1+e^{-2t})^{m(m+1)/2} (1-e^{-2t})^{n(n+1)/2}\). This fact lies on the basis of the proof of the main formula. Namely, the wave function \(\Psi(t,z)= e^{zt}b_{m,n} (t,z)/a_{m,n}(t)\) solves the Schrödinger equation \(((d/dt)^2 +u(t)-z^2) \psi=0\) with the potential \(u(t)=2 (d/dt)^2\log a_{m,n} (t)=m(m+1) (\cosh t)^{-2}- n(n+1)(\sinh t)^{-2}\). This equation turns out to be just the equation which is obtained from the equation for \(\varphi\) mentioned above by passing to the function \(\psi=(\cosh t)^{m+1}(\sinh t)^{n+1} \varphi\).