<p>In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$J_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> for quantum Hamiltonians of the hyperbolic relativistic <i>N</i>-particle Calogero–Moser system. For generic coupling values, they are non-elementary functions that in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> case essentially amount to a ‘relativistic’ generalisation of the conical function specialisation of the Gauss hypergeometric function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${}_2F_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>2</mn> <mrow /> </mmultiscripts> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\psi _N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ψ</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> of Baker–Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$J_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> essentially amounts to the antisymmetrisation of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\psi _N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ψ</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> and, as a byproduct, we obtain a recursive construction of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\psi _N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ψ</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> in terms of an iterated residue formula.</p>

Baker–Akhiezer Specialisation of Joint Eigenfunctions for Hyperbolic Relativistic Calogero–Moser Hamiltonians

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2026年3月6日

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Martin Hallnäs

摘要

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_N$ $J_{N}$ for quantum Hamiltonians of the hyperbolic relativistic N-particle Calogero–Moser system. For generic coupling values, they are non-elementary functions that in the $N=2$ $N = 2$ case essentially amount to a ‘relativistic’ generalisation of the conical function specialisation of the Gauss hypergeometric function ${}_2F_1$ $_{2} F_{1}$ . In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions $\psi _N$ $ψ_{N}$ of Baker–Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that $J_N$ $J_{N}$ essentially amounts to the antisymmetrisation of $\psi _N$ $ψ_{N}$ and, as a byproduct, we obtain a recursive construction of $\psi _N$ $ψ_{N}$ in terms of an iterated residue formula.