Linear systems, spectral curves and determinants

Abstract Let $$(-A,B,C)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> be a continuous-time linear system with state space a separable complex Hilbert space H , where $$-A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> generates a strongly continuous contraction semigroup $$(e^{-tA})_{t\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>t</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> on H , and $$\phi (t)=Ce^{-tA}B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>t</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:msup> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> is the impulse response function. Associated with such a system is a Hankel integral operator $$\Gamma _\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> acting on $$L^2((0, \infty );\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>;</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and a Schrödinger operator whose potential is found via a Fredholm determinant by the Faddeev–Dyson formula. Fredholm determinants of products of Hankel operators also play an important role in Tracy and Widom’s theory of matrix models and asymptotic eigenvalue distributions of random matrices. This paper provides formulas for the Fredholm determinants that arise thus, and determines consequent properties of the associated differential operators. We prove a spectral theorem for self-adjoint linear systems that have scalar input and output: the entries of Kodaira’s characteristic matrix are given explicitly with formulas involving the infinitesimal Darboux addition for $$(-A,B,C)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Under suitable conditions on $$(-A,B,C)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> we give an explicit version of Burchnall–Chaundy’s theorem, showing that the algebra generated by an associated family of differential operators is isomorphic to an algebra of functions on a particular hyperelliptic curve.