Spectrally meromorphic operators and non-linear systems

We consider meromorphic linear ordinary differential operators acting on functions of a real variable $x\in \mathbb R$ and having the form $L=\partial^n_x+\sum_{n\geq i\geq 2} a_{n-i}\partial_x^{n-i}$ with coefficients meromorphic near the axis $\mathbb R\subset \mathbb C$. Such an operator is said to be spectrally meromorphic (or s-meromorphic) at a pole $x_j$ if all the solutions of the equation $L\psi=\alpha\psi$ are meromorphic near $x_j$ for all $\alpha\in \mathbb C$. We shall study operators that are s-meromorphic at all their poles $x_j\in \mathbb R$.

We introduce the necessary function spaces. Given a set of vector subspaces $R_j$ of the form $\sum_{-r_j\leq i <0 }b_i y^i$ in the space of all Laurent polynomials, we define the space $F= F_{(\{x_j;R_j\})}$ of all functions that are meromorphic near the axis $x\in \mathbb R$, with possible poles at a fixed discrete set of points $x_j\in \mathbb R$ and with the principal parts of their Laurent expansions with respect to $y=x-x_j$ belonging to the fixed subspaces $R_j$. The only restrictions on the positive Laurent coefficients at the poles $x_j$ follow from the requirement that the products of functions in $F$ do not have terms of the form $y^{-1}$.

Lemma 1.  The formula $\langle f,g\rangle=\int f(x)\bar{g}(\bar{x})\,dx$ gives a well-defined scalar product for all compactly supported functions $f(x),g(x)$, $x\in \mathbb R$, in the space $F$, where the integral is taken over the axis $x\in \mathbb R$ outside a neighbourhood of the singularities and over a contour going around the singular points and close to them. This scalar product is indefinite, and each singular point $x_j$ gives $p_j$ negative squares (independently of the other points $x_k$), where $p_j$ is the dimension of the subspace $R_j$ of the Laurent principal parts near $x_j$.

The proof follows easily from the fact that, by the definition of the space $F$, a product of functions does not contain powers $-1$ near any pole. According to the BChK construction (Burchnall–Chandy–Krichever [1]), a Riemann surface $\Gamma$ endowed with a distinguished point $P$, a local parameter $z=1/\lambda$, a divisor $D$ of degree $g$ (the genus), and a meromorphic function $\alpha$ with a pole only at $P$ of order $k$ determines a $k$th-order ordinary differential operator $L\psi=\alpha\psi$ which is algebraic of rank 1. This operator is easily seen to be s-meromorphic. It is formally self-adjoint (symmetric) if the following condition $\mathbf{SA}$ holds: we are given an anti-involution $\tau\colon\Gamma\to \Gamma$ with $\tau(P)=P$ such that $\tau(D)+D\sim K+2P$, $\tau(\lambda)=-\bar\lambda$.

The singularities of the operator $L$ determine a class of functions $F=F^L$ (see above). In the periodic case we construct the space $F_{\kappa}^L\subset F^L$ of functions with $f(x+T)=\kappa f(x)$, $|\kappa|=1$. The number of negative squares in the spaces $F^L_{\kappa}$ is finite. We recall that the differential of the quasi-momentum is a meromorphic differential $dp$ having only a second-order pole at $P$ with $dp\sim d\lambda$ and such that the integrals $\oint dp$ over all cycles are purely real. Since $\tau\,dp = \overline{dp}$, we choose $\operatorname{Im} p=p_I$ to be a single-valued real function with $\tau(p_I)=-p_I$. The level $p_I=0$ is called the canonical contour $C_0$.

The projection $\alpha(C_0)$ of the contour $C_0$ on the complex plane $\mathbb C$ coincides with the spectrum of $L$ on the whole axis $x\in \mathbb R$ in the space $F^{L}$.

Theorem 2.  The scalar product in the space $F^L$ for an algebraic operator $L$ of rank 1 satisfying the condition $\mathbf{SA}$ is well defined and indefinite. The operator $L$ is self-adjoint with respect to this scalar product (see above). Its spectrum coincides with the set $\alpha(C_0)$, where $C_0$ is the canonical contour. The scalar product is given by the integral over the contour $C_0$ with the spectral measure $\Omega$. If the basis of eigenfunctions on $C_0$ is complete in the space $F^L_{\kappa}$, then the number of negative squares for the scalar product in the spaces $F^L_{\kappa}$ is determined by the sign of the spectral measure $\Omega$ at the corresponding points of $C_0$ and is an integral of the Gelfand–Dickey systems $\dot L=[L,A]$.

To prove the theorem, we consider the Baker–Akhiezer function $\psi(\lambda,x)\sim e^{\lambda x}$ (near $P$) with pole divisor $D=(\gamma_1,\dotsc,\gamma_g)$ and the dual 1-form $\psi^*(\mu, x)\,d\mu\sim e^{-\mu x}\,d\mu$ (near $P$) with zeros $D=(\gamma_1,\dotsc,\gamma_g)$. In the language of scalar Baker– Akhiezer functions we have $\psi^*(\mu,x)\,d\mu=\psi^+(\mu,x)\Omega$, where $\Omega=(1+o(1))\,d\mu$ near $P$, with zeros $D+\tau(D)$, $\tau(D)+D\sim K+2P$. Here $K$ is the canonical divisor of differential forms and $\Omega$ is the spectral measure on the contour $C_0$. We have $L^*\psi^*=\mu\psi^*$, where $L^*$ is the conjugate of $L$. This fact has perhaps not been mentioned explicitly in the literature. A Cauchy-type kernel on $\Gamma$ was studied in [2]: $\omega(\lambda,\mu,x)\sim -d\mu/(\lambda-\mu)+O(1)$ near the diagonal $\lambda=\mu$, $\omega\sim e^{\lambda x}o(1)$ as $\lambda\to P$, and $\omega\to e^{-\mu x}o(1)\,d\mu$ as $\mu\to P$. It was found that $\partial_x\omega=-\psi(\lambda,x)\psi^*(\mu,x)\,d\mu$. This equality plays a key role here. It follows that the residue with respect to $x$ is equal to zero for any product of eigenfunctions of the self-adjoint operator $L$ near the singularity. Hence the scalar product (see above) is well defined. For any given $\kappa$, the number of negative squares is an invariant of the Riemann surface, which is preserved under evolution in time: the equation $e^{ip(z)t}=\kappa$ on the contour $C_0$ gives a discrete set of points $z_s\in C_0$. The number of points at which the quantity $d\Omega|_{z_s}\in \mathbb R$ is negative, is finite and gives the number of negative squares for the scalar product in $F^L_{\kappa}$. The whole space $F^L$ is a direct integral of the spaces $F^L_{\kappa}$.

Example. [3] For $k=2$ we have $L=-\partial_x^2+u$. Suppose that $u\sim r_j(r_j+1)/(x- x_j)^2+\operatorname{Reg}$. For $\mathrm s$-meromorphy, the regular part of the potential must have zero odd derivatives at $x_j$ up to and including order $2r_j-1$. All the spaces $R_j$ are of the form $b_1y^{-r_j}+b_2y^{-r_j+2}+\dotsb+b_{r_j/2}y^{-2}$ for even $r_j\geq 2$, or $b_1y^{-r_j}+b_2y^{-r_j+2}+\dotsb+b_{(r_j+1)/2}y^{-1}$ for odd $r_j\geq 1$. Each pole gives $[(r_j+1)/2]$ negative squares. The positive terms of the Laurent expansion of functions $f\in F$ at the points $x_j$ have zero coefficients of the odd powers $y^{2i-1}$ with $i\leq r_j/2$ in the first case, and of the even powers $y^{2i}$ with $i\leq (r_j+1)/2$ in the second case. One of the finite components of the spectrum $\alpha(C_0)$ is complex already for the genus $g=1$ and the real elliptic potential corresponding to the rhombic lattice. The completeness of the eigenfunctions is proved in [3]. Hence the number of negative squares is preserved under the action of the KdV and the higher KdVs.

The following are to be proved. 1. If an operator is $\mathrm s$-meromorphic, then so is its conjugate operator. 2. All the singularities $R_j$ of $\mathrm s$-meromorphic operators are realized by algebraic ones. Their classification is an open problem. 3. The completeness theorem for symmetric algebraic operators.