Completeness, norm formulae, and extension to the full non-symmetric Fock space for the non-symmetric Yang–Baxter algebra of the quantum nonlinear Schrödinger model

The quantum nonlinear Schrödinger (Lieb–Liniger) model admits, besides the symmetric Bethe wavefunctions, a family of non-symmetric "pre-wavefunctions" obtained from plane waves by Gutkin's propagation operator, a construction based on the degenerate affine Hecke algebra introduced into this setting by Komori and Hikami. Vlaar (2013) built a non-symmetric quantum inverse scattering formalism — operators $a_\mu$, $b_\mu^\pm$, $c_\mu^\pm$, $d_\mu$ analogous to the QISM integral operators, acting on the non-symmetric Fock space $h=\bigoplus_N L^2(J^N)$ of distinguishable particles — and proved Yang–Baxter-type commutation relations for them, but only on the closed span of the pre-wavefunctions. Whether that span is the whole space (completeness), what the norms and inner products of the pre-wavefunctions are, whether the relations extend to the full finite-particle Fock space (Conjecture 5.3.6), and whether the thermodynamic-limit operators are bounded (Conjecture 5.4.2) were left open. The problem of establishing the analytic (Hilbert-space) foundations of the non-symmetric Yang–Baxter algebra at the standard of the classical symmetric theory is therefore open.

Background

The quantum nonlinear Schrödinger (QNLS) model — also called the Lieb–Liniger model — describes $N$ one-dimensional particles with pairwise delta-function interaction, with Hamiltonian $H_\gamma=-\sum_j\partial_{x_j}^2+2\gamma\sum_{j<k}\delta(x_j-x_k)$ for real coupling $\gamma$ (repulsive for $\gamma>0$), on the line $\mathbb{R}$ or the interval $J=[-L/2,L/2]$. It is a paradigmatic exactly solved many-body system: the coordinate Bethe ansatz writes the physical (bosonic, i.e. permutation-symmetric) eigenfunctions $\Psi_\lambda$ as sums of plane waves whose coefficients are fixed by two-body scattering, and on the interval the allowed momenta $\lambda=(\lambda_1,\dots,\lambda_N)$ are quantized by the Bethe ansatz equations (BAEs). The quantum inverse scattering method (QISM, or algebraic Bethe ansatz) reorganizes this algebraically: the entries of a $2\times2$ monodromy matrix (the matrix of QISM scattering-data operators) — realized for the continuum QNLS model by explicit integral operators $A_\mu,B_\mu,C_\mu,D_\mu$ ($\mu\in\mathbb{C}$) on the bosonic Fock space $H=\bigoplus_N H_N$ with $H_N=L^2(J^N)^{S_N}$ — satisfy the Yang–Baxter (RTT) commutation relations encoded in the Yang–Baxter equation, their trace (the transfer matrix) commutes with all conserved charges, and $B_\mu$ generates the Bethe wavefunctions recursively from the vacuum. The analytic backbone of this picture in the symmetric sector is classical: Gaudin proved closure and orthogonality of the scattering states on the line and conjectured the determinantal norm formula that Korepin proved by QISM techniques, while Dorlas proved completeness and orthogonality of the Bethe wavefunctions on the interval.

A parallel algebraic route uses the degenerate affine Hecke algebra (dAHA): the associative algebra generated by the symmetric group $S_N$ together with commuting polynomial multiplication and partial-derivative operators with cross relations, introduced in the mid-1980s as the rational degeneration of the affine Hecke algebra, and represented on functions by Dunkl-type differential-reflection operators. Gutkin's propagation operator (or intertwiner) $P_\gamma$ — a finite sum of weighted permutations acting chamber-wise on $\mathbb{R}^N$ (i.e. separately on each region where a fixed ordering $x_{\sigma(1)}<\cdots<x_{\sigma(N)}$ of the particle coordinates holds) — intertwines the free and interacting representations of the dAHA. Komori and Hikami used it to construct non-symmetric (not $S_N$-invariant) solutions $\psi_\lambda=P_\gamma e^{i(\lambda,\cdot)}$ of the QNLS eigenvalue problem; symmetrizing $\psi_\lambda$ recovers the Bethe wavefunction $\Psi_\lambda$, so $\psi_\lambda$ is called the pre-wavefunction. These are the rational delta-interaction analogues of the non-symmetric Jack polynomials (a celebrated family of non-symmetric multivariate eigenfunctions) of the trigonometric Calogero–Sutherland model, the Lieb–Liniger model's trigonometric cousin.

Vlaar's thesis and paper connect the two routes by building a non-symmetric QISM on the non-symmetric Fock space $h=\bigoplus_N h_N$ with $h_N=L^2(J^N)$ (no symmetrization: the Hilbert space of $N$ distinguishable particles) and its dense finite-particle subspace $h_{\mathrm{fin}}$. There he defines non-symmetric analogues $a_\mu,b_\mu^\pm,c_\mu^\pm,d_\mu$ of the QISM integral operators; the $b_\mu^\pm$ act as non-symmetric creation operators generating the pre-wavefunctions recursively from the vacuum, and Yang–Baxter-type commutation relations among $a_\mu,b_\mu^\pm,c_\mu^\pm,d_\mu$ are proved (Thm. 5.3.5 of the thesis) — but only on $z_{\mathrm{fin}}=h_{\mathrm{fin}}\cap\bigoplus_{N\ge0}z_N$, where $z_N$ is the closure in $h_N$ of the span of the pre-wavefunctions. Whether $z_N$ equals $h_N$ (completeness) is unknown: the only invertibility result for $P_\gamma$, due to Emsiz–Opdam–Stokman, is a bijection between real-analytic functions and functions with matching derivative-jump conditions across the diagonals $x_j=x_k$, far from an $L^2$ statement. Likewise no norm or inner-product formulae are known for the $\psi_\lambda$, and the boundedness of the thermodynamic-limit operators of the non-symmetric formalism is unproven. The problem below asks to close these gaps, completing for the non-symmetric theory what Gaudin, Korepin and Dorlas proved for the symmetric one.

Problem Statement

For the quantum nonlinear Schrödinger model with real coupling $\gamma$ (in particular the repulsive case $\gamma>0$) on the interval $J=[-L/2,L/2]$, with the line $\mathbb{R}$ entering through the thermodynamic limit $L\to\infty$, establish the analytic (Hilbert-space) foundations of the non-symmetric Yang–Baxter algebra. Concretely, with $\psi_\lambda$ the non-symmetric pre-wavefunctions, $h_N=L^2(J^N)$ the non-symmetric $N$-particle sector on the interval, $h_N(\mathbb{R})=L^2(\mathbb{R}^N)$ its counterpart on the line, $h_{\mathrm{fin}}$ the dense finite-particle subspace of the non-symmetric Fock space, $z_N$ the closure of the span of the pre-wavefunctions in $h_N$, and $a_\mu,b_\mu^\pm,c_\mu^\pm,d_\mu$ the non-symmetric integral operators of Vlaar's formalism:

(1) Completeness. Prove or disprove that $z_N=h_N$ for all $N$, i.e. that the pre-wavefunctions $\{\psi_\lambda:\lambda\in\mathbb{C}^N\}$ are complete in $h_N=L^2(J^N)$, and decide whether the smaller Bethe-ansatz-equation-constrained family $\{\psi_\lambda:\lambda\in\mathbb{R}_+^N\ \text{satisfying the BAEs}\}$ is complete in $h_N([-L/2,L/2])$. On the line, where the $\psi_\lambda$ are not square-integrable, settle the corresponding spanning question as posed in the source: whether the $\{\psi_\lambda:\lambda\in\mathbb{C}^N\}$ span the space $C_B^\infty(\mathbb{R}^N)$ of continuous functions on $\mathbb{R}^N$ that are smooth on each chamber $x_{\sigma(1)}<\cdots<x_{\sigma(N)}$; an $L^2(\mathbb{R}^N)$ version of line completeness may be formulated through wave-packet superpositions or the thermodynamic limit. A route indicated in the source is bounded invertibility of Gutkin's propagation operator $P_\gamma^{(N)}$ on a suitable $L^2$ domain, combined with completeness of the plane waves $e^{i(\lambda,\cdot)}$ in $h_N$.

(2) Norms and inner products. In the interval setting $h_N=L^2(J^N)$, derive explicit (for example determinantal) formulae for $\|\psi_\lambda\|_{L^2(J^N)}$ and for the inner products $(\psi_\lambda,\psi_\mu)$, and settle the orthogonality pattern suggested by the source's $N=2$ calculations, namely that $(\psi_\lambda,\psi_\mu)=0$ only if $\|\lambda\|=\|\mu\|$ (Euclidean norm on momentum space) when both $\lambda$ and $\mu$ satisfy the Bethe ansatz equations.

(3) Operator-algebra extension (Conjecture 5.3.6 of the thesis). Prove that the non-symmetric Yang–Baxter commutation relations of Thm. 5.3.5 — established only on $z_{\mathrm{fin}}$ — hold as identities in $\mathrm{End}(h_{\mathrm{fin}})$, and hence extend to the full non-symmetric Fock space; either via (1) and density, or by establishing commutation relations for the coefficient operators $b^\pm_{\lambda;n}$ appearing in the expansions of the $b^\pm_\lambda$ and summing the resulting series.

(4) Thermodynamic limit (Conjecture 5.4.2 of the thesis). Prove that $\lim_{L\to\infty}e^{i\mu L/2}a_\mu$ for $\operatorname{Im}\mu>0$ and $\lim_{L\to\infty}e^{-i\mu L/2}d_\mu$ for $\operatorname{Im}\mu<0$ (the pointwise limits established on pre-wavefunctions in Prop. 5.4.1) extend to operators on $h_N(\mathbb{R})$, and that they are bounded on $h_N(\mathbb{R})$ whenever $\operatorname{Im}\mu\neq0$.

These four deliverables share one objective: to promote the non-symmetric quantum inverse scattering formalism from identities on the pre-wavefunction span to a rigorous operator algebra on the full non-symmetric Fock space, at the standard of the classical symmetric theory (Gaudin's closure, Korepin's norm formula, Dorlas's completeness). Any mathematically rigorous route is admissible; a resolution that leaves one of (1)–(4) untouched is partial.

References

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