For the Lieb-Liniger model of N repulsive delta-interacting bosons in a one-dimensional box (hard-wall, i.e. Dirichlet, boundary conditions), the coordinate Bethe-ansatz eigenfunctions are superpositions over both permutations and sign reversals of the quasi-momenta, and no rigorous formula for their normalization is known. In a 2022 study of classical and quantum metrology of this model, Baak and Fischer proposed an explicit determinant formula for the squared norm of the hard-wall Bethe eigenstates (their Eq. (A.10)): a prefactor 2N2^N, times the product of the factors 1+c2/(kj−kl)21+c^2/(k_j-k_l)^2 and 1+c2/(kj+kl)21+c^2/(k_j+k_l)^2 over all pairs j<lj<l, times the determinant of the Hessian of the hard-wall Yang-Yang-type function whose stationarity conditions are the Bethe equations. They adopted the formula 'without mathematical rigor being used', validating it only against direct numerical integration for N <= 3. The problem is to prove this identity rigorously for all particle numbers, couplings, and Bethe root sets, or to exhibit a certified counterexample - the hard-wall analogue of the norm conjecture that Gaudin stated and Korepin proved for the periodic Lieb-Liniger gas in 1982. No proof or refutation of this formula is known in the literature: norm theorems exist for periodic models and for open spin chains with parametric integrable boundaries, while for the nonlinear Schrödinger equation on the interval only the spectrum has been treated. The problem is open.

Background

The Lieb-Liniger model describes NN identical bosons in one space dimension interacting through a repulsive pointlike two-body force. In units ℏ=2m=1\hbar=2m=1 its Hamiltonian is H=−∑j=1N∂2∂xj2+2c∑j<lδ(xj−xl),c>0,H=-\sum_{j=1}^{N}\frac{\partial^2}{\partial x_j^2}+2c\sum_{j<l}\delta(x_j-x_l),\qquad c>0, where cc is the interaction strength. Lieb and Liniger (1963) diagonalized it on a ring (periodic boundary conditions) by the coordinate Bethe ansatz: in each ordering sector xP1<⋯<xPNx_{P_1}<\cdots<x_{P_N} of configuration space the eigenfunction is a superposition of plane waves ei∑jkPjxje^{i\sum_j k_{P_j}x_j} whose quasi-momenta (Bethe rapidities) k1,…,kNk_1,\dots,k_N are fixed by a system of algebraic Bethe equations encoding repeated two-body scattering. A basic structural fact of the periodic model, conjectured by Gaudin and proved by Korepin (1982), is that the squared norm of a Bethe eigenstate - a priori a double sum over N!N! permutations - collapses to a single determinant of the Gaudin matrix, the N×NN\times N matrix of second derivatives of the Yang-Yang function (the generating function of the logarithmic Bethe equations). This 'Gaudin determinant' is the normalization constant of every Bethe eigenstate and underlies the exact correlation-function and form-factor program for the periodic gas.

For bosons confined to a box (hard-wall, i.e. Dirichlet boundary conditions Ψ(0,⋅)=Ψ(L,⋅)=0\Psi(0,\cdot)=\Psi(L,\cdot)=0), the standard geometry of ultracold-gas box-trap experiments, the coordinate Bethe ansatz is richer: in the ordered sector the wavefunction is a superposition over both permutations PP and sign reversals ϵj=±1\epsilon_j=\pm1 of the quasi-momenta, and the Bethe equations become e2ikjL=∏l≠jkj+kl+ickj+kl−ickj−kl+ickj−kl−ice^{2ik_jL}=\prod_{l\neq j}\frac{k_j+k_l+ic}{k_j+k_l-ic}\frac{k_j-k_l+ic}{k_j-k_l-ic} with 0<k1<⋯<kN0<k_1<\cdots<k_N. The Bethe equations and thermodynamics of the hard-wall model go back to Gaudin (1971) and were developed by Batchelor, Guan, Oelkers and Lee (2005) and Oelkers et al. (2006), but no formula for the norm of its Bethe eigenstates was known.

The question arose concretely in a metrological study. Baak and Fischer (2022) computed the quantum Fisher information F\mathfrak F (the information a quantum state carries about a parameter, here the coupling cc, which bounds the precision of any estimation protocol) and the classical Fisher information FF of position measurements for Lieb-Liniger eigenstates in ring and box geometries; the box calculation requires normalized hard-wall eigenfunctions. By 'modifying Gaudin's conjecture' - applying the periodic Gaudin-determinant recipe to the hard-wall Bethe equations - they proposed the norm formula published as Eq. (A.10): with BB the hard-wall analogue of the Yang-Yang function and H(B)\mathbf H(B) its Hessian, the squared norm of the hard-wall Bethe eigenfunction equals 2N∏j<l[1+c2/(kj−kl)2][1+c2/(kj+kl)2]det⁡H(B)2^N\prod_{j<l}[1+c^2/(k_j-k_l)^2][1+c^2/(k_j+k_l)^2]\det\mathbf H(B). They used it for all reported box-geometry results, checked it against direct numerical integration of ∣ψ∣2|\psi|^2 for small particle numbers (N≤3N\le3; the sign-permutation double sums cost ∼22N(N!)2\sim2^{2N}(N!)^2 operations), and stated explicitly that they take it as exact 'without mathematical rigor being used'. Whether the formula is true is the hard-wall analogue of the question Korepin settled for the periodic model.

Nearby rigorous results do not settle it. Norms and scalar products are proven for periodic quantum integrable models (Korepin's theorem; Slavnov's determinant formula for scalar products), and for open spin chains with integrable parametric boundaries (Wang for the boundary XXX chain, i.e. the isotropic Heisenberg spin-1/21/2 chain with boundary fields; Kitanine, Maillet, Niccoli and Terras via separation of variables, a reformulation of the algebraic Bethe ansatz in which eigenstates and their scalar products become explicit), and the spectrum of the generalized nonlinear Schrödinger model on the interval has been derived. However, a hard wall is an energy-independent reflection with reflection phase exactly −1-1, a degenerate case outside the parametric boundary families covered by those theorems, and no norm theorem for the Dirichlet-interval Lieb-Liniger model exists. Whether the Baak-Fischer formula is a theorem-in-waiting or fails at some particle number or root configuration is unknown.

Problem Statement

Prove or disprove the Baak-Fischer norm formula for Bethe eigenstates of the Lieb-Liniger model with hard-wall (Dirichlet) boundary conditions. Consider, in units ℏ=2m=1\hbar=2m=1, the Hamiltonian H=−∑j=1N∂2∂xj2+2c∑j<lδ(xj−xl)H=-\sum_{j=1}^{N}\frac{\partial^2}{\partial x_j^2}+2c\sum_{j<l}\delta(x_j-x_l) with c>0c>0, acting on symmetric wavefunctions on [0,L]N[0,L]^N with Ψ(0,x2,…,xN)=Ψ(x1,…,xN−1,L)=0\Psi(0,x_2,\dots,x_N)=\Psi(x_1,\dots,x_{N-1},L)=0 (equivalently, Ψ\Psi vanishes whenever any particle coordinate equals 00 or LL). In the ordered sector 0≤x1≤⋯≤xN≤L0\le x_1\le\cdots\le x_N\le L the unnormalized coordinate Bethe eigenfunction is ψ~(x1,…,xN)=∑ϵ1,…,ϵN=±1 ∑Pϵ1ϵ2⋯ϵN A(ϵ1kP1,…,ϵNkPN) ei∑j=1NϵjkPjxj,\tilde\psi(x_1,\dots,x_N)=\sum_{\epsilon_1,\dots,\epsilon_N=\pm1}\ \sum_P \epsilon_1\epsilon_2\cdots\epsilon_N\,A(\epsilon_1k_{P_1},\dots,\epsilon_Nk_{P_N})\,e^{i\sum_{j=1}^{N}\epsilon_jk_{P_j}x_j}, A(q1,…,qN)=∏j<l(1−icqj+ql)(1+icqj−ql),A(q_1,\dots,q_N)=\prod_{j<l}\Big(1-\frac{ic}{q_j+q_l}\Big)\Big(1+\frac{ic}{q_j-q_l}\Big), where PP runs over permutations, ϵj=±1\epsilon_j=\pm1, and 0<k1<⋯<kN0<k_1<\cdots<k_N solve the hard-wall Bethe equations e2ikjL=∏l≠jkj+kl+ickj+kl−ic⋅kj−kl+ickj−kl−ic,j=1,…,N.e^{2ik_jL}=\prod_{l\neq j}\frac{k_j+k_l+ic}{k_j+k_l-ic}\cdot\frac{k_j-k_l+ic}{k_j-k_l-ic},\qquad j=1,\dots,N. Let I1<I2<⋯<INI_1<I_2<\cdots<I_N be the positive integers labelling the state, and define B(k1,…,kN)=L2∑j=1Nkj2−π∑j=1NIjkj+12∑j=1N∑l≠j[∫0kj−kldu tan⁡−1(uc)+∫0kj+kldu tan⁡−1(uc)],B(k_1,\dots,k_N)=\frac{L}{2}\sum_{j=1}^{N}k_j^2-\pi\sum_{j=1}^{N}I_jk_j+\frac{1}{2}\sum_{j=1}^{N}\sum_{l\neq j}\Big[\int_0^{k_j-k_l}du\,\tan^{-1}\Big(\frac{u}{c}\Big)+\int_0^{k_j+k_l}du\,\tan^{-1}\Big(\frac{u}{c}\Big)\Big], whose stationarity conditions are the logarithmic Bethe equations, and let H(B)\mathbf H(B) be its Hessian with entries [H(B)]ij=δij(L+∑l≠i[c(ki−kl)2+c2+c(ki+kl)2+c2])+(1−δij)(−c(ki−kj)2+c2+c(ki+kj)2+c2).[\mathbf H(B)]_{ij}=\delta_{ij}\Big(L+\sum_{l\neq i}\Big[\frac{c}{(k_i-k_l)^2+c^2}+\frac{c}{(k_i+k_l)^2+c^2}\Big]\Big)+(1-\delta_{ij})\Big(-\frac{c}{(k_i-k_j)^2+c^2}+\frac{c}{(k_i+k_j)^2+c^2}\Big). Conjecture (Baak-Fischer, Eq. (A.10) of arXiv:2209.06193): the squared norm of ψ~\tilde\psi, i.e. N2=∫0LdxN∫0xNdxN−1⋯∫0x2dx1 ∣ψ~∣2\mathcal N^2=\int_0^Ldx_N\int_0^{x_N}dx_{N-1}\cdots\int_0^{x_2}dx_1\,|\tilde\psi|^2, equals N2=2N∏j<l[1+c2(kj−kl)2][1+c2(kj+kl)2] det⁡H(B).\mathcal N^{2}=2^{N}\prod_{j<l}\Big[1+\frac{c^2}{(k_j-k_l)^2}\Big]\Big[1+\frac{c^2}{(k_j+k_l)^2}\Big]\,\det\mathbf H(B). The task: establish this identity rigorously for all particle numbers N≥1N\ge1, all repulsive couplings c>0c>0, all box lengths L>0L>0, and all Bethe root sets solving the hard-wall Bethe equations (ground and excited states) - or exhibit a certified counterexample to it. A proof may proceed by any route (direct evaluation of the sign-permutation double sum, determinant calculus, a proven reduction to existing norm or scalar-product theorems for open-boundary integrable models, separation of variables for the interval problem, etc.), provided it covers the full stated domain; verification only at small NN or special couplings does not suffice. The attractive case c<0c<0 and other (e.g. twisted or mixed) boundary conditions are natural extensions, but settling the repulsive Dirichlet case above already resolves the problem.

References

  1. J.-G. Baak and U. R. Fischer, 'Classical and quantum metrology of the Lieb-Liniger model', Phys. Rev. A 106, 062442 (2022), DOI: 10.1103/PhysRevA.106.062442, arXiv:2209.06193, https://arxiv.org/abs/2209.06193
  2. V. E. Korepin, 'Calculation of norms of Bethe wave functions', Commun. Math. Phys. 86, 391-418 (1982), DOI: 10.1007/BF01212176, https://doi.org/10.1007/BF01212176
  3. M. Gaudin, 'The Bethe Wavefunction' (ed. J.-S. Caux), Cambridge University Press (2014), DOI: 10.1017/CBO9781107053885, https://doi.org/10.1017/CBO9781107053885
  4. M. Gaudin, 'Boundary Energy of a Bose Gas in One Dimension', Phys. Rev. A 4, 386-394 (1971), DOI: 10.1103/PhysRevA.4.386, https://doi.org/10.1103/PhysRevA.4.386
  5. E. H. Lieb and W. Liniger, 'Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State', Phys. Rev. 130, 1605-1616 (1963), DOI: 10.1103/PhysRev.130.1605, https://doi.org/10.1103/PhysRev.130.1605
  6. M. T. Batchelor, X.-W. Guan, N. Oelkers, and C. Lee, 'The 1D interacting Bose gas in a hard wall box', J. Phys. A: Math. Gen. 38, 7787-7806 (2005), DOI: 10.1088/0305-4470/38/36/001, https://doi.org/10.1088/0305-4470/38/36/001
  7. N. Oelkers, M. T. Batchelor, M. Bortz, and X.-W. Guan, 'Bethe ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions', J. Phys. A: Math. Gen. 39, 1073-1098 (2006), DOI: 10.1088/0305-4470/39/5/005, https://doi.org/10.1088/0305-4470/39/5/005
  8. N. A. Slavnov, 'Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz', Theor. Math. Phys. 79, 502-508 (1989), DOI: 10.1007/BF01016531, https://doi.org/10.1007/BF01016531
  9. Yan-Shen Wang, 'The scalar products and the norm of Bethe eigenstates for the boundary XXX Heisenberg spin-1/2 finite chain', Nucl. Phys. B 622, 633-649 (2002), DOI: 10.1016/S0550-3213(01)00610-1, https://doi.org/10.1016/S0550-3213(01)00610-1
  10. N. Kitanine, J.-M. Maillet, G. Niccoli, and V. Terras, 'The open XXX spin chain in the SoV framework: scalar product of separate states', J. Phys. A: Math. Theor. 50, 224001 (2017), DOI: 10.1088/1751-8121/aa6cc9, arXiv:1606.06917, https://arxiv.org/abs/1606.06917
  11. A. Doikou, D. Fioravanti, and F. Ravanini, 'The generalized non-linear Schrödinger model on the interval', Nucl. Phys. B 790, 465-492 (2008), DOI: 10.1016/j.nuclphysb.2007.08.007, arXiv:0706.1515, https://arxiv.org/abs/0706.1515
  12. Hui-Huang Chen, 'Exact overlaps in the Lieb-Liniger model from coordinate Bethe ansatz', Phys. Lett. B 808, 135631 (2020), DOI: 10.1016/j.physletb.2020.135631, https://doi.org/10.1016/j.physletb.2020.135631
  13. J.-G. Baak and U. R. Fischer, 'Self-consistent many-body metrology', Phys. Rev. Lett. 132, 240803 (2024), DOI: 10.1103/PhysRevLett.132.240803, arXiv:2310.02580, https://arxiv.org/abs/2310.02580
  14. Y.-J. Baak and U. R. Fischer, 'Quantum Many-Body Metrology of Rotation Sensing with Strong Interactions', arXiv:2608.04082 (2026), https://arxiv.org/abs/2608.04082
  15. V. Montenegro, C. Mukhopadhyay, R. Yousefjani, S. Sarkar, U. Mishra, M. G. A. Paris, and A. Bayat, 'Review: Quantum metrology and sensing with many-body systems', Physics Reports 1134, 1-62 (2025), DOI: 10.1016/j.physrep.2025.05.005, arXiv:2408.15323, https://arxiv.org/abs/2408.15323
  16. K. D. Agarwal, S. Mondal, A. Sahoo, D. Rakshit, A. Sen(De), and U. Sen, 'Quantum sensing with ultracold simulators in lattice and ensemble systems: a review', Int. J. Mod. Phys. C (2025), DOI: 10.1142/S0129183125430065, arXiv:2507.06348, https://arxiv.org/abs/2507.06348
  17. G. Arutyunov, 'Bethe Ansatz', Cambridge University Press (2026), DOI: 10.1017/9781009664837, https://doi.org/10.1017/9781009664837