Finite-size energy of a one-dimensional Fermi gas beyond non-negative potentials: bound states and the zero-energy-resonance regime

Gebert (Math. Phys. Anal. Geom. 18 (2015)) rigorously computed the $1/L$ finite-size correction to the difference of the ground-state energies of two non-interacting $N$-fermion gases confined to an interval $(0,L)$, one free and one subjected to a local potential $V$, expressing the coefficient through scattering data (the phase shift $\delta$, the spectral shift function $\xi$, and the orthogonality exponent $\zeta(E)=\delta^2(\sqrt E)/\pi^2$) and showing it to be non-universal in the thermodynamic-limit parameter. The theorem is restricted to non-negative short-range potentials $V\ge 0$, a hypothesis adopted explicitly to avoid bound states and zero-energy resonances. The problem is to extend these asymptotics to sign-indefinite short-range potentials: a later preprint of Otte and Spitzer (arXiv:1801.00144) covers the bound-state case with the same $1/L$ structure, but the zero-energy-resonance regime - where the phase shift fails to vanish at threshold, the perturbation determinant (the Fredholm determinant encoding the difference between the perturbed and free resolvents) vanishes at the spectral edge, and the lowest box eigenvalues are anomalous - has remained untreated in the literature. A solution must decide whether the resonance modifies the finite-size energy beyond the Levinson normalization of the spectral shift function, or whether additional discrete contributions or different powers of $L$ appear.

Background

Consider $N$ identical spin-polarized fermions confined to the interval $(0,L)$ with Dirichlet boundary conditions (wave functions vanish at the walls). Because the particles do not interact, the many-body ground-state energy is the sum of the $N$ smallest eigenvalues of the one-particle Schrödinger operator: $E_L^N=\sum_{j=1}^N\lambda_j^L$ for the free operator $H_L=-\Delta_L$, and $E_L^{\prime N}=\sum_{j=1}^N\mu_j^L$ for the perturbed operator $H_L^{\prime}=-\Delta_L+V$, where $V$ is a local external potential supported near the origin. The quantity of interest is the difference $E_L^{\prime N}-E_L^N$ in the thermodynamic limit $N,L\to\infty$ at fixed density, for instance along the family $(N+a)/L\to\sqrt E/\pi$ with a real parameter $a$, where $E>0$ plays the role of the Fermi energy.

Two classical scattering quantities enter. The phase shift $\delta(k)$ of the pair of infinite-volume half-line operators $H=-\Delta$ and $H^{\prime}=-\Delta+V$ on $L^2(0,\infty)$ (with Dirichlet condition at the origin) measures the displacement of the nodes of a scattering solution at energy $k^2$ relative to the free solution. Kreĭn's spectral shift function $\xi$ measures the net redistribution of eigenvalues between the two operators and satisfies the Birman–Yafaev identity $\delta(\sqrt E)=-\pi\xi(E)$ for $E>0$. By Levinson's theorem, the threshold value $\delta(0^+)$ counts the bound states (negative eigenvalues) of $H^{\prime}$, plus one half if $H^{\prime}$ has a zero-energy resonance - a bounded, non-square-integrable solution of $-u^{\prime\prime}+Vu=0$ with $u(0)=0$, also called a half-bound state.

Physically this setting models the sudden switching-on of a local scatterer in a Fermi sea, as in Anderson's orthogonality catastrophe: the overlap of the two many-body ground states decays as $L^{-\zeta(E)/2}$ with $\zeta(E)=\delta^2(\sqrt E)/\pi^2$. Affleck and co-authors argued from conformal field theory that the next-to-leading term of the energy difference - the finite-size energy, of order $1/L$, sitting on top of the order-one Fumi term $\int_{-\infty}^{E}\xi(x)\,dx$ - should be tied to the same exponent $\zeta(E)$.

Gebert (Math. Phys. Anal. Geom. 18 (2015)) proved for non-negative short-range potentials, $V\ge 0$ with $\int_0^\infty V(x)(1+x^2)\,dx<\infty$, that

$$E_L^{\prime N}-E_L^N=-\frac1\pi\int_0^{(N\pi/L)^2}\delta(\sqrt x)\,dx+\frac{\sqrt E}{L}\Bigl(-\delta(\sqrt E)+\frac1\pi\delta^2(\sqrt E)\Bigr)+o\Bigl(\frac1L\Bigr),$$

so that along $(N+a)/L=\sqrt E/\pi$ the expansion takes the form

$$E_L^{\prime N}-E_L^N=\int_{-\infty}^{E}\xi(x)\,dx+\frac{\pi\sqrt E}{L}\,x^a_{FS}(E)+o\Bigl(\frac1L\Bigr),$$

where the finite-size energy $x^a_{FS}(E)=(1-2a)\,\xi(E)+\zeta(E)$ - the coefficient of $\frac{\pi\sqrt E}{L}$, in the normalization of the source - is non-universal in the thermodynamic-limit parameter $a$, and equals the orthogonality exponent $\zeta(E)$ only for the special choice $a=1/2$. The hypothesis $V\ge0$ was adopted, in the paper's own words, to avoid bound states or zero-energy resonances (Remark 2.5(iv)): for such $V$ the phase shift obeys $|\delta(k)|\le c\min\{k,1/k\}$, a small-$k$ bound that drives the proof's eigenvalue estimates in Prüfer variables (an amplitude–phase reformulation of the one-dimensional Schrödinger eigenvalue problem that yields eigenvalue-spacing asymptotics) and its Euler–Maclaurin summation (the classical correction formula comparing discrete sums with the corresponding integrals). For sign-changing $V$ the operator $H^{\prime}$ has finitely many bound states, and at threshold couplings a zero-energy resonance; in both regimes $\delta(0^+)\neq0$ and this small-$k$ control fails. A later preprint of Otte and Spitzer covers the sign-indefinite case without resonance, but the resonant regime has remained open.

Problem Statement

Extend the rigorous $1/L$ asymptotics of the ground-state-energy difference of two non-interacting one-dimensional Fermi gases beyond the non-negative short-range class in which Gebert (2015) proved them. Concretely: let $V$ be a real-valued short-range potential on the half-line $(0,\infty)$, allowed to change sign (for definiteness, $\int_0^\infty(1+x^3)|V(x)|\,dx<\infty$), so that the operator $H^{\prime}=-\Delta+V$ on $L^2(0,\infty)$ with Dirichlet boundary condition at the origin has finitely many bound states and possibly a zero-energy resonance, i.e. a bounded, non-square-integrable solution of $-u^{\prime\prime}+Vu=0$ with $u(0)=0$. On the interval $(0,L)$ with Dirichlet boundary conditions at both ends, let $E_L^N$ and $E_L^{\prime N}$ denote the sums of the $N$ smallest eigenvalues of $-\Delta_L$ and $-\Delta_L+V$ respectively. Prove an expansion

$$E_L^{\prime N}-E_L^N=\int_{-\infty}^{E}\xi(x)\,dx+\frac{\pi\sqrt E}{L}\,C_a(E)+o\Bigl(\frac1L\Bigr)$$

(or an explicitly equivalent normalization, stated) along the thermodynamic-limit family $(N+a)/L\to\sqrt E/\pi$ (with $E>0$ the Fermi energy and $a\in\mathbb R$ fixed), valid over the stated class including the resonant case, with the finite-size energy $C_a(E)$ expressed through the scattering data of the pair $(H,H^{\prime})$: the phase shift $\delta(k)$ and the spectral shift function $\xi$ with an explicitly stated normalization, including its Levinson jump at threshold (Levinson's theorem: $\delta(0^+)$ equals $\pi$ times the number of bound states, plus $\pi/2$ at a resonance). The answer must decide whether the non-resonant formula $C_a(E)=x^a_{FS}(E)=(1-2a)\,\xi(E)+\zeta(E)$, with $\zeta(E)=\delta^2(\sqrt E)/\pi^2$, persists at a zero-energy resonance once $\xi$ is normalized through Levinson's theorem, or whether additional discrete contributions or different powers of $L$ appear; in particular, the anomalous behaviour of the lowest interval eigenvalues at the resonance must be controlled. The sign-indefinite case without resonance has a preprint-level treatment (Otte and Spitzer, arXiv:1801.00144) which a solution may build upon or re-derive; the zero-energy-resonance regime is the genuinely open core. The analogous problem on the whole line with general boundary conditions may optionally be treated with the same answer; the required open core is the half-line Dirichlet case including the zero-energy resonance.

References

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