Magnetic Fractional Cwikel–Lieb–Rozenblum Inequalities with the Improved Corso–Ried Constant

The paper establishes that the improved Cwikel–Lieb–Rozenblum constant obtained by Corso and Ried in 2024 for the fractional Laplacian (−Δ)^s transfers without loss to magnetic fractional Schrödinger operators H_(s,A) = ‖−i∇ + A‖^(2s) for the range 0 < s ≤ 1. The main result is that for every dimension d ≥ 3, every 0 < s ≤ 1, every magnetic vector potential A ∈ L²_loc(ℝ^d; ℝ^d), and every potential V ∈ L^(d/(2s))(ℝ^d), the number of negative eigenvalues of H_(s,A) + V, counted with multiplicity and denoted N_(s,A)(V), satisfies N_(s,A)(V) ≤ C_(d,s)^(CR) ∫_(ℝ^d) V_−(x)^(d/(2s)) dx, where V_− := max(0, −V) is the negative part of V and C_(d,s)^(CR) is the non-magnetic Corso–Ried constant, which is independent of the magnetic field A. The corresponding Lieb–Thirring inequalities also follow: for every γ > 0 and V ∈ L^(γ + d/(2s))(ℝ^d), one has Tr(H_(s,A) + V)_−^γ ≤ L_(γ,d,s)^(CR) ∫_(ℝ^d) V_−(x)^(γ + d/(2s)) dx, where L_(γ,d,s)^(CR) is the Corso–Ried Lieb–Thirring constant, again independent of A. As an application, the paper obtains improved stability bounds for relativistic matter in arbitrary magnetic fields: the critical threshold Zα = 2/π remains unchanged, but the stability constant is strictly improved. The case 1 < s < d/2 remains open and is stated as an open problem. The classical Cwikel–Lieb–Rozenblum inequality bounds the number of negative eigenvalues of a Schrödinger operator −Δ + V in terms of the L^(d/2)-norm of the negative part of V. For d ≥ 3 and V ∈ L^(d/2)(ℝ^d), one has N(−Δ + V) ≤ C_d ∫_(ℝ^d) V_−(x)^(d/2) dx. The optimal constant C_d is not known, and the semiclassical constant L^cl := (2π)^(−d) |B₁| = (2π)^(−d) π^(d/2) / Γ(d/2 + 1) is conjectured to be optimal; this is the CLR conjecture, which remains open for d ≥ 4. The Lieb–Thirring inequalities generalize the CLR inequality: for γ > 0 and V ∈ L^(γ + d/2)(ℝ^d), one has Tr(−Δ + V)_−^γ ≤ L_(γ,d) ∫_(ℝ^d) V_−(x)^(γ + d/2) dx. Both the CLR and Lieb–Thirring inequalities extend to the fractional Laplacian (−Δ)^s with 0 < s < d/2 and to magnetic Schrödinger operators. Early contributions to the CLR inequality include Rozenblum in 1972, Lieb in 1976, and Cwikel in 1977; a modern proof via maximal Fourier multipliers was given by Hundertmark, Kunstmann, Ried, and Vugalter in 2023. The magnetic fractional Schrödinger operator of order 2s is defined as H_(s,A) := ‖−i∇ + A‖^(2s) = ((−i∇ + A)²)^s, defined by the spectral theorem. Equivalently, H_(s,A) is the self-adjoint operator associated with the quadratic form ℰ_A(u) := ⟨u, H_(s,A) u⟩ for u ∈ H^s(ℝ^d), whose precise form is most conveniently expressed through the integral representation H_(s,A) u(x) = c_(d,s) P.V. ∫_(ℝ^d) [u(x) − e^(i(x−y)·A((x+y)/2)) u(y)] / |x−y|^(d+2s) dy, with c_(d,s) > 0 a normalization constant. For A = 0, one recovers H_(s,0) = (−Δ)^s. The functional-analytic framework of magnetic fractional Sobolev spaces is discussed by Bonder and Salort in 2020. For a measurable potential V: ℝ^d → ℝ, one defines V_− := max(0, −V) and N_(s,A)(V) := N(0, H_(s,A) + V) = Tr 1_(−∞,0)(H_(s,A) + V), the number of negative eigenvalues of H_(s,A) + V, counted with multiplicity. Throughout the paper, d ≥ 3, 0 < s ≤ 1, and A ∈ L²_loc(ℝ^d; ℝ^d). The operator H_(s,A) = ‖−i∇ + A‖^(2s) is self-adjoint and bounded below on L²(ℝ^d), with form domain H^s(ℝ^d), and it has order 2s. The potential class V ∈ L^(d/(2s)) is the natural one for the CLR inequality: it is the class for which the kinetic energy |ξ|^(2s) and the potential V scale the same way under the scaling u ↦ u(λ·), x ↦ λx. The proof of the main theorems rests on two published results. The first is the comparison principle of Frank, Lieb, and Seiringer from 2008. For 0 < s ≤ 1, the spectral quantities of the magnetic operator H_(s,A) are dominated by those of the non-magnetic operator (−Δ)^s. More precisely, one has N_(s,A)(V) ≤ N_((−Δ)^s)(V) and Tr(H_(s,A) + V)_−^γ ≤ Tr((−Δ)^s + V)_−^γ. This comparison is uniform in the magnetic field A: the right-hand side does not depend on A. The second result is the Corso–Ried constant from 2024. For 0 < s < d/2, the non-magnetic operator satisfies N_((−Δ)^s)(V) ≤ C_(d,s)^(CR) ∫ V_−^(d/(2s)) and Tr((−Δ)^s + V)_−^γ ≤ L_(γ,d,s)^(CR) ∫ V_−^(γ+d/(2s)). Combining these two inequalities yields the main theorems immediately. The key point is that the comparison principle preserves the constant: the same bound holds for the magnetic operator as for the non-magnetic one, with no loss. Hence the improvement in the Corso–Ried constant transfers entirely to the magnetic setting. The novelty of the paper is that the Corso–Ried constant has not been previously applied to magnetic fractional operators. Frank, Lieb, and Seiringer established the comparison principle in 2008 but used the classical Cwikel constant; they did not have access to the improved Corso–Ried constant, which appeared in 2024. No published work combines the improved constant with the magnetic fractional setting. The transfer argument is elementary, but to the best of the authors' knowledge, it has not appeared in the literature. The paper relates to the literature as follows. Rozenblum (1972), Lieb (1976), and Cwikel (1977) gave the original CLR inequality and the first explicit constants. Frank, Lieb, and Seiringer (2008) proved Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, including magnetic fields, with the classical constant, and established the comparison principle for 0 < s ≤ 1. Frank, Hundertmark, Jex, and Nam (2021) gave a new proof of the Lieb–Thirring inequality via the Rumin method, improving the constant to 1.456 L^cl for γ = 1. Hundertmark, Kunstmann, Ried, and Vugalter (2023) gave a new proof of the CLR inequality via maximal Fourier multipliers, leading to a variational problem. Corso and Ried (2024) solved this variational problem explicitly, giving improved constants C_(d,s)^(CR) and L_(γ,d,s)^(CR). The present paper transfers the Corso–Ried constant to the magnetic fractional setting via the comparison principle, with an application to the stability of relativistic matter. The case 1 < s < d/2 is not covered because the Frank–Lieb–Seiringer argument relies on the positivity of the heat kernel e^(−t(−Δ)^s), which holds if and only if 0 < s ≤ 1. This case remains open. The preliminaries include the diamagnetic inequality and heat-kernel domination. Lemma 2.1 of Frank, Lieb, and Seiringer (2008) states that for 0 < s ≤ 1 and A ∈ L²_loc(ℝ^d; ℝ^d), for all t > 0 and almost every x, y ∈ ℝ^d, one has |e^(−tH_(s,A))(x, y)| ≤ e^(−t(−Δ)^s)(x, y). Moreover, for real V and all t > 0, one has |e^(−t(H_(s,A) + V))(x, y)| ≤ e^(−t((−Δ)^s + V))(x, y). Inequality (2.1) is the diamagnetic inequality for the fractional magnetic Laplacian, and (2.2) follows from the Feynman–Kac formula. The restriction 0 < s ≤ 1 is essential: for s > 1, the heat kernel e^(−t(−Δ)^s) changes sign and (2.1) fails. The proof sketch of (2.2) is as follows. For 0 < s ≤ 1, the semigroup e^(−t(−Δ)^s) is positivity preserving because it is the transition semigroup of a symmetric stable Lévy process. The Feynman–Kac formula gives e^(−t(H_(s,A) + V))(x, y) = E_(x,y)[e^(−∫_0^t V(ω(r)) dr) · phase(A, ω)], where E_(x,y) is the expectation over the stable process from x to y, and phase(A, ω) is a unimodular functional of the magnetic field along the path ω. Taking absolute values and using |phase(A, ω)| = 1 and |e^(−∫ V)| = e^(−∫ V) for real V, one obtains |e^(−t(H_(s,A) + V))(x, y)| ≤ E_(x,y)[e^(−∫_0^t V(ω(r)) dr)] = e^(−t((−Δ)^s + V))(x, y). This establishes (2.2). The comparison principle is Lemma 2.2 of Frank, Lieb, and Seiringer (2008). Let 0 < s ≤ 1, A ∈ L²_loc(ℝ^d; ℝ^d), and V: ℝ^d → ℝ measurable. Then for every γ ≥ 0, one has Tr(H_(s,A) + V)_−^γ ≤ Tr((−Δ)^s + V)_−^γ, with the convention that both sides may be +∞. In particular, for γ = 0, one has N_(s,A)(V) ≤ N_((−Δ)^s)(V). The proof is as follows. For γ = 0, the inequality is an immediate consequence of (2.2) and the Birman–Schwinger principle (Reed–Simon 1978), or directly from the Feynman–Kac formula for the heat kernel. For γ > 0, one uses the Aizenman–Lieb representation Tr(K_−^γ) = γ ∫_0^∞ λ^(γ−1) N(K + λ) dλ, valid for any self-adjoint operator K bounded below with N(K) < ∞. Applying (2.4) to the potential V + λ, one has N(H_(s,A) + V + λ) ≤ N((−Δ)^s + V + λ) for every λ > 0. Substituting into (2.5) and integrating over λ ∈ (0, ∞) yields Tr(H_(s,A) + V)_−^γ = γ ∫_0^∞ λ^(γ−1) N(H_(s,A) + V + λ) dλ ≤ γ ∫_0^∞ λ^(γ−1) N((−Δ)^s + V + λ) dλ = Tr((−Δ)^s + V)_−^γ. This establishes (2.3) for γ > 0. Remark 2.3 notes that the comparison (2.3) is uniform in A: the bound on the right-hand side does not depend on A. This is the key property that allows the Corso–Ried constant to transfer without loss. Remark 2.4 notes that Lemma 2.2 is implicit in Frank–Lieb–Seiringer (2008, Section 2), where the magnetic Lieb–Thirring inequality is derived from the non-magnetic one via the diamagnetic inequality and the Birman–Schwinger principle. The paper states it explicitly for convenience, with a self-contained proof via the Aizenman–Lieb representation. The Corso–Ried constant is the main theorem of Corso–Ried (2024). Theorem 2.5 states that for d ≥ 3 and 0 < s < d/2, for V ∈ L^(d/(2s))(ℝ^d), one has N_((−Δ)^s)(V) ≤ C_(d,s)^(CR) ∫_(ℝ^d) V_−(x)^(d/(2s)) dx, and for γ > 0 and V ∈ L^(γ + d/(2s))(ℝ^d), one has Tr((−Δ)^s + V)_−^γ ≤ L_(γ,d,s)^(CR) ∫_(ℝ^d) V_−(x)^(γ + d/(2s)) dx, where the constants are given by C_(d,s)^(CR) / L^cl ≤ (1/4) · ν^(ν+1) / (ν−2)^(ν−2) · M_ν, with ν := d/s, and M_ν is the solution of the variational problem M_ν := inf_(m₁, m₂) (‖m₁‖ ‖m₂‖)^(ν−2) ∫_0^∞ |(m₁ * m₂)(t) − t|² t^(−ν) dt/t. Here the infimum is over measurable m₁, m₂: (0, ∞) → [0, ∞), ‖m‖² := ∫_0^∞ m(t)² dt/t, and (m₁ * m₂)(t) := ∫_0^∞ m₁(t/u) m₂(u) du/u is the multiplicative convolution on (0, ∞). The variational problem is solved explicitly in Corso–Ried (2024) using the Hadamard three-lines theorem. The resulting