Minimax Principles, Hardy-Dirac Inequalities,

and Operator Cores for Two and Three Dimensional

Coulomb-Dirac Operators

David Müller

Received: March 28, 2016 Revised: August 9, 2016

Communicated by Heinz Siedentop

Abstract. For n ∈ {2, 3} we prove minimax characterisations of eigenvalues in the gap of the n dimensional Dirac operator with an potential, which may have a Coulomb singularity with a coupling constant up to the critical value 1/(4 − n). This result implies a socalled Hardy-Dirac inequality, which can be used to define a distinguished self-adjoint extension of the Coulomb-Dirac operator defined on C_{0}^{∞}(ℝ^{n} \ {0}; ℂ^{2(n−1)}), as long as the coupling constant does not exceed 1/(4 − n). We also find an explicit description of an operator core of this operator.

2010 Mathematics Subject Classification: 49R05, 49J35, 81Q10 Keywords and Phrases: Minimax Principle, Hardy-Dirac Inequality, Coulomb-Dirac Operator

1 Introduction

The relativistic dynamics of an electron moving in an atomic field is described by a Dirac operator with potential V having a Coulomb singularity. Since we want to consider such Dirac operators in two and three dimensions simultaneously, we assume throughout the text that n ∈ {2, 3}. In n dimensions the relativistic electron corresponds to a 2(n − 1) component spinor and V is a 2(n − 1) × 2(n − 1) hermitian matrix function on ℝ^{n}. We say that V belongs to 𝔓_{n} if for some ν ∈ [0, 1/(4 − n)) the inequality 0 ≥ V ≥ −ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}} holds.

This motivates the following question. Does the Dirac operator with potential V ∈ 𝔓_{n} ∪ {−1/((4 − n)| · |) ⊗ 𝕀_{ℂ^{2(n−1)}}}

D̃_{n}(V) := \{ \begin{matrix}−iσ · ∇ + σ_{3} + V if n = 2 \\ −iα · ∇ + β + V if n = 3 \end{matrix} defined on C_{0}^{∞}(ℝ^{n} \ \{0\}; ℂ^{2(n−1)}),

(1)

have a unique self-adjoint extension? In (1) are σ = (σ_{1}, σ_{2}), α = (α_{1}, α_{2}, α_{3})

vectors; σ_{1}, σ_{2}, σ_{3} the standard Pauli matrices; α_{i} = (\begin{matrix}0_{ℂ^{2}} & σ_{i} \\ σ_{i} & 0_{ℂ^{2}} \end{matrix}) for i ∈

\{1, 2, 3\} and β = (\begin{matrix}𝕀_{ℂ^{2}} & 0_{ℂ^{2}} \\ 0_{ℂ^{2}} & −𝕀_{ℂ^{2}} \end{matrix}). It is the uniqueness not the existence of a

self-adjoint extension that is doubtful. For example the Coulomb-Dirac operator D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) is essentially self-adjoint if n = 2, ν = 0 or n = 3, ν ∈ [0, \sqrt{3}/2] but for n = 2, ν ∈ (0, 1/2] or n = 3, ν ∈ (\sqrt{3}/2, 1] there are infinitely many self-adjoint extensions (see Lemma 14). Thus it is also natural to ask, whether there is a physically distinguished self-adjoint extension? In fact for V ∈ 𝔓_{n} there is a unique self-adjoint extension D_{n}(V) of D̃_{n}(V), for which the wave functions in its domain possess finite mean kinetic energy, i.e. 𝔇(D_{n}(V)) ⊂ H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}). The existence of this distinguished selfadjoint extension is proven in Section 3. There we apply some general results developed in [15]. Note that for ν ∈ [0, 1/(4 − n)) the domain of the Coulomb-Dirac operator D_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) is contained in H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}) and for D̃_{n}(((n − 4)| · |)^{−1} ⊗ 𝕀_{ℂ^{2(n−1)}}) there is no self-adjoint extension with this property. In this sense 1/(4−n) is a critical constant. At this point we want to mention that in the context of Theorem 5 we define a distinguished self-adjoint extension of D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) for ν ∈ [0, 1/(4 − n)], i.e. the case of a Coulomb potential with the critical coupling constant 1/(4 − n) is in particular included here. Let V ∈ 𝔓_{n}. As in Proposition 1 in [4] one can prove that there is a gap in the essential spectrum of D_{n}(V). To be more precise

σ_{ess}(D_{n}(V)) = (−∞, −1] ∪ [1, ∞).

In 1986 James D. Talman proposed in [17] a formal minimax characterisation of the lowest eigenvalue in the gap of the essential spectrum of the operator D_{3}(V). In this work we prove a minimax characterisation of eigenvalues in the gap of D_{3}(V) in the spirit of Talman and an analogous result for D_{2}(V). The exact result is:

Theorem 1 (Talman minimax principle). Let V ∈ 𝔓_{n}. If the k^{th} eigenvalue μ_{k} of D_{n}(V) in (−1, 1), counted from below with multiplicity, exists, then it is given by

μ_{k} = inf_{\begin{matrix}𝔐⊂H^{1/2}(ℝ^{n};ℂ^{n−1}) \\ dim 𝔐=k \end{matrix}} sup_{ψ∈(𝔐⊕H^{1/2}(ℝ^{n};ℂ^{n−1}))\\{0\}} \frac{d_{n}[ψ] + v[ψ]}{∥ψ∥^{2}}.

Here d_{n} and v are the quadratic forms associated to the operators D_{n}(0) and V.

About Theorem 1 we want to remark that for n = 3 there is an historical overview of results of the same type in [13] and that for n = 2 there is no comparable result known. Moreover, Theorem 1 improves in the three dimensional case Theorem 3 in [13], which is the best known result for a Dirac operator with an electrostatic potential having strong Coulomb singularity. Furthermore, we give a different proof of the Esteban-Séré minimax principle (see Theorem 2 in [13] and [9]) and prove an analogous result for two dimensional Dirac operators:

Theorem 2 (Esteban-Séré minimax principle). Let V ∈ 𝔓_{n}. If the k^{th} eigenvalue μ_{k} of D_{n}(V) in (−1, 1), counted from below with multiplicity, exists, then it is given by

μ_{k} = inf_{\begin{matrix}𝔐⊂P_{n}^{+}H^{1/2}(ℝ^{n};ℂ^{2(n−1)}) \\ dim 𝔐=k \end{matrix}} sup_{ψ∈(𝔐⊕P_{n}^{−}H^{1/2}(ℝ^{n};ℂ^{2(n−1)}))\\{0\}} \frac{d_{n}[ψ] + v[ψ]}{∥ψ∥^{2}}.

Here P_{n}^{+} is the projector on the non-negative spectral subspace of D_{n}(0) and P_{n}^{−} := 𝕀 − P_{n}^{+}.

A direct application of Theorem 1 is:

Theorem 3 (Hardy-Dirac inequality). Let v be a scalar function on ℝ^{n} such that v ⊗ 𝕀_{ℂ^{2(n−1)}} ∈ 𝔓_{n}. Moreover, we define the operator:

K_{n} := \begin{cases}−i∂_{1} − ∂_{2} if n = 2, \\ −iσ · ∇ if n = 3, \end{cases}

and denote by λ(v) the smallest eigenvalue of D_{n}(v⊗𝕀_{ℂ^{2(n−1)}}) in the gap (−1, 1). Then for all φ ∈ H^{1}(ℝ^{n}; ℂ^{n−1}) the inequality

0 ≤ ∫_{ℝ^{n}} \frac{|K_{n}φ(x)|^{2}}{1 + λ(v) − v(x)}dx + ∫_{ℝ^{n}} (1 − λ(v) + v(x))|φ(x)|^{2}dx  (2)

holds.

We follow the tradition of [5] and call these type of inequality Hardy-Dirac inequality. In [6] it is demonstrated, how one can prove Hardy-Dirac inequalities for n = 3 with the help of the Talman minimax principle. We know that the lowest eigenvalue of D_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) in (−1, 1) is

\sqrt{1 − ((4 − n)ν)^{2}} for ν ∈ (0, 1/(4 − n)) (see [7] and [19]). Thus Theorem 3 implies with a simple limiting argument Corollary 4. Let ν ∈ [0, 1/(4 − n)]. Then

0 ≤∫_{ℝ^{n}} (\frac{|K_{n}φ|^{2}}{1 + \sqrt{1 − ((4 − n)ν)^{2}} + \frac{ν}{|x|}} + (1 − \sqrt{1 − ((4 − n)ν)^{2}} − \frac{ν}{|x|})|φ|^{2})dx

holds for all φ ∈ H^{1}(ℝ^{n}; ℂ^{n−1}).

Let ν ∈ [0, 1/(4 − n)]. With the help of Corollary 4 and Theorem 1 in [8] (D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) corresponds to H there) we know that there is only one self-adjoint extension of D̃_{n}(−ν/|·|⊗𝕀_{ℂ^{2(n−1)}}) with a positive Schur complement. We denote this distinguished self-adjoint extension by D_{n}^{ν}. Now we want to give an explicit description of an operator core of D_{n}^{ν}. For this purpose we introduce polar and spherical coordinates. We denote by the coordinate pair (ρ, ϑ) ∈ [0, ∞) × [0, 2π) the radial and angular polar coordinates in ℝ^{2} and by the coordinate triplet (r, θ, ϕ) ∈ [0, ∞) × [0, π) × [0, 2π) the radial, inclination and azimuthal spherical coordinates in ℝ^{3}. For m ∈ {−1/2, 1/2}^{n−1} we define the function ζ_{n,m}^{ν} in polar coordinates for n = 2

ζ_{2,m}^{ν}(ρ, ϑ) := ξ(ρ)ρ^{\sqrt{1/4−ν^{2}}−1/2} (\begin{matrix}ν \frac{e^{−i(1/2+m)ϑ}}{\sqrt{2π}} \\ −i(\sqrt{1/4 − ν^{2}} + (−1)^{1/2−m}/2)\frac{e^{i(1/2−m)ϑ}}{\sqrt{2π}} \end{matrix} ); (3)

and in spherical coordinates for n = 3

ζ_{3,m}^{ν}(r, θ, ϕ) := ξ(r)r^{\sqrt{1−ν^{2}}−1} (\begin{matrix}νΩ_{\frac{1}{2} +m_{2},m_{1},−m_{2}}(θ, ϕ) \\ −i(\sqrt{1 − ν^{2}} + (−1)^{\frac{1}{2} −m_{2}})Ω_{\frac{1}{2} −m_{2},m_{1},m_{2}}(θ, ϕ) \end{matrix}); (4)

with the spherical spinor Ω_{l,m,s} (see Relation (7) in [10]) and the smooth cut-off function ξ (i.e., ξ ∈ C^{∞}(ℝ_{+}; ℝ_{+}), ξ(t) = 1 for t ∈ (0, 1) and ξ(t) = 0 for t > 2). In the next theorem we give a characterisation of an operator core of D_{n}^{ν} with the help of the functions ζ_{n,m}^{ν} introduced in (3) and (4).

Theorem 5 (Operator core). Let ν ∈ [0, 1/(4 − n)]. The set

ℭ_{n}^{ν} := C_{0}^{∞}(ℝ^{n} \ \{0\}; ℂ^{2(n−1)}) ˙+\begin{cases}\{0\}, if n = 2, ν = 0 or n = 3, ν ∈ [0, \frac{\sqrt{3}}{2}]; \\ span\{ζ_{n,m}^{ν} : m ∈ \{−1/2, 1/2\}^{n−1}\}, else; \end{cases}

(5)

is an operator core for D_{n}^{ν}.

The knowledge of the operator core of D_{n}^{ν} is important for the proof of estimates on the square of the operator, see e.g. [14]. In Remark 15 we show that for ν ∈ [0, 1/(4 − n)) the set ℭ_{n}^{ν} is an operator core for D_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}). A direct consequence is:

Corollary 6. Let ν ∈ [0, 1/(4−n)). The distinguished self-adjoint extensions of D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) in the sense of [15] and [8] coincide, i.e.,

D_{n}^{ν} = D_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}).

The proofs of the minimax characterisations rely on the angular momentum channel decomposition of the Coulomb-Dirac operator in the momentum space. This representation and the corresponding unitary transformations are introduced in the next section. In the remaining sections we prove in the order of enumeration: Theorems 1, 2, 3 and 5.

2 Angular momentum channel decomposition in the momentum space

The Fourier transform connects the quantum mechanical descriptions of a particle in the configuration and momentum space. We use the standard unitary Fourier transform F_{n} in L^{2}(ℝ^{n}) given for φ ∈ L^{1}(ℝ^{n}) ∩ L^{2}(ℝ^{n}) by

F_{n}φ := \frac{1}{(2π)^{n/2}} ∫_{ℝ^{n}} e^{−i⟨·,x⟩}φ(x)dx.  (6)

For the angular momentum channel decomposition in n dimensions we use an orthonormal basis in L^{2}(𝕊^{n−1}; ℂ^{n−1}). For n = 2 this orthonormal basis is ((2π)^{−1/2}e^{im(·)})_{m∈ℤ}. In three dimensions we use spherical spinors Ω_{l,m,s}, which are defined in Relation (7) in [10], with l ∈ ℕ_{0}, m ∈ {−l − 1/2, …, l + 1/2} and s ∈ {−1/2, 1/2}. The corresponding index sets are denoted by

𝔗_{2} := ℤ;  (7)

and

𝔗_{3} := \{(l, m, s) : l ∈ ℕ_{0}, m ∈ \{ − l − \frac{1}{2}, …, l + \frac{1}{2}\}, s = ±\frac{1}{2}, Ω_{l,m,s} ≠ 0\}.(8)

Furthermore, we define subsets 𝔗_{n}^{±} of 𝔗_{n}:

𝔗_{n}^{a} := \begin{cases}2ℤ & if n = 2, a = +; \\ 2ℤ + 1 & if n = 2, a = −; \\ \{(l, m, s) ∈ 𝔗_{3} : s = ±1/2\} & if n = 3, a = ±. \end{cases}  (9)

Note that if (l, m, −1/2) ∈ 𝔗_{3}^{−} then l ∈ ℕ. Moreover, we introduce bijective maps

T_{2} : 𝔗_{2} → 𝔗_{2}, T_{2}k := k + 1;  (10)

and

T_{3} : 𝔗_{3} → 𝔗_{3}, T_{3}(l, m, s) := (l + 2s, m, −s).  (11)

We can represent any φ ∈ L^{2}(ℝ^{2}; ℂ) in polar coordinates and ζ ∈ L^{2}(ℝ^{3}; ℂ^{2}) in spherical coordinates as

φ(ρ, ϑ) = ∑_{k∈𝔗_{2}}(2πρ)^{−1/2}φ_{k}(ρ)e^{ikϑ};  (12)
ζ(r, θ, ϕ) = ∑_{(l,m,s)∈𝔗_{3}} r^{−1}ζ_{(l,m,s)}(r)Ω_{l,m,s}(θ, ϕ);  (13)

with

φ_{k}(ρ) := \sqrt{\frac{ρ}{2π}} ∫_{0}^{2π}φ(ρ, ϑ)e^{−ikϑ}dϑ;  (14)

ζ_{(l,m,s)}(r) := r ∫_{0}^{2π}∫_{0}^{π} ⟨Ω_{l,m,s}(θ, ϕ), ζ(r, θ, ϕ)⟩_{ℂ^{2}} sin(θ)dθdϕ.  (15)

With the help of (14) and (15) we define the unitary operator

U_{n} : L^{2}(ℝ^{n}; ℂ^{n−1}) → ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}); ψ ↦ ⨁_{j∈𝔗_{n}} ψ_{j}.  (16)

For the proof of the following lemma see Theorem 2.2.5 in [1] (based on Lemmata 2.1, 2.2 of [2]) for n = 2 and Section 2.2 in [1] for n = 3.

Lemma 7. For j ∈ (ℕ_{0}/2 − 1/2) and z ∈ (1, ∞) let

Q_{j}(z) = 2^{−j−1} ∫_{−1}^{1}(1 − t^{2})^{j}(z − t)^{−j−1} dt  (17)

be a Legendre function of the second kind (see Section 15.3 in [21]). Let the sesquilinear form q_{j} be defined on L^{2}(ℝ_{+}, (1+p^{2})^{1/2}dp)×L^{2}(ℝ_{+}, (1+p^{2})^{1/2}dp) by

q_{j}[f, g] := π^{−1} ∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)}Q_{j}(\frac{1}{2}(\frac{q}{p} + \frac{p}{q}))g(q) dq dp.  (18)

For the special case f = g we introduce q_{j}[f] := q_{j}[f, f]. Then for every ζ, η ∈ H^{1/2}(ℝ^{n}) the relation

∫_{ℝ^{n}} \frac{\overline{ζ}(x) · η(x)}{|x|}dx = \begin{cases}∑_{k∈𝔗_{2}} q_{|k|−1/2}[(F_{2}ζ)_{k}, (F_{2}η)_{k}] if n = 2, \\ ∑_{(l,m,s)∈𝔗_{3}} q_{l}[(F_{3}ζ)_{(l,m,s)}, (F_{3}η)_{(l,m,s)}] if n = 3, \end{cases}

(19)

holds.

The operators −iσ ·∇ and −iα·∇ are partially diagonalised in the momentum space by the unitary transforms

W_{2} : L^{2}(ℝ^{2}; ℂ^{2}) → ⨁_{k∈𝔗_{2}} L^{2}(ℝ_{+}; ℂ^{2}); (\begin{matrix}φ \\ ψ \end{matrix}) ↦ ⨁_{k∈𝔗_{2}} (\begin{matrix}φ_{k} \\ ψ_{T_{2}k} \end{matrix})  (20)

and

W_{3} : L^{2}(ℝ^{3}; ℂ^{4}) → ⨁_{(l,m,s)∈𝔗_{3}} L^{2}(ℝ_{+}; ℂ^{2}); (\begin{matrix}ψ_{1} \\ ψ_{2} \\ ψ_{3} \\ ψ_{4} \end{matrix}) ↦ ⨁_{(l,m,s)∈𝔗_{3}} (\begin{matrix}ψ_{(l,m,s)}^{+} \\ ψ_{T_{3}(l,m,s)}^{−} \end{matrix}) (21)

with

ψ_{(l,m,s)}^{+} := (\begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix})_{(l,m,s)} and ψ_{(l,m,s)}^{−} := (\begin{matrix}ψ_{3} \\ ψ_{4} \end{matrix})_{(l,m,s)}  (22)

for (l, m, s) ∈ 𝔗_{3}. To be more precise:

Lemma 8. For the self-adjoint operators −iσ · ∇ and −iα · ∇ the relations

(W_{n}F_{n})^{∗}( ⨁_{j∈𝔗_{n}} (\begin{matrix}0 & (·) \\ (·) & 0 \end{matrix}) )(W_{n}F_{n}) = \begin{cases}−iσ · ∇ if n = 2, \\ −iα · ∇ if n = 3, \end{cases}  (23)

hold.

Proof. By a straightforward calculation and Relation 2.1.28 in [1] the relations

σ · x = (\begin{matrix}0 & e^{−iϑ}ρ \\ e^{iϑ}ρ & 0 \end{matrix}) for x ∈ ℝ^{2};  (24)
σ · \frac{x}{|x|}Ω_{l,m,s} = Ω_{l+2s,m,−s} for x ∈ ℝ^{3} and (l, m, s) ∈ 𝔗_{3};  (25)

hold. The set C_{0}^{∞}(ℝ^{n}; ℂ^{2(n−1)}) is dense in H^{1}(ℝ^{n}; ℂ^{2(n−1)}). Thus it is enough to work with ψ ∈ C_{0}^{∞}(ℝ^{2}; ℂ^{2}) and ζ ∈ C_{0}^{∞}(ℝ^{3}; ℂ^{4}). Moreover, the Fourier transform diagonalises differential operators:

⟨ψ, −iσ · ∇ψ⟩ = ⟨F_{2}ψ, σ · pF_{2}ψ⟩,  (26)
⟨ζ, −iα · ∇ζ⟩ = ⟨F_{3}ζ, α · pF_{3}ζ⟩.  (27)

Here we denote by p the independent variable of multiplication in L^{2}(ℝ^{n}; dp). Now we prove (23) for n = 3. We obtain by the representation (13) of the upper and lower bispinor of F_{3}ζ and the notation introduced in (22) that the right hand side of (27) is equal to

\begin{matrix}2 ∑_{\begin{matrix}(l′,m′,s′)∈𝔗_{3} \\ (l,m,s)∈𝔗_{3} \end{matrix}} ℜ (⟨|p|^{−1}(F_{3}ζ)_{(l′,m′,s′)}^{+}Ω_{l′,m′,s′}, (σ · p)|p|^{−1}(F_{3}ζ)_{(l,m,s)}^{−}Ω_{l,m,s}⟩) . \\ (28) \end{matrix}

The expression in (28) is equal to

2 ∑_{(l,m,s)∈𝔗_{3}} ℜ (⟨(F_{3}ζ)_{(l+2s,m,−s)}^{+}, (·)(F_{3}ζ)_{(l,m,s)}^{−}⟩)

= ∑_{(l,m,s)∈𝔗_{3}} ⟨(\begin{matrix}(F_{3}ζ)_{(l,m,s)}^{+} \\ (F_{3}ζ)_{T_{3}(l,m,s)}^{−} \end{matrix}) , (\begin{matrix}0 & (·) \\ (·) & 0 \end{matrix}) (\begin{matrix}(F_{3}ζ)_{(l,m,s)}^{+} \\ (F_{3}ζ)_{T_{3}(l,m,s)}^{−} \end{matrix})⟩

= ⟨W_{3}F_{3}ζ, ( ⨁_{(l,m,s)∈𝔗_{3}} (\begin{matrix}0 & (·) \\ (·) & 0 \end{matrix}) )W_{3}F_{3}ζ⟩  (29)

by the sequential application of (25), (21) and (6). Thus the claim of Lemma 8 is a consequence of (27), (28) and (29). For n = 2 we obtain (23) by an analogous procedure, i.e., we represent the upper and lower component of F_{2}ψ by (12) in (26) and perform a calculation, which involves (24).

3 Proof of Theorem 1

Let V ∈ 𝔓_{n}. We use the abstract minimax principle Theorem 1 of [13] to prove the Talman minimax principle. We apply the theorem with q := d_{n} (quadratic form associated to D_{n}(0)), B := D_{n}(V) and Λ_{±} as the projector T_{n}^{±} on the upper and lower (n − 1) components of a 2(n − 1) spinor, i.e.,

T_{n}^{+}\binom{φ}{ψ} = \binom{φ}{0}, T_{n}^{−}\binom{φ}{ψ} = \binom{0}{ψ}, for φ, ψ ∈ L^{2}(ℝ^{n}; ℂ^{n−1}).

That D_{n}(V) plays the role of B in [13] is a consequence of Theorem 2.1 in [15] and the following lemma.

Lemma 9. Let V ∈ 𝔓_{n}. Then the quadratic form v associated to the operator V is a form perturbation of D_{n}(0) in the sense of Definition 2.1 in [15].

Proof. V is D_{n}(0) form bounded by the Herbst inequality (see Theorem 2.5 in [11]). Moreover, the inequality

∥r^{−1/2}D_{n}(0)^{−1}r^{−1/2}∥ ≤ 4 − n

holds. This is proven in Section 2 in [12] for n = 3. The same arguments also apply for n = 2 (see Step 1 in the proof of Theorem 1 in [4]). Thus

∥V^{1/2}D_{n}(0)^{−1}V^{1/2}∥ ≤ ∥V^{1/2}r^{1/2}∥^{2} · ∥r^{−1/2}D_{n}(0)^{−1}r^{−1/2}∥ < 1.

Hence 1 + V^{1/2}D_{n}(0)^{−1}V^{1/2} has a bounded inverse by the Neumann series. Now the claim follows from Theorem 2.2 in [15] with A := D_{n}(0) and t := 0.

Since the assumptions (i) and (ii) of Theorem 1 in [13] are obviously fulfilled, it remains to check assumption (iii). Thus it is enough to find an operator L_{n} : H^{1/2}(ℝ^{n}; ℂ^{n−1}) → H^{1/2}(ℝ^{n}; ℂ^{n−1}) such that

inf_{φ∈H^{1/2}(ℝ^{n};ℂ^{n−1})\\{0\}} \frac{d_{n}[\binom{φ}{L_{n}φ}] + v[\binom{φ}{L_{n}φ}]}{∥\binom{φ}{L_{n}φ}∥2} > −1.

Now we give in three steps an explicit construction of L_{n} and show that L_{n} satisfies the requirements. For k ∈ 𝔗_{2} and (l, m, s) ∈ 𝔗_{3} we define in the first step various constants:

c_{n} := 2(4 − n)\frac{Γ(\frac{n+1}{4})^{2}}{Γ(\frac{n−1}{4})^{2}} ;  (30)

c_{2,k} := \begin{cases}c_{2}^{−1} if k ∈ 𝔗_{2}^{−}, \\ c_{2} if k ∈ 𝔗_{2}^{+}; \end{cases}  (31)
c_{3,(l,m,s)} := c_{3}^{2s}.  (32)

In the second step we define the operator R_{n}

R_{n} : ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}) → ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}); ⨁_{j∈𝔗_{n}} ψ_{j} ↦ ⨁_{j∈𝔗_{n}} c_{n,j}ψ_{T_{n}^{−1}j}.  (33)

Finally we define

L_{n} := (U_{n}F_{n})^{∗}R_{n}(U_{n}F_{n}).  (34)

The desired properties of L_{n} are proven in the following lemma:

Lemma 10. Let φ ∈ H^{1/2}(ℝ^{n}; ℂ^{n−1}) then L_{n}φ ∈ H^{1/2}(ℝ^{n}; ℂ^{n−1}) and the following inequality

\frac{c_{n}^{2} − 1}{c_{n}^{2} + 1}∥\binom{φ}{L_{n}φ}∥^{2} ≤ d_{n}[\binom{φ}{L_{n}φ}] − \frac{1}{4 − n} ∫_{ℝ^{n}} \frac{1}{|x|}|\binom{φ(x)}{(L_{n}φ)(x)}|^{2}dx (35)

holds.

Proof. We recall that

H^{1/2}(ℝ^{n}) = {ψ ∈ L^{2}(ℝ^{n}) : (1 + | · |^{2})^{1/4}F_{n}ψ ∈ L^{2}(ℝ^{n})}.

Thus the unitarity of U_{n} implies

H^{1/2}(ℝ^{n}) = {ψ ∈ L^{2}(ℝ^{n}) : ⨁_{j∈𝔗_{n}}(1 + (·)^{2})^{1/4}(F_{n}ψ)_{j} ∈ ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+})}. (36)

Moreover we observe that the operator R_{n} is bounded, which together with (36) and (34) implies that L_{n}φ ∈ H^{1/2}(ℝ^{n}). Now we define the quadratic form p on L^{2}(ℝ_{+}, (1 + p^{2})^{1/2}dp) by

p[χ] := ∫_{0}^{∞}p|χ(p)|^{2}dp.

For the proof of (35) we recall that the quadratic form (18) satisfy the inequalities

q_{k+1/2}[ζ] ≤ q_{k−1/2}[ζ];
q_{k+1}[ζ] ≤ q_{k}[ζ];
q_{0}[ζ] ≤ c_{3}^{−1}p[ζ], q_{1}[ζ] ≤ c_{3}p[ζ];
q_{−1/2}[ζ] ≤ 2c_{2}^{−1}p[ζ], q_{1/2}[ζ] ≤ 2c_{2}p[ζ];

(37)

for k ∈ ℕ_{0} and ζ ∈ L^{2}(ℝ_{+}, (1 + p^{2})^{1/2}dp) (see [2] and [10]). By Lemma 7 we obtain

∫_{ℝ^{n}} \frac{|φ(x)|^{2}}{|x|} dx = \begin{cases}∑_{k∈𝔗_{2}} q_{|k|−1/2}[(F_{2}φ)_{k}] if n = 2; \\ ∑_{(l,m,s)∈𝔗_{3}} q_{l}[(F_{3}φ)_{(l,m,s)}] if n = 3; \end{cases}  (38)

and by (31) - (34)

∫_{ℝ^{n}} \frac{|(L_{n}φ)(x)|^{2}}{|x|} dx

= \{ ∑_{k∈𝔗_{2}^{+}} c_{2}^{2}q_{|k|− \frac{1}{2}} [(F_{2}φ)_{k−1}] + ∑_{k∈𝔗_{2}^{−}} c_{2}^{−2}q_{|k|− \frac{1}{2}} [(F_{2}φ)_{k−1}] if n = 2; ∑_{(l,m, \frac{1}{2} )∈𝔗_{3}^{+}} c_{3}^{2}q_{l}[(F_{3}φ)_{(l+1,m,− \frac{1}{2} )}] + ∑_{(l,m,− \frac{1}{2} )∈𝔗_{3}^{−}} c_{3}^{−2}q_{l}[(F_{3}φ)_{(l−1,m, \frac{1}{2} )}] if n = 3.

(39)

Note that (l, m, s) ∈ 𝔗_{3}^{−} implies l ∈ ℕ. Hence (37) implies that the right hand sides of (38) can be estimated by

(4 − n) ( ∑_{j∈𝔗_{n}^{+}} c_{n}^{−1}p[(F_{n}φ)_{j}] + ∑_{j∈𝔗_{n}^{−}} c_{n}p[(F_{n}φ)_{j}]);  (40)

and the right hand side of (39) by

(4 − n) ( ∑_{j∈𝔗_{n}^{+}} c_{n}p[(F_{n}φ)_{T_{n}^{−1}j}] + ∑_{j∈𝔗_{n}^{−}} c_{n}^{−1}p[(F_{n}φ)_{T_{n}^{−1}j}]) .  (41)

By T_{n}(𝔗_{n}^{±}) = 𝔗_{n}^{∓} we conclude that (41) is equal to (40). This together with the relation

(F_{n}L_{n}φ)_{T_{n}j} = c_{n,T_{n}j}(F_{n}φ)_{j} for all j ∈ 𝔗_{n},

implies

\begin{matrix}\frac{1}{4 − n} ∫_{ℝ^{n}} \frac{1}{|x|}|\binom{φ(x)}{(L_{n}φ)(x)}|^{2}dx ≤ \\ ∑_{j∈𝔗_{n}} ∫_{ℝ_{+}} ⟨(\begin{matrix}(F_{n}φ)_{j}(p) \\ (F_{n}L_{n}φ)_{T_{n}j}(p) \end{matrix}) , (\begin{matrix}0 & p \\ p & 0 \end{matrix}) (\begin{matrix}(F_{n}φ)_{j}(p) \\ (F_{n}L_{n}φ)_{T_{n}j}(p) \end{matrix})⟩_{ℂ^{2}} dp. \end{matrix} (42)

A straightforward calculation using (31) - (34) gives

\begin{matrix}⟨\binom{φ}{L_{n}φ}, (\begin{matrix}𝕀_{ℂ^{n−1}} & 0 \\ 0 & ∓𝕀_{ℂ^{n−1}} \end{matrix}) \binom{φ}{L_{n}φ}⟩ \\ = (1 ∓ c_{n}^{−2}) ∑_{j∈𝔗_{n}^{+}} ∥(F_{n}φ)_{j}∥^{2} + (1 ∓ c_{n}^{2}) ∑_{j∈𝔗_{n}^{−}} ∥(F_{n}φ)_{j}∥^{2}. \end{matrix}  (43)

By Lemma 8 we know that the right hand side of Relation (42) plus the minus case of the left hand side of (43) is equal to d_{n}[\binom{φ}{L_{n}φ}]. Thus we obtain (35) by (42) and (43).

4 Proof of Theorem 2

We proceed analogously to the proof of Theorem 1. Thus it is enough to find an operator G_{n} : P_{n}^{+}H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}) → P_{n}^{−}H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}) such that

inf_{φ∈P_{n}^{+}H^{1/2}(ℝ^{n};ℂ^{2(n−1)})\\{0\}} \frac{d_{n}[φ + G_{n}φ] + v[φ + G_{n}φ]}{∥φ + G_{n}φ∥^{2}} > −1  (44)

holds. In the following lemma we prove that a possible choice of G_{n} is

G_{n} := (W_{n}F_{n})^{∗}E_{n}(W_{n}F_{n}),  (45)

with

E_{n} : ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}; ℂ^{2}) → ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}; ℂ^{2});  (46)

⨁_{j∈𝔗_{n}} Ψ_{j} ↦ ⨁_{j∈𝔗_{n}} \frac{1 − c_{n,j}(·) + \sqrt{1 + (·)^{2}}}{c_{n,j} + (·) + c_{n,j}\sqrt{1 + (·)^{2}}} (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) Ψ_{j}. (47)

Lemma 11. Let φ ∈ P_{n}^{+}H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}) then G_{n}φ ∈ P_{n}^{−}H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}) and the relation

L_{n}(φ + G_{n}φ)_{1} = (φ + G_{n}φ)_{2}  (48)

holds.

Remark 12. By Lemma 10 and Relation (48) we conclude (44).

Proof of Lemma 11. By Lemma 8 we deduce that ψ ∈ P_{n}^{±}H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}) if and only if there exists ⨁_{j∈𝔗_{n}} ζ_{j} ∈ ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}; (1 + p^{2})^{1/2}dp) such that

(W_{n}F_{n}ψ)_{j}(p) = \begin{cases}ζ_{j}(p) ( 1_{\frac{p}{1+\sqrt{1+p^{2}}}} ) (”+” case); \\ ζ_{j}(p) ( 1^{\frac{−p}{1+\sqrt{1+p^{2}}}} ) (”-” case); \end{cases}  (49)

holds for every j ∈ 𝔗_{n} and p ∈ ℝ_{+}. Hence we get G_{n}φ ∈ P_{n}^{−}H^{1/2}(ℝ^{n}; ℂ^{2(n−1)}). By (49),(46) we obtain that there exists ⨁_{j∈𝔗_{n}} χ_{j} ∈ ⨁_{j∈𝔗_{n}} L^{2}(ℝ_{+}; (1 + p^{2})^{1/2}dp)

such that

(W_{n}F_{n}φ)_{j}(p) = χ_{j}(p) ( 1_{\frac{p}{1+\sqrt{1+p^{2}}}} )

and

\begin{matrix}((𝕀 + E_{n})W_{n}F_{n}φ)_{j} = ( χ̃_{j} ; c_{n,T_{n}j}χ̃_{j}) with \\ χ̃_{j}(p) := \frac{c_{n,j}(p^{2} + (1 + \sqrt{1 + p^{2}})^{2})}{(1 + \sqrt{1 + p^{2}})(c_{n,j} + p + c_{n,j}\sqrt{1 + p^{2}})}χ_{j}(p) for p ∈ ℝ_{+}, \end{matrix} (50)

hold for every j ∈ 𝔗_{n}. Hence we get by (45),(33) and (34) the relation

φ + G_{n}φ = (W_{n}F_{n})^{∗} ⨁_{j∈𝔗_{n}} ( χ̃_{j} ; c_{n,T_{n}j}χ̃_{j}) = (\begin{matrix}(U_{n}F_{n})^{∗} ⨁_{j∈𝔗_{n}} χ̃_{j} \\ L_{n}(U_{n}F_{n})^{∗} ⨁_{j∈𝔗_{n}} χ̃_{j} \end{matrix}) .

Thus we have proven Relation (48).

5 Proof of Theorem 3

Since the right hand side of (2) is continuous in the H^{1}(ℝ^{n}; ℂ^{n−1}) norm (see Theorem 2.5 in [11]), we can assume that φ ∈ C_{0}^{∞}(ℝ^{n} \ {0}; ℂ^{n−1}) \ {0} by the density of C_{0}^{∞}(ℝ^{n} \ {0}; ℂ^{n−1}) in H^{1}(ℝ^{n}; ℂ^{n−1}).

By the application of Theorem 1 we obtain

λ(v) ≤ sup_{ψ∈H^{1}(ℝ^{n}\{0};ℂ^{n−1})} I_{n,v,φ}(ψ) with  (51)
I_{n,v,φ} : H^{1}(ℝ^{n} \ {0}; ℂ^{n−1}) → ℝ;  (52)

I_{n,v,φ}(ψ) := \frac{⟨ (\begin{matrix}φ \\ ψ \end{matrix}) , (\begin{matrix}(1 + v) ⊗ 𝕀_{ℂ^{n−1}} & K_{n} \\ K_{n} & (−1 + v) ⊗ 𝕀_{ℂ^{n−1}} \end{matrix}) (\begin{matrix}φ \\ ψ \end{matrix}) ⟩}{∥ (\begin{matrix}φ \\ ψ \end{matrix}) ∥^{2}}. (53)

Note that we calculate the suprema in (51) over H^{1}(ℝ^{n} \ {0}; ℂ^{n−1}) instead of H^{1/2}(ℝ^{n}; ℂ^{n−1}). This is justified by a density argument, which makes use of the form boundedness of v ⊗ 𝕀_{ℂ^{2(n−1)}} with respect to D_{n}(0) (see Lemma 9) and the density of H^{1}(ℝ^{n} \ {0}; ℂ^{n−1}) in H^{1/2}(ℝ^{n}; ℂ^{n−1}). Thus the proof of Theorem 3 basically follows from the following lemma.

Lemma 13. We define

J_{n,v,φ} : (−1, ∞) → ℝ;
J_{n,v,φ}(λ) := ∫_{ℝ^{n}} (\frac{|K_{n}φ(x)|^{2}}{1 + λ − v(x)} + (1 − λ + v(x))|φ(x)|^{2}) dx.

For λ ∈ (−1, ∞), J_{n,v,φ}(λ) ≤ 0 implies

sup_{ψ∈H^{1}(ℝ^{n}\{0};ℂ^{n−1})} I_{n,v,φ}(ψ) ≤ λ.

Proof. We introduce

ψ_{n,v,φ} : (−1, ∞) → H^{1}(ℝ^{n} \ \{0\}; ℂ^{n−1}); ψ_{n,v,φ}(λ) := \frac{K_{n}φ}{1 + λ − v} . (54)

For every ζ ∈ H^{1}(ℝ^{n} \ {0}; ℂ^{n−1}) the inequality

(I_{n,v,φ}(ψ_{n,v,φ}(λ) + ζ) − λ)(∥φ∥^{2} + ∥ψ_{n,v,φ}(λ) + ζ∥^{2})
= J_{n,v,φ}(λ) + 2ℜ⟨ζ, K_{n}φ − (1 + λ − v)ψ_{n,v,φ}(λ)⟩+
⟨K_{n}φ − (1 + λ − v)ψ_{n,v,φ}(λ), ψ_{n,v,φ}(λ)⟩ − ⟨ζ, (1 + λ − v)ζ⟩ ≤ J_{n,v,φ}(λ)

holds, and thus we conclude the claim.

By Lemma 13 and (51) we obtain

J_{n,v,φ}(λ(v) − ε) > 0 for ε ∈ (0, 1 + λ(v)).  (55)

Letting ε ↘ 0 in (55) we obtain Theorem 3.

6 Proof of Theorem 5

The proof is based on:

Lemma 14. Let ν ∈ [0, 1/(4 − n)]. The restriction of (D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}))^{∗} to ℭ_{n}^{ν} is essentially self-adjoint.

Proof. For m ∈ 𝔗_{2} and (l, m, s) ∈ 𝔗_{3} we define

κ_{m} := m + 1/2;
κ_{(l,m,s)} := 2sl + s + 1/2.

Furthermore we introduce for every j ∈ 𝔗_{n} the operator D^{j,ν} in L^{2}(ℝ_{+}; ℂ^{2}) by the differential expression

d^{j,ν} := (\begin{matrix}−\frac{ν}{r} & −\frac{d}{dr} − \frac{κ_{j}}{r} \\ \frac{d}{dr} − \frac{κ_{j}}{r} & −\frac{ν}{r} \end{matrix} )

on C_{0}^{∞}(ℝ_{+}; ℂ^{2}). Now we observe that any solution of the equation d^{j,ν}φ = 0 in ℝ_{+} is a linear combination of the two functions

φ_{j,1}^{ν}(r) := \begin{cases}(\begin{matrix}1 \\ 0 \end{matrix}) r^{κ_{j}} if ν = 0, \\ (\begin{matrix}ν \\ \sqrt{κ_{j}^{2} − ν^{2}} − κ_{j} \end{matrix}) r^{\sqrt{κ_{j}^{2}−ν^{2}}} else, \end{cases}

and

φ_{j,2}^{ν}(r) := \begin{cases}(\begin{matrix}0 \\ 1 \end{matrix}) r^{−κ_{j}} if ν = 0, \\ (\begin{matrix}ν \\ −\sqrt{κ_{j}^{2} − ν^{2}} − κ_{j} \end{matrix}) r^{−\sqrt{κ_{j}^{2}−ν^{2}}} if 0 < ν^{2} < κ_{j}^{2}, \\ (\begin{matrix}ν ln(r) \\ 1 − κ_{j} ln(r) \end{matrix}) if ν^{2} = κ_{j}^{2}. \end{cases}

Through the application of the results of [20] as in Section 2 in [14] we obtain that the closure D_{ex}^{j,ν} of the restriction of (D^{j,ν})^{∗} to ℭ^{𝔧,ν} is self-adjoint with

ℭ^{𝔧,ν} := \begin{cases}C_{0}^{∞}(ℝ_{+}; ℂ^{2}) ˙+ span\{ξφ_{j,1}^{ν}\} if κ_{j}^{2} − ν^{2} < 1/4; \\ C_{0}^{∞}(ℝ_{+}; ℂ^{2}) else. \end{cases}

Here ξ is a smooth cut-off function with ξ ∈ C^{∞}(ℝ_{+}; ℝ_{+}), ξ(t) = 1 for t ∈ (0, 1) and ξ(t) = 0 for t > 2. Thus we conclude the claim by

(D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}))^{∗} = (W_{n}M_{n})^{∗} ( ⨁_{j∈𝔗_{n}} (D^{j,ν} + σ_{3})^{∗}) W_{n}M_{n} with, (56)

M_{n} : = diag(1, i) ⊗ 𝕀_{ℂ^{n−1}}

(see Section 7.3.3 in [19] for n = 2 and Section 2.1 in [1] for n = 3) and the fact that σ_{3} is a bounded operator in L^{2}(ℝ_{+}; ℂ^{2}).

Remark 15. Let ν ∈ [0, 1/(4 − n)) and j ∈ 𝔗_{n}. By the embedding

H^{1/2}(ℝ^{n}) ⊂ L^{2}(ℝ^{n}, (1 + |x|^{−1})dx)

and (56) we obtain that the domain of (W_{n}M_{n}D_{n}(−ν/|·|⊗𝕀_{ℂ^{2(n−1)}}) (W_{n}M_{n})^{∗} )_{j} is in L^{2}(ℝ_{+}, (1 + r^{−1})dr). Hence there is a self-adjoint extension of D^{j,ν} with domain in L^{2}(ℝ_{+}, (1 + r^{−1})dr). By ξφ_{j,2}^{ν} ∉ L^{2}(ℝ_{+}, (1 + r^{−1})dr) for ν > 0 and Theorem 1.5 in [20] we get that D_{ex}^{j,ν} is the unique self-adjoint extension of D^{j,ν} with domain in L^{2}(ℝ_{+}, (1 + r^{−1})dr). Therefore, we obtain

(W_{n}M_{n}D_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}) (W_{n}M_{n})^{∗} )_{j} = D_{ex}^{j,ν}.

We conclude that the closure of (D̃_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}))^{∗} restricted to ℭ_{n}^{ν} is D_{n}(−ν/| · | ⊗ 𝕀_{ℂ^{2(n−1)}}).

As a consequence of Lemma 14 it remains to prove that ζ_{n,m}^{ν} ∈ 𝔇(D_{n}^{ν}) for m ∈ \{−1/2, 1/2\}^{n−1} and (n, ν) ∈ (\{2\} × (0, 1/2]) ∪ (\{3\} × (\sqrt{3}/2, 1]). We introduce the symmetric and non-negative (by Corollary 4) quadratic form q_{n}^{ν} on C_{0}^{∞}(ℝ^{n} \ {0}; ℂ^{n−1}) by

q_{n}^{ν}[φ] := ∫_{ℝ^{n}} (\frac{|K_{n}φ|^{2}}{1 + \sqrt{1 − ((4 − n)ν)^{2}} + \frac{ν}{|x|}} +

(1 − \sqrt{1 − ((4 − n)ν)^{2}} − \frac{ν}{|x|})|φ|^{2})dx.

Note that q_{n}^{ν} is closable by Theorem X.23 in [16]. We denote the domain of the closure of q_{n}^{ν} by 𝔔_{n}^{ν}. By the characterisation of 𝔇(D_{n}^{ν}) in Theorem 1 in [8], it is enough to show that for all m ∈ {−1/2, 1/2}^{n−1} the upper (n − 1) spinor of ζ_{n,m}^{ν} is in 𝔔_{n}^{ν}, i.e., ς_{n,m}^{ν} ∈ 𝔔_{n}^{ν} with ς_{2,m}^{ν} given in polar coordinates by

ς_{2,m}^{ν}(ρ, ϑ) := ξ(ρ)ρ^{\sqrt{1/4−ν^{2}}−1/2}e^{−i(m+1/2)ϑ};

and ς_{3,m}^{ν} in spherical coordinates by

ς_{3,m}^{ν}(r, θ, ϕ) := ξ(r)r^{\sqrt{1−ν^{2}}−1}Ω_{1/2+m_{2},m_{1},−m_{2}}(θ, ϕ).

We achieve this goal by the application of the following abstract lemma Lemma 16. Let q be a closable and non-negative quadratic form on a dense linear subspace 𝔔 of the Hilbert space ℌ and ψ ∈ ℌ. If there is a sequence (ψ_{n})_{n∈ℕ} ⊂ 𝔔 with sup_{n∈ℕ} q[ψ_{n}] < ∞ which converges weakly in ℌ to ψ, then ψ is

in the domain of the closure of q.

Proof. We denote by \overline{q} the closure of q and by \overline{𝔔} the domain of \overline{q}. There is a unique self-adjoint operator B : \overline{𝔔} → ℌ with

\overline{q}[φ] = ∥Bφ∥^{2} for all φ ∈ \overline{𝔔}

by Theorem 2.13 in [18] (B^{2} corresponds to A there). Thus we know that

sup_{n∈ℕ} ∥Bψ_{n}∥^{2} < ∞.

Hence there is a Ψ ∈ ℌ and a subsequence (Bψ_{n_{k}})_{n_{k}∈ℕ} of (Bψ_{n})_{n∈ℕ} ⊂ ℌ that converges weakly to Ψ by the Banach-Alaoglu Theorem. This implies that ((ψ_{n_{k}}, Bψ_{n_{k}}))_{n_{k}∈ℕ} converges weakly to (ψ, Ψ) ∈ ℌ ⊕ ℌ. By the closedness of the graph of B and Theorem 8 in Chapter 1 of [3] we deduce the claim.

Now we pick υ ∈ C_{0}^{∞}(ℝ_{+}) such that υ(r) = ξ(r) for all r ∈ [1, ∞) and 0 ≤ υ(r) ≤ 1 for r ∈ (0, 1). Let k ∈ ℕ. We define

υ_{k}(r) := \begin{cases}υ(kr) & if r ∈ (0, 1/k]; \\ 1 & if r ∈ (1/k, 1]; \\ ξ(r) & else ; \end{cases}

and the function ς_{2,m,k}^{ν} in the polar coordinates by

ς_{2,m,k}^{ν}(ρ, ϑ) := υ_{k}(ρ)ρ^{\sqrt{1/4−ν^{2}}−1/2}e^{−i(m+1/2)ϑ},

and ς_{3,m,k}^{ν} in the spherical coordinates by

ς_{3,m,k}^{ν}(r, θ, ϕ) := υ_{k}(r)r^{\sqrt{1−ν^{2}}−1}Ω_{1/2+m_{2},m_{1},−m_{2}}(θ, ϕ).

The sequence (ς_{n,m,k}^{ν})_{k∈ℕ} converges to ς_{n,m}^{ν} in L^{2}(ℝ^{n}; ℂ^{n−1}). By Lemma 16 it is thus enough to prove that

sup_{k∈ℕ} q_{n}^{ν}[ς_{n,m,k}^{ν}] < ∞.  (57)

Let φ ∈ C_{0}^{∞}(ℝ^{n} \ {0}; ℂ^{n−1}). At first we observe that

q_{n}^{ν}[φ] ≤ ∫_{ℝ^{n}} (\frac{|x|}{ν}|K_{n}φ|^{2} − \frac{ν}{|x|}|φ|^{2} + |φ|^{2})dx.  (58)

A tedious calculation shows

K_{n} = \begin{cases}−ie^{iϑ}(∂_{ϱ} − \frac{1}{ρ}A_{2}) with A_{2} := −i∂_{ϑ} if n = 2; \\ −i (σ · \frac{x}{|x|}) (∂_{r} − \frac{1}{r}A_{3}) with A_{3} := σ · ( − ix ∧ ∇) if n = 3. \end{cases} (59)

Using (59) and integration by parts we obtain that the right hand side of (58) is equal to

∫_{ℝ^{n}} (\frac{|x|}{ν} |∂_{|x|}φ|^{2}+ \frac{1}{ν|x|} |(1/(4 − n) + A_{n})φ|^{2} − \frac{(ν + \frac{1}{(4−n)^{2}ν} )}{|x|} |φ|^{2} + |φ|^{2}) dx. (60)

By (60) and Relation 2.1.37 in [1] we obtain

∫_{ℝ^{n}} (\frac{|x|}{ν}|K_{n}ς_{n,m,k}^{ν}|^{2} − \frac{ν}{|x|}|ς_{n,m,k}^{ν}|^{2} + |ς_{n,m,k}^{ν}|^{2})dx

= ∫_{0}^{∞}(\frac{t^{n}}{ν} |∂_{t}υ_{k}(t)t^{\sqrt{(4−n)^{−2}−ν^{2}}−(4−n)^{−1}} |^{2}  (61)

− νυ_{k}(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}−1} + υ_{k}(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}})dt.

A straightforward calculation shows that (61) is equal to

\begin{aligned}& ∫_{0}^{∞}(ν^{−1}υ′_{k}(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}+1} + υ_{k}(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}}) dt \\ = & ∫_{0}^{1/k}ν^{−1}k^{2}υ′(kt)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}+1}dt + ∫_{1}^{∞}ν^{−1}υ′(t)^{2}r^{2\sqrt{(4−n)^{−2}−ν^{2}}+1}dt \\ + & ∫_{0}^{∞}υ_{k}(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}}dt. \end{aligned} (62)

An upper bound for the expression in (62) is

∫_{0}^{∞}ν^{−1}υ′(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}+1}dt + ∫_{0}^{∞}ξ(t)^{2}t^{2\sqrt{(4−n)^{−2}−ν^{2}}}dt.  (63)

The combination of (63), (62) (61) and (58) implies (57).

References

[1] Alexander A. Balinsky and William D. Evans. Spectral analysis of relativistic operators. Imperial College Press, 2011.

[2] Abdelkader Bouzouina. Stability of the two-dimensional Brown-Ravenhall operator. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 132(05):1133–1144, 2002.

[3] Ward Cheney. Analysis for applied mathematics, volume 208. Springer Science & Business Media, 2013.

[4] Jean-Claude Cuenin and Heinz Siedentop. Dipoles in graphene have infinitely many bound states. Journal of Mathematical Physics, 55(12), 2014.

[5] Jean Dolbeault, Maria J. Esteban, Michael Loss, and Luis Vega. An analytical proof of Hardy-like inequalities related to the Dirac operator. Journal of Functional Analysis, 216(1):1–21, 2004.

[6] Jean Dolbeault, Maria J. Esteban, and Eric Séré. On the eigenvalues of operators with gaps. Application to Dirac operators. Journal of Functional Analysis, 174(1):208–226, 2000.

[7] Shi-Hai Dong and Zhong-Qi Ma. Exact solutions to the Dirac equation with a Coulomb potential in 2+1 dimensions. Physics Letters A, 312(1):78– 83, 2003.

[8] Maria J. Esteban and Michael Loss. Self-adjointness via partial Hardy-like inequalities. In Mathematical results in quantum mechanics, pages 41–47. World Sci. Publ., Hackensack, NJ, 2008.

[9] Maria J. Esteban and Eric Séré. Existence and multiplicity of solutions for linear and nonlinear Dirac problems. In Partial Differential Equations and their Applications, volume 12 of CRM Proceedings and Lecture Notes, pages 107–118. American Mathematical Society, 1997.

[10] William D. Evans, Peter Perry, and Heinz Siedentop. The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Communications in Mathematical Physics, 178(3):733–746, 1996.

[11] Ira W. Herbst. Spectral theory of the operator (p^{2} + m^{2})^{1/2} − Ze^{2}/r. Communications in Mathematical Physics, 53(3):285–294, 1977.

[12] Tosio Kato. Holomorphic families of Dirac operators. Mathematische Zeitschrift, 183(3):399–406, 1983.

[13] Sergey Morozov and David Müller. On the minimax principle for Coulomb-Dirac operators. Mathematische Zeitschrift, 280:733–747, 2015.

[14] Sergey Morozov and David Müller. Lieb-Thirring and Cwickel-Lieb-Rozenblum inequalities for perturbed graphene with a Coulomb impurity. Preprint, 2016.

[15] Gheorghe Nenciu. Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms. Communications in Mathematical Physics, 48(3):235–247, 1976.

[16] Michael Reed and Barry Simon. Methods of modern mathematical physics II: Fourier analysis, self-adjointness, volume 2. Academic Press, 1975.

[17] James D. Talman. Minimax principle for the Dirac equation. Physical Review Letters, 57(9):1091–1094, 1986.

[18] Gerald Teschl. Mathematical methods in quantum mechanics, volume 99. American Mathematical Society, 2009.

[19] Bernd Thaller. The Dirac equation. Springer-Verlag, Berlin, 1992.

[20] Joachim Weidmann. Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen. Mathematische Zeitschrift, 119:349–373, 1971.

[21] Edmund T. Whittaker and George N. Watson. A course of modern analysis. Cambridge University Press, 1996.

David Müller Mathematik, LMU Theresienstr. 39 80333 München Germany dmueller@math.lmu.de 1170

Documenta Mathematica 21 (2016)