Ground state of the polaron hydrogenic atom in a strong magnetic field

Ghanta, Rohan

doi:10.1063/5.0012192

The ground-state electron density of a polaron bound to a Coulomb potential in a homogeneous magnetic field—the transverse coordinates integrated out—converges pointwise and weakly in the strong magnetic field limit to the square of a hyperbolic secant function.

I. INTRODUCTION

A non-relativistic hydrogen atom in a strong magnetic field interacting with the quantized longitudinal optical modes of an ionic crystal is considered within the framework of Fröhlich’s 1950 polaron model.²⁶ Starting with Platzman’s variational treatment in 1962, the polaron hydrogenic atom has been of interest for describing an electron bound to a donor impurity in a semiconductor.⁴⁹ Its first rigorous examination, however, came much later in 1988 from Löwen who disproved several longstanding claims about a self-trapping transition.⁴⁵

A study of the polaron hydrogenic atom in strong magnetic fields was initiated by Larsen in 1968 for interpreting cyclotron resonance measurements in InSb.³⁵ The model has since been considered in formal analogy to the hydrogen atom in a magnetic field though the latter was understood rigorously again much later in 1981 by Avron et al. who proved several properties including the non-degeneracy of the ground state.³ Whether or not these hydrogenic properties indeed persist when a coupling to a quantized field is turned on remains to be seen.

Polarons are the simplest quantum field theory models, yet their most basic features such as the effective mass, ground-state energy, and wave function cannot be evaluated explicitly. And, quite unfortunately, while several successful theories have been proposed over the years to approximate the energy and effective mass of various polarons, usually they are built entirely on unjustified, even questionable, Ansätze for the wave function (see Refs. 15, 26, and 48); the only exceptions are the rigorous treatments^{1,2,6–8,13,17,19–23,27,30,42,43,45–47,53,57} including the recent work on the effective mass.^14,38,39 The paper provides now for the first time an explicit description of the wave function.

For the polaron hydrogenic atom in a homogeneous magnetic field, its ground-state electron density in the magnetic-field direction is shown to converge in the strong field limit to the square of a hyperbolic secant function: a sharp contrast to the paradigmatic Gaussian variational wave functions (see Ref. 58 and also Ref. 54 and references therein). The explicit limiting function is realized as a density of the minimizer of a one-dimensional problem with a delta-function potential describing the second leading-order term of the ground-state energy (cf. Refs. 5, 17, and 41).

II. MODEL AND MAIN RESULT

The Fröhlich model is defined by the Hamiltonian

H (B) ≔ H_{B} - \partial_{3}^{2} - β {|x|}^{- 1} + N + \frac{\sqrt{α}}{2 π} \int_{R^{3}} (\frac{a_{k} e^{i k \cdot x}}{|k|} + \frac{a_{k}^{†} e^{- i k \cdot x}}{|k|}) d k

(2.1)

acting on the Hilbert space $H : = L^{2} (R^{3}) \otimes F$ ⁠, where $F : = \oplus_{n \geq 0} \otimes_{s}^{n} L^{2} (R^{3})$ is a symmetric phonon Fock space over $L^{2} (R^{3})$ ⁠. The creation and annihilation operators for a phonon mode $a_{k}^{†}$ and a_k act on $F$ and satisfy $[a_{k}, a_{k^{'}}^{†}] = δ (k - k^{'})$ ⁠. The energy of the phonon field is described by the operator $N = \int_{R^{3}} a_{k}^{†} a_{k} d k$ ⁠. The kinetic energy of the electron is described by the operator $H_{B} - \partial_{3}^{2}$ acting on $L^{2} (R^{3})$ ⁠, where $H_{B} = \sum_{j = 1, 2} {(- i \partial_{j} + A_{j} (x))}^{2}$ is the two-dimensional Landau Hamiltonian with the magnetic vector potential $A (x_{1}, x_{2}, x_{3}) = B / 2 (- x_{2}, x_{1}, 0)$ corresponding to a homogeneous magnetic field of strength B ≥ 0 in the x₃-direction; the transverse coordinates are denoted by x_⊥ = (x₁, x₂). Furthermore, inf spec H_B = B. The parameters α ≥ 0 and β > 0 denote the strengths of the Coulombic electron–phonon interaction and the localizing Coulomb potential; the coupling function $1 / |k|$ is proportional to the square root of the Fourier transform of the Coulomb interaction. The ground-state energy is

E_{0} (B) ≔ \inf \{⟨Ψ, H (B) Ψ⟩ : ‖ Ψ ‖ = 1, Ψ \in H_{A}^{1} (R^{3}) \otimes dom (\sqrt{N})\},

(2.2)

where $H_{A}^{1} (R^{3})$ is a magnetic Sobolev space of order one. A ground state exists since −i∇ − β|x|⁻¹ has a negative energy bound state in $L^{2} (R^{3})$ ⁠.²⁹

Unlike previous treatments, here, the arguments remain valid for all values of the parameters α ≥ 0 and β > 0. First, the large B asymptotics of the ground-state energy is derived; the main result is given as Theorem 2.2. Since the pioneering work of Larsen, the model has been considered only in the perturbative regime α ≪ β, and the ground-state energy E₀(B) has been approximated as the hydrogenic energy,

E_{H} (B) ≔ inf spec H_{B} - \partial_{3}^{2} - β {|x|}^{- 1},

with a supposedly small correction from the electron–phonon interaction. The large B asymptotics of the hydrogenic energy was derived rigorously in 1981 by Avron et al.³ using ideas from Refs. 9 and 55,

\begin{align} E_{H} (B) = B & - \frac{β^{2}}{4} {(\ln B)}^{2} + β^{2} \ln B \ln \ln B - β^{2} (- γ_{E} / 2 + \ln 2) \ln B - β^{2} {(\ln \ln B)}^{2} \\ + 2 β^{2} (- γ_{E} / 2 - 1 + \ln 2) \ln \ln B + O (1) as B \to \infty, \end{align}

(2.3)

with γ_E being the Euler–Mascheroni constant, and the expansion can be carried out to arbitrary order. The first three terms are understood heuristically: For large B, the electron is tightly bound in the transverse plane to the lowest Landau orbit while localized in the magnetic-field direction by a one-dimensional effective Coulomb potential that behaves to leading order like a delta well of strength β ln(B/(ln B)²)^12,51,52 [see (4.1) and Appendix B below]. The electron motion is effectively one-dimensional (cf. Ref. 11). The second and third leading-order terms describe the dominant asymptotic behavior of the ground-state energy of this one-dimensional electron confined along the magnetic field. The pronounced anisotropy in the system is reflected by the characteristic length scales of the electron density in the transverse and the magnetic-field direction $1 / \sqrt{B}$ and 1/ln B, respectively.

The above hydrogenic heuristics still apply when a coupling to the phonon field is introduced, i.e., α > 0. For large B, the phonons cannot follow the electron’s rapid motion in the transverse plane and so resign themselves to dressing its entire Landau orbit: not only is the electron again localized in the magnetic-field direction by the one-dimensional effective Coulomb potential but also the electron–phonon coupling function is now proportional to the square root of the Fourier transform of the same effective Coulomb interaction [cf. Refs. 34 and 56 and property (k) in Ref. 51]; the system behaves as a one-dimensional strongly coupled polaron to the leading order with interaction strength α ln(B/(ln B)²) confined along the magnetic field by a delta well of strength β ln(B/(ln B)²), i.e., in the effective one-dimensional model, the electron–phonon coupling is mediated by the magnetic field. The analogous large B asymptotics of the polaron hydrogenic energy is derived to second order.

Theorem 2.1.

Let E₀(B) be as defined in (2.2) above. Then,

E_{0} (B) = B + e_{0} {(\ln B)}^{2} + O ({(\ln B)}^{3 / 2}) as B \to \infty, with

(2.4)

e_{0} : = \inf \{\int_{R} {|φ^{'}|}^{2} d x - \frac{α}{2} \int_{R} {|φ|}^{4} d x - β {|φ (0)|}^{2} : \int_{R} {|φ|}^{2} d x = 1\}

(2.5)

= - \frac{1}{48} (α^{2} + 6 α β + 12 β^{2}) .

(2.6)

Here, the second leading-order term describes the dominant asymptotic behavior of the ground-state energy of the effective one-dimensional strongly coupled polaron confined along the magnetic field. It is evaluated explicitly by minimizing a nonlinear functional. Furthermore, the cross term in (2.6) indicates that for large B, the effect of the electron–phonon interaction is not perturbative.

The large B asymptotics for the polaron hydrogenic energy is argued differently from the proof of (2.3) given by Avron et al. and generalizes the result of Frank and Geisinger who proved (2.4)–(2.6) when β = 0 using upper and lower bounds to the ground-state energy.¹⁷ Their upper bound is established with a trial wave function. Their lower bound is established by showing that the Hamiltonian when restricted to the lowest Landau level is bounded from below in the sense of quadratic forms by an essentially one-dimensional strong-coupling Hamiltonian; the strategy from Ref. 42 is then used to arrive at the nonlinear minimization problem for the second leading-order term along with lower order error terms (cf. Ref. 28).

For proving Theorem 2.1, Frank and Geisinger's strategy in Ref. 17 applies mutatis mutandis. Here, the argument for the upper bound is simplified using the effective Coulomb potential, given in (4.1), which plays an essential role in the proof of (2.3) by Avron et al. but is conspicuously absent from Ref. 17. Here, the argument for the lower bound uses the bathtub principle³⁷ to extract a delta function from the Coulomb potential in the lowest Landau level; just like in Ref. 17, the error terms are bounded by $O ({(\ln B)}^{3 / 2})$ ⁠, but—and this has been demonstrated recently in Ref. 24 for a strongly coupled polaron—the error terms in the lower bound should be much smaller. Moreover, the upper bound suggests that the third leading-order term again analogously to its hydrogenic counterpart in (2.3) is $- 4 e_{0} \ln B \ln \ln B$ ⁠.

The expansion in (2.4) should be carried out to higher order. I conjecture the first six leading-order terms behave analogously to their hydrogenic counterparts in (2.3): these should describe the leading asymptotics for the minimization of a nonlinear functional arising naturally in the proof of the upper bound and given below in (4.3), which is a classical approximation to the Fröhlich model of the strongly coupled one-dimensional polaron confined along the magnetic field: in this classical approximation, the electron–phonon interaction is replaced with the one-dimensional effective Coulomb self-interaction of the electron. The second leading-order term above arises from this classical approximation by estimating the one-dimensional effective Coulomb interaction as a delta interaction of strength ln B. Furthermore, in the seventh leading-order term, there should be an order-one quantum correction to the classical approximation (cf. Refs. 24 and 32). These higher-order asymptotics should be provable with better control of the error terms when arguing the lower bound (cf. Ref. 24) and by making full use of the one-dimensional effective Coulomb potential: the electron–phonon interaction of the one-dimensional Hamiltonian that is derived both here and in Frank and Geisinger's proof of the lower bound is described using an artificial coupling function, given in (5.9) below, when instead it should really be argued that the coupling is proportional to the square-root of the Fourier transform of the effective Coulomb potential in (4.1); then, the strategy from Ref. 42 can be used to arrive at the classical approximation in (4.3) now as a lower bound.

The two-term asymptotics of the ground-state energy achieved in Theorem 2.1 suffices for arguing the main result.

Theorem 2.2.

Let

Ψ^{(B)} \in H

be any approximate ground-state wave function satisfying

⟨Ψ^{(B)}, H (B) Ψ^{(B)}⟩ = E_{0} (B) + o ({(\ln B)}^{2})

. The one-dimensional minimization problem in (2.5) admits up to complex phase a unique minimizer,

ϕ_{0} (x_{3}) = \frac{α + 2 β}{\sqrt{8 α} \cosh ((\frac{α + 2 β}{4}) |x_{3}| + \tanh^{- 1} (\frac{2 β}{α + 2 β}))},

and, for W, a sum of a bounded Borel measure on the real line and a

L^{\infty} (R)

function,

\begin{align} \lim_{B \to \infty} \frac{1}{(\ln B)} \int_{R} W (x_{3}) (\int_{R^{2}} ‖ Ψ^{(B)} ‖_{F}^{2} (x_{⊥}, \frac{x_{3}}{(\ln B)}) d x_{⊥}) d x_{3} \\ = \int_{R} W (x_{3}) ϕ_{0} {(x_{3})}^{2} d x_{3} . \end{align}

(2.7)

By choosing W as a delta-function potential, pointwise convergence is obtained. When α = 0, the limiting density in (2.7) is $\sqrt{(β / 2)} \exp (- β | x_{3} | / 2)$ (cf. Refs. 25 and 50).

For proving Theorem 2.2, the strategy is to add to the Hamiltonian ϵ times the one-dimensional potential W scaled appropriately in the magnetic-field direction (cf. Refs. 4, 18, 33, 40, 41, and 44). For

H_{ϵ} (B) ≔ H (B) - ϵ {(\ln B)}^{2} W ((\ln B) x_{3})

(2.8)

and E_ϵ(B) being the corresponding ground-state energy, it is argued vis-à-vis Theorem 2.1 that

E_{ϵ} (B) = B + e_{ϵ} {(\ln B)}^{2} + O ({(\ln B)}^{3 / 2}) as B \to \infty,

(2.9)

with

e_{ϵ} ≔ \inf_{‖ φ ‖_{2} = 1} \{\int_{R} {|φ^{'}|}^{2} d x - \frac{α}{2} \int_{R} {|φ|}^{4} d x - β {|φ (0)|}^{2} - ϵ \int_{R} W (x) {|φ|}^{2} d x\} .

(2.10)

By the variational principle, $E_{ϵ} (B) \leq ⟨Ψ^{(B)}, H_{ϵ} (B) Ψ^{(B)}⟩$ ⁠, and the expectation value on the right-hand side evaluates to

E_{0} (B) + o ({(\ln B)}^{2}) - ϵ (\ln B) \int_{R} W (x_{3}) (\int_{R^{2}} ‖) Ψ^{(B)} {(‖}_{F}^{2} (x_{⊥}, \frac{x_{3}}{\ln B}) d x_{⊥}) d x_{3} .

Then, for ϵ > 0,

\frac{E_{0} (B) - E_{ϵ} (B)}{ϵ {(\ln B)}^{2}} \geq \frac{1}{(\ln B)} \int_{R} W (x_{3}) (\int_{R^{2}} ‖) Ψ^{(B)} {(‖}_{F}^{2} (x_{⊥}, \frac{x_{3}}{\ln B}) d x_{⊥}) d x_{3} + o (1),

and taking the limit B → ∞, by Theorem 2.1 and (2.9),

\frac{e_{0} - e_{ϵ}}{ϵ} \geq \underset{B \to \infty}{lim sup} \frac{1}{(\ln B)} \int_{R} W (x_{3}) (\int_{R^{2}} ‖) Ψ^{(B)} {(‖}_{F}^{2} (x_{⊥}, \frac{x_{3}}{\ln B}) d x_{⊥}) d x_{3} .

For ϵ < 0, the above inequality is reversed with “lim sup” replaced by “lim inf.” Hence, the theorem follows if the quotient on the left-hand side has a limit as ϵ → 0. The map $ϵ \mapsto e_{ϵ}$ is differentiable at ϵ = 0 for all W if and only if the one-dimensional problem for the energy $e_{0}$ in (2.5) admits up to the complex phase a unique minimizer: Uniqueness is established by explicitly solving the corresponding Euler–Lagrange equation (cf. Ref. 41) and by the variational principle and a compactness argument,

\lim_{ϵ \to 0} \frac{e_{0} - e_{ϵ}}{ϵ} = \int_{R} W (x_{3}) ϕ_{0} {(x_{3})}^{2} d x_{3} .

(2.11)

In Sec. III, the differentiation of the one-dimensional energy (2.11) is proved as Theorem 3.3. In Sec. IV, an upper bound to the ground-state energy is proved as Theorem 4.1. In Sec. V, a lower bound to the ground-state energy is proved as Theorem 5.1, and Theorem 2.1 follows from Theorems 4.1 and 5.1. In Sec. VI, the main result Theorem 2.2 is proved.

III. DIFFERENTIATING THE ONE-DIMENSIONAL ENERGY

The one-dimensional problem for the energy $e_{0}$ in (2.5) is denoted as

e_{0} ≔ \inf_{‖ φ ‖_{2} = 1} E_{0} (φ),

with

E_{0} (φ) ≔ \int_{R} {|φ^{'}|}^{2} d x - \frac{α}{2} \int_{R} {|φ|}^{4} d x - β {|φ (0)|}^{2} .

(3.1)

In the following, Lemmas 3.1, 3.2, and Theorem 3.3 apply for all α ≥ 0 and β > 0.

Lemma 3.1.

The minimization problem in (2.5) for the energy

e_{0}

admits up to the complex phase a unique minimizer,

ϕ_{0} (x) = \frac{α + 2 β}{\sqrt{8 α} \cosh ((\frac{α + 2 β}{4}) |x| + \tanh^{- 1} (\frac{2 β}{α + 2 β}))}

(3.2)

and

e_{0} = - \frac{1}{48} (α^{2} + 6 α β + 12 β^{2}) .

When α = 0, the corresponding minimizer in (3.2) is $\sqrt{(β / 2)} \exp (- β | x | / 2)$ ⁠.

Proof.

The existence of a minimizer is shown in Appendix A. Any minimizer up to multiplication by a complex phase is a nonnegative

C^{2} (R \ \{0\})

function in

H^{1} (R)

solving the Euler–Lagrange equation −ψ″ − αψ³ − βδ(x)ψ = −λψ with

- λ = e_{0} - \frac{α}{2} \int_{R} ψ^{4} d x < 0 .

(3.3)

Or equivalently, it must solve

- ψ^{″} - α ψ^{3} = - λ ψ for |x| > 0

(3.4)

and satisfy the boundary condition

\lim_{ϵ \to 0^{+}} [ψ^{'} (- ϵ) - ψ^{'} (ϵ)] = β ψ (0) .

(3.5)

The first integral of () is

ψ^{' 2} = - \frac{α}{2} ψ^{4} + λ ψ^{2} for |x| > 0 .

(3.6)

When α = 0, the lemma is obvious. When α > 0, any nonnegative

C^{2} (R \ {0})

solution of () in

H^{1} (R)

satisfying the boundary condition in () must be of the form

ψ = \sqrt{\frac{2 λ}{α}} \frac{1}{\cosh (\sqrt{λ} (|x| - τ))}

(3.7)

for some τ (cf. Ref. 16). The boundary condition and that ‖ψ‖₂ = 1 require

\tanh (τ \sqrt{λ}) = - \frac{β}{2 \sqrt{λ}} and \frac{α}{4 \sqrt{λ}} = 1 + \tanh (τ \sqrt{λ}),

respectively. Any minimizer up to complex phase must therefore be of the form in () with

λ = {(\frac{α + 2 β}{4})}^{2} and τ = - \frac{4}{α + 2 β} \tanh^{- 1} (\frac{2 β}{α + 2 β}) .

The explicit calculation of

e_{0}

now follows from ().□

Lemma 3.2.

If ${\{ϕ_{n}\}}_{n = 1}^{\infty}$ is a minimizing sequence for $e_{0}$ , then $|ϕ_{n}|$ converges in $H^{1} (R)$ to the minimizer ϕ₀ given in (3.2).

Proof.

By Theorem 7.8 in Ref. 37, ${\{|ϕ_{n}|\}}_{n = 1}^{\infty}$ is also a minimizing sequence for $e_{0}$ ⁠. It is argued in Appendix A that every minimizing sequence for $e_{0}$ has a subsequence converging in $H^{1} (R)$ to some minimizer. Since ϕ₀ is up to complex phase the unique minimizer, every subsequence of ${\{|ϕ_{n}|\}}_{n = 1}^{\infty}$ must converge in $H^{1} (R)$ to ϕ₀.□

Theorem 3.3.

Let W be a sum of a bounded Borel measure on the real line and a

L^{\infty} (R)

function. For ϵ a real parameter, consider the one-dimensional energy

e_{ϵ} ≔ \inf_{‖ φ ‖_{2} = 1} E_{ϵ} (φ),

(3.8)

where

E_{ϵ} (φ) ≔ E_{0} (φ) - ϵ \int_{R} W (x) {|φ (x)|}^{2} d x

(3.9)

with the functional

E_{0}

as given in (3.1). Then, the map

ϵ \mapsto e_{ϵ}

is differentiable at ϵ = 0 and

{(\frac{d}{d ϵ}|}_{ϵ = 0} e_{ϵ} = - \int_{R} W (x) ϕ_{0} {(x)}^{2} d x,

with ϕ₀ being the minimizer given in (3.2) for the energy

e_{0}

.

Proof.

Writing W = μ + ω, where μ is a signed bounded measure on the real line and

ω \in L^{\infty} (R)

⁠, it follows from Hölder’s inequality, the Sobolev inequality, and completion of the square that for

φ \in H^{1} (R)

⁠,

\begin{align} E_{ϵ} (φ) & \geq ‖ φ^{'} ‖_{2}^{2} - ‖ φ ‖_{\infty}^{2} (\frac{α}{2} ‖ φ ‖_{2}^{2} + |ϵ| |μ| (R) + β) - |ϵ| ‖ ω ‖_{\infty} ‖ φ ‖_{2}^{2} \\ \geq \frac{3}{4} ‖ φ^{'} ‖_{2}^{2} - ‖ φ ‖_{2}^{2} {(\frac{α}{2} ‖ φ ‖_{2}^{2} + |ϵ| |μ| (R) + β)}^{2} - |ϵ| ‖ ω ‖_{\infty} ‖ φ ‖_{2}^{2} . \end{align}

(3.10)

Hence,

e_{ϵ} > - \infty

⁠, and for each ϵ, there is a

ϕ_{ϵ} \in H^{1} (R)

satisfying

E_{ϵ} (ϕ_{ϵ}) \leq e_{ϵ} + ϵ^{2} and ‖ ϕ_{ϵ} ‖_{2} = 1 .

(3.11)

By the variational principle,

e_{0} \leq E_{0} (ϕ_{ϵ}) = E_{ϵ} (ϕ_{ϵ}) + ϵ \int_{R} W (x) {|ϕ_{ϵ} (x)|}^{2} d x \leq e_{ϵ} + ϵ^{2} + ϵ \int_{R} W (x) {|ϕ_{ϵ} (x)|}^{2} d x

and

e_{ϵ} \leq E_{ϵ} (ϕ_{0}) = e_{0} - ϵ \int_{R} W (x) ϕ_{0} {(x)}^{2} d x .

Then, for ϵ > 0,

- \int_{R} W (x) {|ϕ_{ϵ} (x)|}^{2} d x - ϵ \leq \frac{e_{ϵ} - e_{0}}{ϵ} \leq - \int_{R} W (x) ϕ_{0} {(x)}^{2} d x .

(3.12)

For ϵ < 0, the inequalities in () are reversed. It suffices therefore to show for any sequence

{\{ϵ_{n}\}}_{n = 1}^{\infty}

⁠,

|ϵ_{n}| > 0

and ϵ_n → 0 that

\lim_{n \to \infty} \int_{R} W (x) {|ϕ_{ϵ_{n}} (x)|}^{2} d x = \int_{R} W (x) ϕ_{0} {(x)}^{2} d x .

(3.13)

Since

e_{ϵ} > - \infty

⁠, the concave map

ϵ \mapsto e_{ϵ}

is continuous. It follows from (), the Sobolev inequality, and () that

‖ ϕ_{ϵ_{n}} ‖_{\infty} < C

⁠. Thus,

\lim_{n \to \infty} ϵ_{n} \int_{R} W (x) {|ϕ_{ϵ_{n}} (x)|}^{2} d x = 0,

\begin{array}{l} e_{0} = \lim_{n \to \infty} e_{ϵ_{n}} & \geq \underset{n \to \infty}{lim sup} (E_{0} (ϕ_{ϵ_{n}}) - ϵ_{n} \int_{R} W (x) {|ϕ_{ϵ_{n}} (x)|}^{2} d x - ϵ_{n}^{2}) \\ = \underset{n \to \infty}{lim sup} E_{0} (ϕ_{ϵ_{n}}), \end{array}

and

{\{ϕ_{ϵ_{n}}\}}_{n = 1}^{\infty}

is a minimizing sequence for

e_{0}

⁠. By Lemma 3.2,

|ϕ_{ϵ_{n}}|

converges in

H^{1} (R)

to ϕ₀, and by Theorem 8.6 in Ref. 37,

|ϕ_{ϵ_{n}}|

converges also pointwise uniformly to ϕ₀ on bounded sets. The convergence in () now follows.□

IV. UPPER BOUND TO THE GROUND-STATE ENERGY

Theorem 4.1.

There is a constant C > 0 such that for B > 1,

E_{0} (B) \leq B + e_{0} {(\ln B)}^{2} - 4 e_{0} \ln B \ln \ln B + C \ln B .

Theorem 4.1 will follow from Lemmas 4.2 and 4.3.

Lemma 4.2.

E_{0} (B) \leq E_{0}^{c} (B)

, where

E_{0}^{c} (B) : = \inf \{P (ψ) : ‖ ψ ‖_{2} = 1\}

is the classical Pekar energy with

P

denoting the three-dimensional magnetic Pekar functional

{⟨ψ, (H_{B} - \partial_{3}^{2}) ψ⟩}_{L^{2}} - \frac{α}{2} \int_{R^{3}} \int_{R^{3}} \frac{{|ψ (x)|}^{2} {|ψ (y)|}^{2}}{|x - y|} d x d y - β \int_{R^{3}} \frac{{|ψ (x)|}^{2}}{|x|} d x .

Proof.

By the variational principle,

E_{0} (B) \leq \inf \{⟨Ψ, H (B) Ψ⟩ : ‖ Ψ ‖ = 1, Ψ = φ (x) Φ and Φ \in F\} = E_{0}^{c} (B),

with the equality following from the completion of the square in annihilation and creation operators.□

Lemma 4.3.

There is a constant C > 0 such that for B > 1,

E_{0}^{c} (B) \leq B + e_{0} {(\ln B)}^{2} - 4 e_{0} \ln B \ln \ln B + C \ln B .

Proof.

With the lowest Landau state in the zero angular momentum sector,

γ_{B} (x_{⊥}) ≔ \sqrt{\frac{B}{2 π}} \exp (- \frac{B}{4} {|x_{⊥}|}^{2}), i.e., H_{B} γ_{B} = B γ_{B},

by an elementary calculation,¹²

V_{U}^{B} (x_{3}) ≔ \int_{R^{2}} \frac{{|γ_{B} (x_{⊥})|}^{2}}{\sqrt{{|x_{⊥}|}^{2} + x_{3}^{2}}} d x_{⊥} = \int_{0}^{\infty} \frac{e^{- u}}{\sqrt{x_{3}^{2} + \frac{2 u}{B}}} d u

(4.1)

and

\int_{R^{2}} \int_{R^{2}} \frac{{|γ_{B} (x_{⊥})|}^{2} {|γ_{B} (y_{⊥})|}^{2}}{\sqrt{{|x_{⊥} - y_{⊥}|}^{2} + {(x_{3} - y_{3})}^{2}}} d x_{⊥} d y_{⊥} = \frac{1}{\sqrt{2}} V_{U}^{B} (\frac{x_{3} - y_{3}}{\sqrt{2}}) .

(4.2)

For L > 0 and with ϕ₀ being the minimizer for the energy

e_{0}

⁠, μ(B): = ln B − 2 ln ln B and

f_{B} (x_{3}) : = \sqrt{μ (B)} ϕ_{0} (μ (B) x_{3})

by the variational principle, Lemma B.1, and Corollary B.2,

\begin{align} E_{0}^{c} (B) & \leq P (γ_{B} (x_{⊥}) f_{B} (x_{3})) \\ = B + \int_{R} {|f_{B}^{'}|}^{2} d x_{3} - \frac{α}{\sqrt{8}} \iint_{R \times R} {|f_{B} (x_{3})|}^{2} V_{U}^{B} (\frac{x_{3} - y_{3}}{\sqrt{2}}) {|f_{B} (y_{3})|}^{2} d x_{3} d y_{3} \\ - β \int_{R} V_{U}^{B} (x_{3}) {|f_{B} (x_{3})|}^{2} d x_{3} \\ \leq B + \int_{R} {|f_{B}^{'}|}^{2} d x_{3} - (α / 2) μ (B) \int_{R} {|f_{B} (x_{3})|}^{4} d x_{3} - β μ (B) {|f_{B} (0)|}^{2} \\ + (α / 2 + β) (L^{- 1} + 8 \sqrt{L} ‖ f_{B}^{'} ‖_{2}^{3 / 2} + |G (B, L / \sqrt{2})| ‖ f_{B}^{'} ‖_{2}), \end{align}

(4.3)

where

G (B, L) : = 2 \ln L + 2 \ln \ln B + 2 \int_{0}^{\infty} e^{- u} \ln (\sqrt{\frac{1}{u} + \frac{2}{B L^{2}}} + \sqrt{\frac{1}{u}}) d u - \ln 2

⁠. Now choosing L = 1/ln B, it can be verified that

|G (B, L / \sqrt{2})| < C

for B > 1. Since ‖f_B′‖₂ = μ(B)‖ϕ₀′‖₂ and

\int_{R} {|f_{B}^{'}|}^{2} d x_{3} - (α / 2) μ (B) \int {|f_{B}|}^{4} d x_{3} - β μ (B) {|f_{B} (0)|}^{2} = {(μ (B))}^{2} e_{0},

(4.4)

the lemma follows.□

Corollary 4.4.

Let W be a sum of a bounded Borel measure on the real line and a

L^{\infty} (R)

function. For ϵ a real parameter, E_ϵ(B) the ground-state energy of the Hamiltonian

H_{ϵ} (B)

in (2.8) and $e_{ϵ}$ the one-dimensional energy in (2.10) there is a constant C > 0 such that for B > 1,

E_{ϵ} (B) \leq B + e_{ϵ} {(\ln B)}^{2} + C \ln B |\ln \ln B| + C \ln B .

Proof.

By the estimate in (),

e_{ϵ} > - \infty

⁠. Then, with the functional

E_{ϵ}

as given in () for B > 1, there is a

ϕ_{B} \in H^{1} (R)

⁠, ‖ϕ_B‖₂ = 1, satisfying

E_{ϵ} (ϕ_{B}) < e_{ϵ} + 1 / \ln B and ‖ ϕ_{B}^{'} ‖_{2} < C .

(4.5)

For L > 0 and with

g_{B} (x_{3}) : = \sqrt{\ln B} ϕ_{B} ((\ln B) x_{3})

by trivial modifications to Lemma B.1 and Corollary B.2, the arguments in the Proofs of Lemmas 4.2 and 4.3 apply mutatis mutandis, and

E_{ϵ} (B) \leq E_{ϵ}^{c} (B)

with

\begin{array}{l} E_{ϵ}^{c} (B) ≔ \inf_{‖ ψ ‖_{2} = 1} \{P (ψ) - ϵ {(\ln B)}^{2} \int_{R^{3}} W ((\ln B) x_{3}) {|ψ (x_{⊥}, x_{3})|}^{2} d x_{⊥} d x_{3}\} \\ \leq B + \int_{R} {|g_{B}^{'}|}^{2} d x_{3} - (α / 2) \ln B \int_{R} {|g_{B}|}^{4} d x_{3} - β \ln B {|g_{B} (0)|}^{2} \\ - ϵ {(\ln B)}^{2} \int_{R} W ((\ln B) x_{3}) {|g_{B} (x_{3})|}^{2} d x_{3} \\ + (α / 2 + β) (L^{- 1} + 8 \sqrt{L} ‖ g_{B}^{'} ‖_{2}^{3 / 2} + |\tilde{G} (B, L / \sqrt{2})| ‖ g_{B}^{'} ‖_{2}), \end{array}

where

\tilde{G} (B, L) : = 2 \ln L + 2 \int_{0}^{\infty} e^{- u} \ln (\sqrt{\frac{1}{u} + \frac{2}{B L^{2}}} + \sqrt{\frac{1}{u}}) d u - \ln 2

⁠. Now choosing L = 1/ln B, it can be verified that

| \tilde{G} (B, L / \sqrt{2}) | < 2 |\ln \ln B| + C

for B > 1. Since

‖ g_{B}^{'} ‖_{2} = (\ln B) ‖ ϕ_{B}^{'} ‖_{2}

⁠, the corollary follows from () and scaling as in ().□

V. LOWER BOUND TO THE GROUND-STATE ENERGY

Theorem 5.1.

There is a constant C > 0 such that for B ≥ C,

E_{0} (B) \geq B + e_{0} {(\ln B)}^{2} - C {(\ln B)}^{3 / 2} .

The Proof of Theorem 5.1 is given at the end of Subsection V B.

A. Extracting a delta-function potential

Below in Lemma 5.2, the Coulomb potential restricted to the lowest Landau level is approximated as a delta well of strength ln(B/(ln B)²). First, some remarks about the notation are in order. For $Ψ \in H_{A}^{1} (R^{3}) \otimes F$ ⁠, its electron density in the magnetic-field direction is denoted as ${\tilde{Ψ}}^{2} (x_{3})$ ⁠, i.e.,

\tilde{Ψ} (x_{3}) ≔ {(\int_{R^{2}} ‖ Ψ ‖_{F}^{2} (x_{⊥}, x_{3}) d x_{⊥})}^{1 / 2} .

By Hölder’s inequality, $‖ \partial_{3} \tilde{Ψ} ‖_{2} \leq ‖ \partial_{3} Ψ ‖$ and $\tilde{Ψ} \in H^{1} (R)$ ⁠. Furthermore, the integral operator $P_{0}^{B}$ acting on $L^{2} (R^{2})$ with kernel

P_{0}^{B} (x_{⊥}, y_{⊥}) ≔ \frac{B}{2 π} e^{- \frac{B}{4} {|x_{⊥} - y_{⊥}|}^{2}} e^{\frac{i B}{2} (x_{1} y_{2} - x_{2} y_{1})}

(5.1)

is the projection onto the lowest Landau level, i.e., the ground state of the Landau Hamiltonian H_B, and $P_{>}^{B} : = 1 - P_{0}^{B}$ ⁠. The operators $P_{0}^{B} \otimes 1$ and $P_{0}^{B} \otimes 1 \otimes 1$ acting on $L^{2} (R^{2}) \otimes L^{2} (R)$ and $L^{2} (R^{2}) \otimes L^{2} (R) \otimes F$ ⁠, respectively, are also denoted $P_{0}^{B}$ ⁠.

Lemma 5.2.

Let L > 0. For B > 1 and

Ψ \in H_{A}^{1} (R^{3}) \otimes F

⁠,

\begin{array}{l} \int_{R^{3}} \frac{‖ P_{0}^{B} Ψ ‖_{F}^{2} (x_{⊥}, x_{3})}{\sqrt{{|x_{⊥}|}^{2} + x_{3}^{2}}} d x_{⊥} d x_{3} - (\ln B - 2 \ln \ln B) {(\tilde{Ψ} (0))}^{2} \\ \leq L^{- 1} ‖ \tilde{Ψ} ‖_{2}^{2} + 8 \sqrt{L} ‖ \partial_{3} \tilde{Ψ} ‖_{2}^{3 / 2} ‖ \tilde{Ψ} ‖_{2}^{1 / 2} + |D (B, L)| ‖ \partial_{3} \tilde{Ψ} ‖_{2} ‖ \tilde{Ψ} ‖_{2}; \end{array}

D (B, L) ≔ 2 \ln L + 2 \ln \ln B + \frac{2}{\sqrt{\frac{2}{B L^{2}} + 1} + 1} + 2 \ln (\sqrt{1 + \frac{2}{B L^{2}}} + 1) - \ln 2 .

(5.2)

Proof.

By Hölder’s inequality,

‖ P_{0}^{B} Ψ ‖_{F}^{2} (x_{⊥}, x_{3}) \leq (\frac{B}{2 π}) {(\tilde{Ψ} (x_{3}))}^{2},

(5.3)

and since

P_{0}^{B}

is a projection,

\int_{R^{3}} ‖ P_{0}^{B} Ψ ‖_{F}^{2} (x_{⊥}, x_{3}) d x_{⊥} d x_{3} \leq \int_{R} {(\tilde{Ψ} (x_{3}))}^{2} d x_{3} .

(5.4)

It follows from the bathtub principle^37,10 that the maximum of the expression

\int_{R^{2}} \frac{G (x_{⊥}, x_{3})}{\sqrt{{|x_{⊥}|}^{2} + x_{3}}} d x_{⊥}

over all functions G satisfying the conditions () and () above is attained by the function

G_{max} (x_{⊥}, x_{3}) = \{\begin{cases} (\frac{B}{2 π}) {(\tilde{Ψ} (x_{3}))}^{2} & when |x_{⊥}| \leq R, \\ 0 & when |x_{⊥}| > R, \end{cases})

where

R = \sqrt{2 / B}

⁠. Therefore,

\begin{array}{l} \int_{R} \int_{R^{2}} \frac{‖ P_{0}^{B} Ψ ‖_{F}^{2} (x_{⊥}, x_{3})}{\sqrt{{|x_{⊥}|}^{2} + x_{3}^{2}}} d x_{⊥} d x_{3} & \leq \int_{R} \int_{|x_{⊥}| \leq \sqrt{2 / B}} \frac{G_{max} (x_{⊥}, x_{3})}{\sqrt{{|x_{⊥}|}^{2} + x_{3}^{2}}} d x_{⊥} d x_{3} \\ = \int_{R} V_{L}^{B} (x_{3}) {(\tilde{Ψ} (x_{3}))}^{2} d x_{3}, \end{array}

with

V_{L}^{B} (x_{3}) ≔ \frac{2}{\sqrt{\frac{2}{B} + x_{3}^{2}} + |x_{3}|} .

(5.5)

The lemma follows from Corollary B.3 in Appendix B.□

Lemma 5.2 is used to prove the following corollary.

Corollary 5.3.

Let τ > 0. There is a constant C > 0 such that for B ≥ C and

Ψ \in H_{A}^{1} (R^{3}) \otimes F

⁠,

‖ \partial_{3} Ψ ‖^{2} - τ ⟨Ψ, P_{0}^{B} {|x|}^{- 1} P_{0}^{B} Ψ⟩ \geq (- \frac{τ^{2}}{4} {(\ln B)}^{2} + τ^{2} \ln B \ln \ln B - C (\ln B)) ‖ Ψ ‖^{2} .

Proof.

Recall that for ϵ > 0,

‖ \partial_{3} \tilde{Ψ} ‖_{2} ‖ \tilde{Ψ} ‖_{2} \leq ϵ ‖ \partial_{3} \tilde{Ψ} ‖_{2}^{2} + ϵ^{- 1} ‖ \tilde{Ψ} ‖_{2}^{2}

⁠. With

D (B, L)

as given in (), under the assumption that

2 τ ϵ |D (B, L)| < 1 / 2

and denoting

μ (B) = (\ln B - 2 \ln \ln B)

⁠, it follows from Lemma 5.2 that

\begin{array}{l} ‖ \partial_{3} Ψ ‖^{2} - τ ⟨Ψ, P_{0}^{B} {|x|}^{- 1} P_{0}^{B} Ψ⟩ \\ \geq (1 - 2 τ ϵ |D (B, L)|) ‖ \partial_{3} \tilde{Ψ} ‖_{2}^{2} - τ μ (B) {(\tilde{Ψ} (0))}^{2} - τ (L^{- 1} + ϵ^{- 1} |D (B, L)|) ‖ \tilde{Ψ} ‖_{2}^{2} \\ + τ ϵ |D (B, L)| ‖ \partial_{3} \tilde{Ψ} ‖_{2}^{2} - 8 τ \sqrt{L} ‖ \tilde{Ψ} ‖_{2}^{1 / 2} ‖ \partial_{3} \tilde{Ψ} ‖_{2}^{3 / 2} \\ \geq \frac{- τ^{2} {(μ (B))}^{2}}{4 (1 - 2 τ ϵ |D (B, L)|)} ‖ \tilde{Ψ} ‖_{2}^{2} - τ (\frac{432 L^{2}}{{|D (B, L)|}^{3} ϵ^{3}} + L^{- 1} + ϵ^{- 1} |D (B, L)|) ‖ \tilde{Ψ} ‖_{2}^{2} \end{array}

\geq ((- \frac{τ^{2}}{4} - τ^{3} ϵ |D (B, L)|) {(μ (B))}^{2} - \frac{432 τ L^{2}}{{|D (B, L)|}^{3} ϵ^{3}} - \frac{τ}{L} - \frac{τ |D (B, L)|}{ϵ}) ‖ \tilde{Ψ} ‖_{2}^{2} .

At the second inequality,

‖ \partial_{3} \tilde{Ψ} ‖_{2}^{2} - ζ {(\tilde{Ψ} (0))}^{2} \geq - (1 / 4) ζ^{2} ‖ \tilde{Ψ} ‖_{2}^{2}

for ζ ≥ 0 and ax² − bx^3/2 ≥−(27/256)b⁴/a³ for a > 0, b ≥ 0, and x ≥ 0 are used. Now choosing ϵ = 1/ln B and L = 1/ln B, the above assumption that

2 τ ϵ |D (B, L)| < 1 / 2

can be verified to be true for B ≥ C. The lemma follows.□

Corollary 5.3 is used to prove Lemma 5.6 in Subsection V B. The arguments for Lemma 5.2 and Corollary 5.3 are used to prove Lemma 5.12 in Subsection V C.

B. Restricting to the lowest Landau level

Below in Lemma 5.8, a Hamiltonian restricted to the lowest Landau level is derived as a lower bound. First, an ultraviolet cutoff is needed. With $K > 0$ and $Γ_{K} : = \{k \in R^{3} : \max (|k_{⊥}|, |k_{3}|) \leq K\}$ ⁠, the cutoff Hamiltonian is denoted as

\begin{array}{l} h_{K}^{co} ≔ (1 - \frac{8 α}{π K}) (H_{B} - \partial_{3}^{2}) & + \frac{1}{2} \int_{Γ_{K}} a_{k}^{†} a_{k} d k + \frac{1}{2} N \\ + \frac{\sqrt{α}}{2 π} \int_{Γ_{K}} (\frac{a_{k} e^{i k \cdot x}}{| k |} + \frac{a_{k}^{†} e^{- i k \cdot x}}{| k |}) d k . \end{array}

Lemma 5.4.

If $K > 8 α / π$ , then $H (B) \geq h_{K}^{co} - β {|x|}^{- 1} - 1 / 4$ .

Proof.

The lemma follows from Lemma 5.1 in Ref. 17.□

Lemma 5.4 is used to prove the following lemma.

Lemma 5.5.

There exists a constant C > 0 such that for B ≥ C,

\begin{array}{l} H (B) \geq (1 - \frac{8 α {(\ln B)}^{2}}{π B}) H_{B} & + (\frac{1}{2} - \frac{8 α {(\ln B)}^{2}}{π B}) (- \partial_{3}^{2}) \\ - β {|x|}^{- 1} + \frac{1}{2} N - C {(\ln B)}^{2} . \end{array}

Proof.

Let

K = B / {(\ln B)}^{2}

and

K_{3} = 16 α |\ln B| / π

⁠. By completion of the square in annihilation and creation operators,

\begin{align} \int_{| k_{3} | \leq K_{3}} \int_{| k_{⊥} | \leq K} (\frac{1}{2} a_{k}^{†} a_{k} + \frac{\sqrt{α}}{2 π} \frac{a_{k}}{| k |} e^{i k \cdot x} + \frac{\sqrt{α}}{2 π} \frac{a_{k}^{†}}{| k |} e^{- i k \cdot x}) d k_{⊥} d k_{3} \\ \geq - \frac{α}{2 π^{2}} \int_{| k_{3} | \leq K_{3}} \int_{| k_{⊥} | \leq K} \frac{1}{| k |^{2}} d k_{⊥} d k_{3} \\ = - \frac{α}{π} \int_{0}^{K_{3}} \ln (\frac{K^{2} + k_{3}^{2}}{k_{3}^{2}}) d k_{3} \\ = - \frac{α}{π} (K_{3} \ln (\frac{K^{2}}{K_{3}^{2}} + 1) + 2 K \arctan (\frac{K_{3}}{K})) \\ \geq - \frac{α}{π} K_{3} (\ln (\frac{K^{2}}{K_{3}^{2}} + 1) + 2 + \frac{2}{3} \frac{K_{3}^{2}}{K^{2}}) \\ \geq - C {(\ln B)}^{2}, \end{align}

(5.6)

valid for some constant C and B ≥ C. The argument corrects a mistake in the Proof of Lemma 5.2 in Ref. 17.

Denoting

Λ (B) = \{(k_{⊥}, k_{3}) \in Γ_{K} : K_{3} \leq |k_{3}| \leq K and |k_{⊥}| \leq K\}

⁠, it can be argued as in the Proof of Lemma 5.2 in Ref. 17 that for large B,

\frac{\sqrt{α}}{2 π} \int_{Λ (B)} (\frac{a_{k}}{|k|} e^{i k \cdot x} + \frac{a_{k}^{†}}{|k|} e^{- i k \cdot x}) d k \geq \frac{1}{2} \partial_{3}^{2} - \frac{1}{2} (\int_{Λ (B)} a_{k}^{†} a_{k} d k + \frac{1}{2}) .

(5.7)

The lemma follows from Lemma 5.4 and the estimates () and ().□

Lemma 5.5 and Corollary 5.3 from Subsection V A are used to prove the following lemma.

Lemma 5.6.

There exists a constant C > 0 such that for B ≥ C and all

Ψ \in H_{A}^{1} (R^{3}) \otimes dom (\sqrt{N})

with ‖Ψ‖ = 1,

⟨Ψ, H (B) Ψ⟩ \geq B + \frac{B}{2} ‖ P_{>}^{B} Ψ ‖^{2} + \frac{1}{2} ⟨Ψ, N Ψ⟩ - C {(\ln B)}^{2} .

Proof.

Recall that by the diamagnetic inequality for τ > 0,

⟨Ψ, (H_{B} - \partial_{3}^{2}) Ψ⟩ \geq τ ⟨Ψ, β | x |^{- 1} Ψ⟩ - 4^{- 1} β^{2} τ^{2} ‖ Ψ ‖^{2} .

Then, with 0 < η < 1,

A > 1

⁠, and

θ : = (\frac{1}{2} - \frac{8 α {(\ln B)}^{2}}{π B})

for B large,

\begin{array}{l} θ ⟨Ψ, (H_{B} - \partial_{3}^{2}) Ψ⟩ - ⟨Ψ, β | x |^{- 1} Ψ⟩ \\ = θ ⟨P_{0}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{0}^{B} Ψ⟩ - ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩ \\ + θ (1 - η) ⟨P_{>}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{>}^{B} Ψ⟩ + θ η ⟨P_{>}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{>}^{B} Ψ⟩ \\ - ⟨P_{>}^{B} Ψ, β | x |^{- 1} P_{>}^{B} Ψ⟩ - ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{>}^{B} Ψ⟩ - ⟨P_{>}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩ \\ \geq θ ⟨P_{0}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{0}^{B} Ψ⟩ - ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩ \\ + θ (1 - η) ⟨P_{>}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{>}^{B} Ψ⟩ - A^{2} β^{2} {(4 θ η)}^{- 1} ‖ P_{>}^{B} Ψ ‖^{2} \\ + (A - 1) ⟨P_{>}^{B} Ψ, β | x |^{- 1} P_{>}^{B} Ψ⟩ - 2 {⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩}^{\frac{1}{2}} {⟨P_{>}^{B} Ψ, β | x |^{- 1} P_{>}^{B} Ψ⟩}^{\frac{1}{2}} \\ \geq θ ⟨P_{0}^{B} Ψ, H_{B} P_{0}^{B} Ψ⟩ + [θ ⟨P_{0}^{B} Ψ, - \partial_{3}^{2} P_{0}^{B} Ψ⟩ - (A / (A - 1)) ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩] \\ + θ (1 - η) ⟨P_{>}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{>}^{B} Ψ⟩ - A^{2} β^{2} {(4 θ η)}^{- 1} ‖ P_{>}^{B} Ψ ‖^{2} \\ \geq θ B + [θ ⟨P_{0}^{B} Ψ, - \partial_{3}^{2} P_{0}^{B} Ψ⟩ - (A / (A - 1)) ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩] \\ + θ B ‖ P_{>}^{B} Ψ ‖^{2} + [θ (B - 3 B η) - A^{2} β^{2} {(4 θ η)}^{- 1}] ‖ P_{>}^{B} Ψ ‖^{2} . \end{array}

Choosing η = 1/4 and

A = \sqrt{B} θ / 2 β

for B large

(A / (A - 1)) < 2

and by Corollary 5.3, there exists some C > 0 such that for B ≥ C,

θ ⟨P_{0}^{B} Ψ, - \partial_{3}^{2} P_{0}^{B} Ψ⟩ - (A / (A - 1)) ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩ \geq - C {(\ln B)}^{2} .

The lemma now follows from Lemma 5.5.□

The following observation is immediate from Theorem 4.1: For every $M > e_{0}$ and large B, there exist wave functions $Ψ \in H_{A}^{1} (R^{3}) \otimes dom (\sqrt{N})$ satisfying

⟨Ψ, H (B) Ψ⟩ \leq B + M {(\ln B)}^{2} and ‖ Ψ ‖ = 1 .

(5.8)

The estimate in (5.8) and Lemma 5.6 are used to prove the following corollary.

Corollary 5.7.

For every

M \in R

⁠, there exists a constant C_M > 0 such that for B ≥ C_M and all

Ψ \in H_{A}^{1} (R^{3}) \otimes dom (\sqrt{N})

satisfying (5.8),

‖ P_{>}^{B} Ψ ‖^{2} \leq C_{M} {(\ln B)}^{2} B^{- 1} and ⟨Ψ, N Ψ⟩ \leq C_{M} {(\ln B)}^{2} .

Proof.

The corollary follows from Lemma 5.6.□

Corollary 5.7 and Lemma 5.4 are used to prove Lemma 5.8.

Lemma 5.8.

Let

K > 8 α / π

and

1 < A < \sqrt{B / \ln B}

. Denoting

κ = 1 - (8 α / π K)

for every

M \in R

⁠, there exists a constant C_M > 0 such that for B ≥ C_M and all

Ψ \in H_{A}^{1} (R^{3}) \otimes dom (\sqrt{N})

satisfying (5.8),

\begin{array}{l} ⟨Ψ, H (B) Ψ⟩ \geq & ⟨P_{0}^{B} Ψ, (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} Ψ⟩ + κ B ‖ P_{>}^{B} Ψ ‖^{2} \\ - C_{M} {(\ln B)}^{2} (K B^{- 1} + \sqrt{K B^{- 1}}) - C_{M} κ^{- 1} \ln B - 1 / 4 . \end{array}

Proof.

It can be argued that as in the Proof of Lemma 5.6 with 0 < η < 1,

\begin{array}{l} κ ⟨Ψ, (H_{B} - \partial_{3}^{2}) Ψ⟩ - ⟨Ψ, β | x |^{- 1} Ψ⟩ \\ \geq κ ⟨P_{0}^{B} Ψ, (H_{B} - \partial_{3}^{2}) P_{0}^{B} Ψ⟩ - (A / (A - 1)) ⟨P_{0}^{B} Ψ, β | x |^{- 1} P_{0}^{B} Ψ⟩ \\ + κ B ‖ P_{>}^{B} Ψ ‖^{2} + [κ (2 B - 3 B η) - A^{2} β^{2} {(4 η κ)}^{- 1}] ‖ P_{>}^{B} Ψ ‖^{2} . \end{array}

It now follows from Lemma 5.4 that

\begin{array}{l} ⟨Ψ, H (B) Ψ⟩ \geq & ⟨P_{0}^{B} Ψ, (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} Ψ⟩ + κ B ‖ P_{>}^{B} Ψ ‖^{2} \\ + [κ (2 B - 3 B η) - A^{2} β^{2} {(4 η κ)}^{- 1}] ‖ P_{>}^{B} Ψ ‖^{2} - \frac{1}{4} \\ + ⟨P_{>}^{B} Ψ, (\int_{Γ_{K}} a_{k}^{†} a_{k} + \frac{\sqrt{α}}{2 π} \frac{a_{k} e^{i k \cdot x}}{| k |} + \frac{\sqrt{α}}{2 π} \frac{a_{k}^{†} e^{- i k \cdot x}}{| k |} d k) P_{>}^{B} Ψ⟩ \\ + ⟨P_{0}^{B} Ψ, (\frac{\sqrt{α}}{2 π} \int_{Γ_{K}} \frac{a_{k} e^{i k \cdot x}}{| k |} + \frac{a_{k}^{†} e^{- i k \cdot x}}{| k |} d k) P_{>}^{B} Ψ⟩ \\ + ⟨P_{>}^{B} Ψ, (\frac{\sqrt{α}}{2 π} \int_{Γ_{K}} \frac{a_{k} e^{i k \cdot x}}{| k |} + \frac{a_{k}^{†} e^{- i k \cdot x}}{| k |} d k) P_{0}^{B} Ψ⟩ . \end{array}

By completion of the square in annihilation and creation operators,

\begin{array}{l} \int_{Γ_{K}} (a_{k}^{†} a_{k} + \frac{\sqrt{α}}{2 π} \frac{a_{k}}{| k |} e^{i k \cdot x} + \frac{\sqrt{α}}{2 π} \frac{a_{k}^{†}}{| k |} e^{- i k \cdot x}) d k \\ \geq - \frac{α}{4 π^{2}} \int_{Γ_{K}} \frac{1}{| k |^{2}} d k_{⊥} d k_{3} \\ = - \frac{α (2 \ln (2) + π)}{4 π} K, \end{array}

and by Corollary 5.7 for B ≥ C_M,

\begin{array}{l} ⟨P_{>}^{B} Ψ, (\int_{Γ_{K}} a_{k}^{†} a_{k} + \frac{\sqrt{α}}{2 π} \frac{a_{k} e^{i k \cdot x}}{| k |} + \frac{\sqrt{α}}{2 π} \frac{a_{k}^{†} e^{- i k \cdot x}}{| k |} d k) P_{>}^{B} Ψ⟩ \\ \geq - \frac{α (2 \ln (2) + π)}{4 π} K ‖ P_{>}^{B} Ψ ‖^{2} \\ \geq - C_{M} K B^{- 1} {(\ln B)}^{2} . \end{array}

Furthermore, it can be argued as in the Proof of Lemma 5.4 in Ref. 17 and using Corollary 5.7 that for B ≥ C_M,

|⟨P_{0}^{B} Ψ, (\frac{\sqrt{α}}{2 π} \int_{Γ_{K}} \frac{a_{k}}{| k |} e^{i k \cdot x} d k) P_{>}^{B} Ψ⟩|

\begin{array}{l} \leq C \sqrt{K} ‖ P_{>}^{B} Ψ ‖ ‖ \sqrt{N + 1} P_{0}^{B} Ψ ‖ \\ \leq C_{M} \sqrt{K B^{- 1}} {(\ln B)}^{2} . \end{array}

The remaining interaction terms are estimated similarly. Choosing η = 2/3, the lemma follows from Corollary 5.7.□

The Hamiltonian $P_{0}^{B} (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B}$ realized as a lower bound in Lemma 5.8 is estimated from below.

Proposition 5.9.

There exists a constant C > 0 such that for B, κ, and

K

satisfying B ≥ C,

C {(\ln B)}^{- 1 / 2} \leq κ \leq C^{- 1} \ln B

,

K \geq \sqrt{B}

⁠, and some

1 < A < \sqrt{B / \ln B}

⁠,

\begin{array}{l} P_{0}^{B} (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} \\ \geq (κ B + κ^{- 1} {(\ln B)}^{2} e_{0} - C κ^{- 1 / 2} {(\ln B)}^{3 / 2} - C (1 + κ^{- 2}) \ln B) P_{0}^{B} . \end{array}

Proposition 5.9 and Lemma 5.8 are used to prove Theorem 5.1.

Proof of Theorem 5.1.

Fix

M > e_{0}

⁠. For large B by Theorem 4.1, there exist wave functions satisfying (). It suffices to argue the desired lower bound on

⟨Ψ, H (B) Ψ⟩

with those wave functions. By Lemma 5.8 and Proposition 5.9,

\begin{array}{l} ⟨Ψ, H (B) Ψ⟩ \\ \geq κ B + (κ^{- 1} {(\ln B)}^{2} e_{0} - C κ^{- 1 / 2} {(\ln B)}^{3 / 2} - C (1 + κ^{- 2}) \ln B) ‖ P_{0}^{B} Ψ ‖^{2} \\ - C {(\ln B)}^{2} (K B^{- 1} + \sqrt{K B^{- 1}}) - C κ^{- 1} \ln B - C, \end{array}

with

κ = 1 - (8 α / π K)

⁠. Choosing

K = B {(\ln B)}^{- 4 / 3}

and since

‖P_{0}^{B} Ψ‖ \leq 1

⁠,

⟨Ψ, H (B) Ψ⟩ \geq B + e_{0} {(\ln B)}^{2} - C {(\ln B)}^{3 / 2},

yielding the claimed lower bound.□

Proposition 5.9 is proved in Subsection V C.

C. Proof of Proposition 5.9

1. Reduction to one dimension

Below in Lemma 5.10, a one-dimensional Hamiltonian is derived as a lower bound. In Ref. 17, the authors consider a one-dimensional Hamiltonian with $0 < K_{3} \leq K$ and $1 \leq K_{⊥} \leq K$ ⁠,

\begin{array}{l} h^{1 d} ≔ \\ κ_{1} (- \partial_{3}^{2}) + \int_{| k_{3} | \leq K_{3}} {\hat{a}}_{k_{3}}^{†} {\hat{a}}_{k_{3}} d k_{3} + \frac{\sqrt{α}}{2 π} \int_{| k_{3} | \leq K_{3}} ν (k_{3}) ({\hat{a}}_{k_{3}} e^{i k_{3} x_{3}} + {\hat{a}}_{k_{3}}^{†} e^{- i k_{3} x_{3}}) d k_{3}, \end{array}

acting on $L^{2} (R^{3}) \otimes F$ with $κ_{1} : = κ - (8 α / π K_{3}) \int_{0}^{\infty} {(1 + t)}^{- 1} \exp (- t K_{3}^{2} / 2 B) d t$ and κ as in the statement of Proposition 5.9, and

{\hat{a}}_{k_{3}} ≔ \frac{1}{ν (k_{3})} \int_{1 \leq |k_{⊥}| \leq K_{⊥}} \frac{a_{k}}{|k|} e^{i k_{⊥} \cdot x_{⊥}} d k_{⊥},

ν (k_{3}) ≔ {(\int_{1 \leq |k_{⊥}| \leq K_{⊥}} {|k|}^{- 2} d k_{⊥})}^{\frac{1}{2}} = \sqrt{π} {(\ln (K_{⊥}^{2} + k_{3}^{2}) - \ln (1 + k_{3}^{2}))}^{\frac{1}{2}},

(5.9)

satisfying $[{\hat{a}}_{k_{3}}, {\hat{a}}_{k_{3}^{'}}^{†}] = δ (k_{3} - k_{3}^{'})$ and $[{\hat{a}}_{k_{3}}, {\hat{a}}_{k_{3}^{'}}] = [{\hat{a}}_{k_{3}}^{†}, {\hat{a}}_{k_{3}^{'}}^{†}] = 0$ for $k_{3}, k_{3}^{'} \in R$ ⁠.

Lemma 5.10.

Denoting

κ_{2} = κ - 2 α π^{- 1} K_{3} K_{⊥}^{- 2}

⁠,

\begin{array}{l} P_{0}^{B} (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} \\ \geq κ_{2} B P_{0}^{B} + P_{0}^{B} (h^{1 d} - β (1 / (1 - A^{- 1})) P_{0}^{B} {|x|}^{- 1} P_{0}^{B}) P_{0}^{B} - (1 + \frac{α}{2}) P_{0}^{B} . \end{array}

Proof.

The lemma follows from Lemmas 6.1–6.4 in Ref. 17. The Proof of Lemma 6.3 in Ref. 17 uses an incorrect vector operator for cutting off high modes in the k_⊥-direction (cf. Ref. 43); the mistake can be fixed, and the lemma stands true.□

2. Localization and decomposition

Below in Lemma 5.11, the one-dimensional Hamiltonian is bounded from below by a Hamiltonian describing an electron interacting with only finitely many modes. In Ref. 17, the authors decompose the mode space into $M$ intervals, indexed with b, each of length $2 K_{3} / M$ ⁠, and consider for $u \in R$ and 0 < γ < 1 the block Hamiltonian

h_{γ}^{(u)} ≔ κ_{1} (- \partial_{3}^{2}) + \sum_{b} [(1 - γ) A_{b}^{(u) *} A_{b}^{(u)} + \frac{\sqrt{α}}{2 π} V (b) (A_{b}^{(u)} e^{i k_{b} x_{3}} + A_{b}^{(u) *} e^{- i k_{b} x_{3}})],

with k_b being a mode in block b and the block creation and annihilation operators $A_{b}^{(u) *}$ and $A_{b}^{(u)}$ acting on $F (L^{2} (R^{3}))$ ⁠, where

A_{b}^{(u)} ≔ \frac{1}{V (b)} \int_{b} ν (k_{3}) e^{i (k_{3} - k_{b}) u} {\hat{a}}_{k_{3}} d k_{3},

$V (b) : = {(\int_{b} ν {(k_{3})}^{2} d k_{3})}^{\frac{1}{2}}$ ⁠, satisfying

[A_{b}^{(u)}, A_{b^{'}}^{(u) *}] = δ_{b b^{'}}, [A_{b}^{(u)}, A_{b^{'}}^{(u)}] = [A_{b}^{(u) *}, A_{b^{'}}^{(u) *}] = 0 for all blocks b, b^{'} .

Furthermore, localizing the electron in the x₃-direction on an interval of length J, the following lemma is proved.

Lemma 5.11.

For

χ \in C_{0}^{\infty} (R), ‖ χ ‖_{2} = 1

being a nonnegative function supported on the interval [−1/2, 1/2] and for J > 0, denoting

χ_{u}^{J} (x_{3}) = J^{- 1 / 2} χ (J^{- 1} (x_{3} - u))

,

\begin{array}{l} h^{1 d} - β (1 / (1 - A^{- 1})) P_{0}^{B} {|x|}^{- 1} P_{0}^{B} \\ \geq \int_{R} χ_{u}^{J} [h_{γ}^{(u)} - β (1 / (1 - A^{- 1})) P_{0}^{B} {|x|}^{- 1} P_{0}^{B}] χ_{u}^{J} d u - \frac{α K_{3}^{2} J^{2}}{4 π^{2} γ M^{2}} R - ‖ χ^{'} ‖_{2}^{2} J^{- 2}, \end{array}

with

R ≔ \int_{| k_{3} | \leq K_{3}} ν {(k_{3})}^{2} d k_{3} = π \int_{| k_{3} | \leq K_{3}} (\ln (K_{⊥} + k_{3}^{2}) - \ln (1 + k_{3}^{2})) d k_{3} .

Proof.

The lemma follows from Lemma 6.5 in Ref. 17.□

3. Error estimates

The block Hamiltonian is bounded from below. Similarly, as in Refs. 17, 28, and 42 representing the block creation and annihilation operators by coherent state integrals and completing the square, it follows for a suitably chosen k_b that

h_{γ}^{(u)} - β (1 / (1 - A^{- 1})) P_{0}^{B} {|x|}^{- 1} P_{0}^{B} \geq I - M,

(5.10)

where

\begin{array}{l} I ≔ \inf_{‖ ϕ ‖_{2} = 1} [κ_{1} ‖ \partial_{3} ϕ ‖_{2}^{2} - \frac{α}{4 π^{2} (1 - γ)}) & \int_{R} ν {(k_{3})}^{2} {|\int_{R^{3}} e^{i k_{3} x_{3}} {|ϕ (x_{⊥}, x_{3})|}^{2} d x|}^{2} d k_{3} \\ (- \frac{β}{1 - A^{- 1}} \int_{R^{3}} {|x|}^{- 1} {|(P_{0}^{B} ϕ) (x_{⊥}, x_{3})|}^{2} d x] \end{array}

will be estimated from below in Lemma 5.12. Combining (5.10) with Lemma 5.11,

h^{1 d} - β (1 / (1 - A^{- 1})) P_{0}^{B} {|x|}^{- 1} P_{0}^{B} \geq I - M - \frac{α K_{3}^{2} J^{2}}{4 π^{2} γ M^{2}} R - ‖ χ^{'} ‖_{2}^{2} J^{- 2} .

Now, it follows from Lemma 5.10 that for some constant C > 0,

\begin{align} P_{0}^{B} (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} \\ \geq κ B P_{0}^{B} + I P_{0}^{B} - C (\frac{K_{3} B}{K_{⊥}^{2}} + M + \frac{K_{3}^{2} J^{2}}{γ M^{2}} R + \frac{1}{J^{2}}) P_{0}^{B} . \end{align}

(5.11)

The argument for Lemma 5.2 from Subsection V A is used to prove Lemma 5.12.

Lemma 5.12.

For any L > 0 and ϵ > 0 and with

D (B, L)

as given in (5.2) assuming

\frac{2 \ln K_{⊥}}{1 - γ} = \frac{μ (B)}{1 - A^{- 1}} and {\tilde{κ}}_{1} ≔ κ_{1} - \frac{4 β ϵ |D (B, L)| \ln K_{⊥}}{μ (B) (1 - γ)} > 0

(5.12)

with μ(B): = ln B − 2 ln ln B,

I \geq \frac{4 {(\ln K_{⊥})}^{2} e_{0}}{{\tilde{κ}}_{1} {(1 - γ)}^{2}} - \frac{2 β \ln K_{⊥}}{μ (B) (1 - γ)} (\frac{432 L^{2}}{{|D (B, L)|}^{3} ϵ^{3}} + \frac{1}{L} + \frac{|D (B, L)|}{ϵ}) .

Proof.

For

ϕ \in L^{2} (R^{3})

⁠, ‖ϕ‖₂ = 1, and

\tilde{ϕ} (x_{3}) : = {(\int_{R^{2}} {|ϕ (x_{⊥}, x_{3})|}^{2} d x_{⊥})}^{1 / 2}

⁠,

\begin{array}{l} κ_{1} & ‖ \partial_{3} ϕ ‖_{2}^{2} - \frac{α}{4 π^{2} (1 - γ)} \int_{R} ν {(k_{3})}^{2} {|\int_{R^{3}} e^{i k_{3} x_{3}} {|ϕ (x_{⊥}, x_{3})|}^{2} d x|}^{2} d k_{3} \\ - \frac{β}{1 - A^{- 1}} \int_{R^{3}} \frac{{|(P_{0}^{B} ϕ) (x_{⊥}, x_{3})|}^{2}}{|x|} d x \\ \geq & κ_{1} ‖ \partial_{3} ϕ ‖_{2}^{2} - \frac{α \ln K_{⊥}}{1 - γ} \int_{R} {|\int_{R^{2}} {|ϕ (x_{⊥}, x_{3})|}^{2} d x_{⊥}|}^{2} d x_{3} \\ - \frac{β}{1 - A^{- 1}} \int_{R^{3}} \frac{{|(P_{0}^{B} ϕ) (x_{⊥}, x_{3})|}^{2}}{|x|} d x \\ \geq & (κ_{1} - \frac{2 β ϵ |D (B, L)|}{1 - A^{- 1}}) ‖ \partial_{3} \tilde{ϕ} ‖_{2}^{2} - \frac{α \ln K_{⊥}}{1 - γ} ‖ \tilde{ϕ} ‖_{4}^{4} - \frac{β μ (B)}{1 - A^{- 1}} {(\tilde{ϕ} (0))}^{2} \\ - \frac{β}{1 - A^{- 1}} (\frac{432 L^{2}}{{|D (B, L)|}^{3} ϵ^{3}} + \frac{1}{L} + \frac{|D (B, L)|}{ϵ}) \\ \geq & \frac{4 {(\ln K_{⊥})}^{2} e_{0}}{{\tilde{κ}}_{1} {(1 - γ)}^{2}} - \frac{2 β \ln K_{⊥}}{μ (B) (1 - γ)} (\frac{432 L^{2}}{{|D (B, L)|}^{3} ϵ^{3}} + \frac{1}{L} + \frac{|D (B, L)|}{ϵ}) . \end{array}

At the first inequality,

ν {(k_{3})}^{2} = π (\ln (K_{⊥}^{2} + k_{3}^{2}) - \ln (1 + k_{3}^{2})) \leq 2 π \ln K_{⊥}

(5.13)

is used along with Plancherel’s identity. At the second inequality, the bathtub principle in Lemma 5.2 and the argument for Corollary 5.3 apply mutatis mutandis. At the third inequality, the assumptions in () are used.□

From Lemma 5.12 and further assuming that

κ - {\tilde{κ}}_{1} \leq \frac{κ}{2} and γ \leq \frac{1}{2},

(5.14)

it can be seen that there is a constant C > 0 such that

\begin{array}{l} I \geq & 4 κ^{- 1} {(\ln K_{⊥})}^{2} e_{0} \\ - C (\frac{{(\ln K_{⊥})}^{2}}{κ} (\frac{κ - {\tilde{κ}}_{1}}{κ} + γ) + \frac{\ln K_{⊥}}{μ (B)} (\frac{L^{2}}{{|D (B, L)|}^{3} ϵ^{3}} + \frac{1}{L} + \frac{|D (B, L)|}{ϵ})) . \end{array}

With the above bound and (5.11), the argument in Ref. 17 applies mutatis mutandis, and choosing $J^{2} = κ^{1 / 5} K_{3}^{- 3 / 5} {(\ln K_{⊥})}^{- 3 / 5}$ ⁠, $M = [J^{- 2}]$ ⁠, and $γ = κ^{4 / 5} K_{3}^{3 / 5} {(\ln K_{⊥})}^{- 7 / 5}$ yields

\begin{array}{l} P_{0}^{B} & (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} \\ \geq & (κ B + \frac{4 {(\ln K_{⊥})}^{2}}{κ} e_{0}) P_{0}^{B} \\ - C (κ^{- 1} (κ - κ_{1}) {(\ln K_{⊥})}^{2} + κ^{- 1 / 5} K_{3}^{3 / 5} {(\ln K_{⊥})}^{3 / 5} + B K_{3} K_{⊥}^{- 2}) P_{0}^{B} \\ - C \frac{\ln K_{⊥}}{μ (B)} (\frac{L^{2} ϵ^{- 3}}{{|D (B, L)|}^{3}} + \frac{1}{L} + \frac{|D (B, L)|}{ϵ} + \frac{ϵ |D (B, L)| {(\ln K_{⊥})}^{2}}{κ^{2}}) P_{0}^{B} . \end{array}

It is shown in Ref. 17 choosing $K_{⊥} = B^{1 / 2}$ and $K_{3} = κ^{- 1 / 2} {(\ln B)}^{3 / 2}$ that

κ^{- 1} (κ - κ_{1}) {(\ln K_{⊥})}^{2} + κ^{- 1 / 5} K_{3}^{3 / 5} {(\ln K_{⊥})}^{3 / 5} + B K_{3} K_{⊥}^{- 2} \leq C κ^{- 1 / 2} {(\ln B)}^{3 / 2} .

Now choosing L = 1/ln B and ϵ = 1/ln B, since $1 \leq |D (B, L)| \leq C$ for B ≥ C,

\frac{\ln K_{⊥}}{μ (B)} (\frac{L^{2} ϵ^{- 3}}{{|D (B, L)|}^{3}} + \frac{1}{L} + \frac{|D (B, L)|}{ϵ} + \frac{ϵ |D (B, L)| {(\ln K_{⊥})}^{2}}{κ^{2}}) \leq C (1 + κ^{- 2}) \ln B .

With the above choice of parameters and the assumptions $C {(\ln B)}^{- 1 / 2} \leq κ \leq C^{- 1} \ln B$ and $K \geq \sqrt{B}$ ⁠, the conditions in (5.12) and (5.14) and that $0 < K_{3} \leq K$ ⁠, $1 \leq K_{⊥} \leq K$ and $1 < A < \sqrt{B / \ln B}$ are verified, and

\begin{array}{l} P_{0}^{B} (h_{K}^{co} - β (1 / (1 - A^{- 1})) {|x|}^{- 1}) P_{0}^{B} \\ \geq (κ B + κ^{- 1} {(\ln B)}^{2} e_{0} - C κ^{- 1 / 2} {(\ln B)}^{3 / 2} - C (1 + κ^{- 2}) \ln B) P_{0}^{B}, \end{array}

thereby concluding the Proof of Proposition 5.9.

Corollary 5.13.

Let W be a sum of a bounded Borel measure on the real line and a

L^{\infty} (R)

function. For ϵ a real parameter and E_ϵ(B) and

e_{ϵ}

as in Corollary 4.4, there is a constant C > 0 such that for B ≥ C,

E_{ϵ} (B) \geq B + e_{ϵ} {(\ln B)}^{2} - C {(\ln B)}^{3 / 2} .

Proof.

The above arguments apply mutatis mutandis (see also Ref. 31).□

VI. PROOF OF THEOREM 2.2

Proof of Theorem 2.2.

Let E_ϵ(B) be the ground-state energy of the Hamiltonian

H_{ϵ} (B)

in () and let the one-dimensional energy

e_{ϵ}

be as defined in (). The large B asymptotics of E_ϵ(B) in () follows from Corollaries 4.4 and 5.13. As explained in the Introduction, it follows from the variational principle, Theorem 2.1, and () that for ϵ > 0,

\frac{e_{0} - e_{ϵ}}{ϵ} \geq \underset{B \to \infty}{lim sup} \frac{1}{(\ln B)} \int_{R} W (x_{3}) (\int_{R^{2}} ‖) Ψ^{(B)} {(‖}_{F}^{2} (x_{⊥}, \frac{x_{3}}{\ln B}) d x_{⊥}) d x_{3},

and for ϵ < 0,

\frac{e_{0} - e_{ϵ}}{ϵ} \leq {lim inf}_{B \to \infty} \frac{1}{(\ln B)} \int_{R} W (x_{3}) (\int_{R^{2}} ‖) Ψ^{(B)} {(‖}_{F}^{2} (x_{⊥}, \frac{x_{3}}{\ln B}) d x_{⊥}) d x_{3} .

Theorem 2.2 now follows from Theorem 3.3.□

ACKNOWLEDGMENTS

This paper is based on the work supported by the NSF (Grant No. DMS-1600560). I thank Michael Loss for several valuable suggestions, Rupert Frank for the general strategy used in the Proof of Theorem 2.2, and the referee for helpful comments about the presentation.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX A: COMPACTNESS OF MINIMIZING SEQUENCES

In the following, Theorem A.1 applies for all α ≥ 0 and β > 0.

Theorem A.1.

If a sequence ${\{ψ_{n}\}}_{n = 1}^{\infty}, ‖ ψ_{n} {(‖}_{2} = 1$ satisfies $\lim_{n \to \infty} E_{0} (ψ_{n}) = e_{0}$ with the functional $E_{0}$ as given in (3.1), then there exists a subsequence ${ψ_{n_{k}}}_{k = 1}^{\infty}$ and some $ψ \in H^{1} (R)$ such that $‖ ψ {(‖}_{2} = 1$ , $E_{0} (ψ) = e_{0}$ ⁠, and $‖ ψ_{n_{k}} - ψ ‖_{H^{1}} \to 0$ as k → ∞.

Proof.

For

φ \in H^{1} (R)

⁠, it can be argued as in the Proof of Theorem 3.3 that

E_{0} (φ) \geq \frac{3}{4} ‖ φ^{'} ‖_{2}^{2} - ‖ φ ‖_{2}^{2} {(\frac{α}{2} ‖ φ ‖_{2}^{2} + β)}^{2} .

Furthermore, since ψ_n is a minimizing sequence,

E_{0} (ψ_{n}) < e_{0} + 1

for n large. Then,

‖ ψ_{n} ‖_{H^{1}} < C

and there exists a subsequence

{ψ_{n_{k}}}_{k = 1}^{\infty}

and some

ψ \in H^{1} (R)

such that

ψ_{n_{k}}

converges to ψ weakly in

H^{1} (R)

⁠.

Step 1 (compactness): it is argued that the subsequence

{ψ_{n_{k}}}_{k = 1}^{\infty}

satisfies

\forall δ > 0, \exists R < \infty s.t. ‖ ψ_{n_{k}} ‖_{L^{2} (\{|x| < R\})}^{2} > 1 - δ .

(A1)

Essential to the argument is the binding inequality

e_{0} < e_{T}

⁠, where

e_{T} : = \inf_{‖ φ ‖_{2} = 1} E_{T} (φ)

and

E_{T} (φ) : = \int_{R} {|φ^{'}|}^{2} d x - (α / 2) \int_{R} {|φ|}^{4} d x

is the translation-invariant problem admitting when α > 0 a symmetric decreasing minimizer ϕ_T.¹⁷ When α = 0, the binding inequality is obvious, and when α > 0,

e_{0} \leq E_{0} (ϕ_{T}) = E_{T} (ϕ_{T}) - β ϕ_{T}^{2} (0) = e_{T} - β ϕ_{T}^{2} (0) < e_{T} .

Moreover, it should be noted that

E_{0} (φ) \geq e_{0} ‖ φ ‖_{2}^{2} and E_{T} (φ) \geq e_{T} ‖ φ ‖_{2}^{2} when ‖ φ ‖_{2} \leq 1 .

(A2)

Also a quadratic partition of unity is chosen,

χ^{2} + {\tilde{χ}}^{2} \equiv 1

⁠, where 0 ≤ χ ≤ 1 is a smooth function with χ(x) = 1 when |x| < 1/2 and χ(x) = 0 when |x| > 1. Denoting

χ_{R} = χ (R^{- 1} \cdot)

⁠, it follows from () that

\begin{align} E_{0} (ψ_{n_{k}}) = & E_{0} (χ_{R} ψ_{n_{k}}) + E_{T} ({\tilde{χ}}_{R} ψ_{n_{k}}) \\ - α \int_{R} χ_{R}^{2} {\tilde{χ}}_{R}^{2} {|ψ_{n_{k}}|}^{4} d x - \int_{R} {|ψ_{n_{k}}|}^{2} ({|χ_{R}^{'}|}^{2} + {|{\tilde{χ}}_{R}^{'}|}^{2}) d x \\ \geq & (e_{0} - e_{T}) ‖ χ_{R} ψ_{n_{k}} ‖_{2}^{2} + e_{T} \\ - α \int_{R} χ_{R}^{2} {\tilde{χ}}_{R}^{2} {|ψ_{n_{k}}|}^{4} d x - \int_{R} {|ψ_{n_{k}}|}^{2} ({|χ_{R}^{'}|}^{2} + {|{\tilde{χ}}_{R}^{'}|}^{2}) d x . \end{align}

(A3)

Since

χ, \tilde{χ}

have bounded derivatives,

\int_{R} {|ψ_{n_{k}}|}^{2} ({|χ_{R}^{'}|}^{2} + {|{\tilde{χ}}_{R}^{'}|}^{2}) d x < C R^{- 2}

for some C > 0. Furthermore, with

D_{R} : = \{R / 2 \leq | x | \leq R\}

⁠,

\int_{R} χ_{R}^{2} {\tilde{χ}}_{R}^{2} {|ψ_{n_{k}}|}^{4} d x \leq ‖ ψ_{n_{k}} ‖_{L^{4} (D_{R})}^{4} and ‖ ψ_{n_{k}} ‖_{L^{4} (D_{R})}^{4} \to ‖ ψ ‖_{L^{4} (D_{R})}^{4}

by Rellich-Kondrashov, so the first term in () can also be made arbitrarily small with R chosen to be large enough uniformly in k. Hence, for any δ > 0, there is some R such that for all k,

E_{0} (ψ_{n_{k}}) \geq (e_{0} - e_{T}) ‖ χ_{R} ψ_{n_{k}} ‖_{2}^{2} + e_{T} - δ (e_{T} - e_{0}) / 2 .

(A4)

Since

{ψ_{n_{k}}}_{k = 1}^{\infty}

is a minimizing sequence for

e_{0}

⁠, for large k,

E_{0} (ψ_{n_{k}}) \leq e_{0} + δ (e_{T} - e_{0}) / 2 .

(A5)

Compactness now follows from () and ().

Step 2 (weak limit is a minimizer): by Rellich–Kondrashov and the compactness property in (),

‖ ψ_{n_{k}} - ψ ‖_{2} \to 0 and ‖ ψ ‖_{2} = 1 .

(A6)

Since

‖ ψ_{n} ‖_{H^{1}} < C

⁠, by Sobolev and Hölder’s inequalities,

\int_{R} (| ψ_{n_{k}} |^{4} - {|ψ|}^{4}) d x \leq C ‖ ψ_{n_{k}} - ψ ‖_{2} \to 0 .

Furthermore, by Theorem 8.6 in Ref. 37,

ψ_{n_{k}} (0) \to ψ (0)

⁠, so

\frac{α}{2} \int_{R} | ψ_{n_{k}} |^{4} d x + β | ψ_{n_{k}} (0) |^{2} \to \frac{α}{2} \int_{R} {|ψ|}^{4} d x + β {|ψ (0)|}^{2} .

(A7)

Then, since

\underset{k \to \infty}{lim inf} ‖ ψ_{n_{k}}^{'} ‖_{2} \geq ‖ ψ^{'} ‖_{2}

⁠,

e_{0} = \lim_{k \to \infty} E_{0} (ψ_{n_{k}}) \geq E_{0} (ψ) \geq e_{0}

and

E_{0} (ψ) = e_{0}

⁠.

Step 3 (convergence in

H^{1} (R)

⁠): from (),

\begin{array}{l} \lim_{k \to \infty} ‖ ψ_{n_{k}}^{'} ‖_{2}^{2} & = \lim_{k \to \infty} (E_{0} (ψ_{n_{k}}) + \frac{α}{2} ‖ ψ_{n_{k}} ‖_{4}^{4} + β {|ψ_{n_{k}} (0)|}^{2}) \\ = e_{0} + \frac{α}{2} ‖ ψ ‖_{4}^{4} + β {|ψ (0)|}^{2} = ‖ ψ^{'} ‖_{2}^{2}, \end{array}

and since

ψ_{n_{k}} ⇀ ψ

in H¹,

‖ ψ_{n_{k}}^{'} - ψ^{'} ‖_{2} \to 0

⁠. Strong convergence in H¹ now follows from ().□

APPENDIX B: BOUND ON THE EFFECTIVE COULOMB POTENTIAL

Recall the effective Coulomb potential $V_{U}^{B}$ from (4.1).

Lemma B.1.

For any L > 0 and

ϕ \in H^{1} (R)

⁠, one has for B > 1,

\begin{array}{l} |\int_{R} V_{U}^{B} (x) {|ϕ (x)|}^{2} d x - (\ln B - 2 \ln \ln B) {|ϕ (0)|}^{2}| \\ \leq L^{- 1} ‖ ϕ ‖_{2}^{2} + 8 \sqrt{L} ‖ ϕ^{'} ‖_{2}^{3 / 2} ‖ ϕ ‖_{2}^{1 / 2} + |G (B, L)| ‖ ϕ^{'} ‖_{2} ‖ ϕ ‖_{2}; \end{array}

G (B, L) ≔ 2 \ln L + 2 \ln \ln B + 2 \int_{0}^{\infty} e^{- u} \ln (\sqrt{\frac{1}{u} + \frac{2}{B L^{2}}} + \sqrt{\frac{1}{u}}) d u - \ln 2 .

Proof.

Writing

\int_{R} V_{U}^{B} (x) {|ϕ (x)|}^{2}

as

| ϕ (0) |^{2} \int_{| x | \leq L} V_{U}^{B} (x) + \int_{| x | \leq L} V_{U}^{B} (x) (| ϕ (x) |^{2} - | ϕ (0) |^{2}) + \int_{| x | \geq L} V_{U}^{B} (x) | ϕ (x) |^{2},

it is possible to bound

\int_{| x | \geq L} V_{U}^{B} (x) {|ϕ (x)|}^{2} d x \leq L^{- 1} \int_{| x | \geq L} | ϕ (x) |^{2} d x,

(B1)

|\int_{| x | \leq L} V_{U}^{B} (x) ({|ϕ (x)|}^{2} - {|ϕ (0)|}^{2}) d x| \leq 8 \sqrt{L} ‖ ϕ^{'} ‖_{2}^{3 / 2} ‖ ϕ ‖_{2}^{1 / 2}

(B2)

and to evaluate the integral

\int_{| x | \leq L} V_{U}^{B} (x) d x = \ln B - 2 \ln \ln B + G (B, L) .

To arrive at the bound in (), the following inequalities are used:

|ϕ (x) - ϕ (0)| \leq \sqrt{|x|} ‖ ϕ^{'} ‖_{2} and ‖ ϕ ‖_{\infty}^{2} \leq ‖ ϕ^{'} ‖_{2} ‖ ϕ ‖_{2} .

(B3)

The lemma now follows from (), (), and the rightmost inequality of ().□

Corollary B.2.

For any L > 0 and

ϕ \in H^{1} (R)

⁠, one has for B > 1,

\begin{array}{l} |\iint_{R \times R} {|ϕ (x)|}^{2} \frac{1}{\sqrt{2}} V_{U}^{B} (\frac{x - y}{\sqrt{2}}) {|ϕ (y)|}^{2} d x d y - (\ln B - 2 \ln \ln B) ‖ ϕ ‖_{4}^{4}| \\ \leq L^{- 1} ‖ ϕ ‖_{2}^{4} + 8 \sqrt{L} ‖ ϕ^{'} ‖_{2}^{3 / 2} ‖ ϕ ‖_{2}^{5 / 2} + |G (B, L / \sqrt{2})| ‖ ϕ^{'} ‖_{2} ‖ ϕ ‖_{2}^{3} \end{array}

with

G (B, L)

as above.

Proof.

The corollary follows from Lemma B.1.□

Now, recall the potential $V_{L}^{B}$ from (5.5).

Corollary B.3.

For any L > 0 and

ϕ \in H^{1} (R)

⁠, one has for B > 1,

\begin{array}{l} |\int_{R} V_{L}^{B} (x) {|ϕ (x)|}^{2} d x - (\ln B - 2 \ln \ln B) {|ϕ (0)|}^{2}| \\ \leq L^{- 1} ‖ ϕ ‖_{2}^{2} + 8 \sqrt{L} ‖ ϕ^{'} ‖_{2}^{3 / 2} ‖ ϕ ‖_{2}^{1 / 2} + |D (B, L)| ‖ ϕ^{'} ‖_{2} ‖ ϕ ‖_{2}; \end{array}

D (B, L) ≔ 2 \ln L + 2 \ln \ln B + \frac{2}{\sqrt{\frac{2}{B L^{2}} + 1} + 1} + 2 \ln (\sqrt{1 + \frac{2}{B L^{2}}} + 1) - \ln 2 .

Proof.

Evaluating the integral

\int_{|x| < L} V_{L}^{B} (x) d x = \ln B - 2 \ln \ln B + D (B, L),

the argument follows the Proof of Lemma B.1.□

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2021

Author(s)

Ground state of the polaron hydrogenic atom in a strong magnetic field

I. INTRODUCTION

II. MODEL AND MAIN RESULT

III. DIFFERENTIATING THE ONE-DIMENSIONAL ENERGY

IV. UPPER BOUND TO THE GROUND-STATE ENERGY

V. LOWER BOUND TO THE GROUND-STATE ENERGY

A. Extracting a delta-function potential

B. Restricting to the lowest Landau level

C. Proof of Proposition 5.9

1. Reduction to one dimension

2. Localization and decomposition

3. Error estimates

VI. PROOF OF THEOREM 2.2

ACKNOWLEDGMENTS

DATA AVAILABILITY

APPENDIX A: COMPACTNESS OF MINIMIZING SEQUENCES

APPENDIX B: BOUND ON THE EFFECTIVE COULOMB POTENTIAL

REFERENCES

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Ground state of the polaron hydrogenic atom in a strong magnetic field

I. INTRODUCTION

II. MODEL AND MAIN RESULT

III. DIFFERENTIATING THE ONE-DIMENSIONAL ENERGY

IV. UPPER BOUND TO THE GROUND-STATE ENERGY

V. LOWER BOUND TO THE GROUND-STATE ENERGY

A. Extracting a delta-function potential

B. Restricting to the lowest Landau level

C. Proof of Proposition 5.9

1. Reduction to one dimension

2. Localization and decomposition

3. Error estimates

VI. PROOF OF THEOREM 2.2

ACKNOWLEDGMENTS

DATA AVAILABILITY

APPENDIX A: COMPACTNESS OF MINIMIZING SEQUENCES

APPENDIX B: BOUND ON THE EFFECTIVE COULOMB POTENTIAL

REFERENCES

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