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Math Phys Anal Geom
Math Phys Anal Geom
Mathematical Physics, Analysis, and Geometry
1385-0172
1572-9656
Springer Netherlands Dordrecht

39050929
9484
10.1007/s11040-024-09484-x
Article
Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities
Behrndt Jussi
Frymark Dale
http://orcid.org/0000-0001-8071-481X
Holzmann Markus holzmann@math.tugraz.at

Stelzer-Landauer Christian
https://ror.org/00d7xrm67 grid.410413.3 0000 0001 2294 748X Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
22 7 2024
22 7 2024
2024
27 3 128 3 2024
25 6 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
For a family of self-adjoint Dirac operators -ic(α·∇)+c22 subject to generalized MIT bag boundary conditions on domains in R3, it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large c.

Keywords

Dirac operator
Generalized MIT bag boundary conditions
Nonrelativistic limit
Spectral inequalities
Mathematics Subject Classification

Primary: 81Q10
58J50
Secondary: 35Q40
http://dx.doi.org/10.13039/501100002428 Austrian Science Fund P33568-N P33568-N P33568-N P33568-N Behrndt Jussi Frymark Dale Holzmann Markus Stelzer-Landauer Christian http://dx.doi.org/10.13039/501100000921 European Cooperation in Science and Technology CA 18232 MAT-DYN-NET CA 18232 MAT-DYN-NET CA 18232 MAT-DYN-NET Behrndt Jussi Frymark Dale Holzmann Markus Graz University of TechnologyOpen access funding provided by Graz University of Technology.

issue-copyright-statement© Springer Nature B.V. 2024
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pmcIntroduction

The MIT bag operator and more general types of self-adjoint Dirac operators on domains Ω⊂R3 have attracted a lot of attention in the last years. The MIT bag model itself originates from the investigation of quarks in hadrons from the 1970s [22, 26, 28, 34] and has been studied from a more mathematical perspective in [3–5, 16, 40, 42, 44, 47, 48]. The present paper is inspired by the recent contribution [7], where spectral properties of the family HκΩ, κ∈R, of self-adjoint Dirac operators1.1 HκΩf=-ic(α·∇)f+c22βf,domHκΩ={f∈H1(Ω;C4):f=i(sinh(κ)I4-cosh(κ)β)(α·ν)fon∂Ω},

in L2(Ω;C4) were studied. Here α·∇=α1∂1+α2∂2+α3∂3 with the usual Dirac matrices α1,α2,α3,β∈C4×4 (see (1.5) and (1.7) below), c>0 is the speed of light, Ω is a C2-domain with unit normal vector ν, and H1(Ω;C4) is the first order L2-based Sobolev space. The operators HκΩ model the propagation of a relativistic spin 12 particle with mass m=12 subject to the boundary conditions in (1.1), which are a three-dimensional counterpart of the quantum dot boundary conditions; cf. [19, 20], the introduction in [7] for more references in dimension two, and Sect. 2.2 for a further motivation of these boundary conditions. In particular, for κ=0 the standard MIT bag boundary conditions are recovered. If Ω is bounded, then the spectrum of HκΩ is purely discrete and consists of eigenvalues1.2 ⋯≤λ2-(HκΩ)≤λ1-(HκΩ)≤-c22<c22≤λ1+(HκΩ)≤λ2+(HκΩ)≤⋯,

that accumulate at ±∞. The main objective in [7] is the analysis of the eigenvalue curves κ↦λj±(HκΩ) and their asymptotic behaviour, which then leads to spectral geometry results for HκΩ with κ sufficiently large. The most remarkable result therein is a variant of the Faber–Krahn inequality for κ sufficiently large minimizing the first positive eigenvalue when Ω is a ball. For related spectral geometry results for two-dimensional Dirac operators with infinite mass boundary conditions we refer to [2, 20, 23, 39, 51].

In this paper we propose a different approach to obtain spectral inequalities and spectral geometry results for the Dirac operators HκΩ, which is based on the analysis of a nonrelativistic limit. This allows us to conclude for all sufficiently large c and all κ∈R, e.g., the Faber–Krahn inequality for the first two positive eigenvalues λ1+(HκΩ), λ2+(HκΩ), the Hong–Krahn–Szegö inequality minimizing the second two positive eigenvalues λ3+(HκΩ), λ4+(HκΩ), or the Payne–Pólya–Weinberger inequality for the ratios λj+(HκΩ)/λl+(HκΩ), j=1,2, l=3,4, of the first two and the second two positive eigenvalues, relying on classical counterparts for the Dirichlet Laplacian [8, 29, 33, 36, 37, 45]; here the spectral inequalities come for pairs of eigenvalues, as all eigenvalues of HκΩ have even multiplicity, and remain valid in an analogous form also for the first two pairs of negative eigenvalues, see Remark 3.7. In the same spirit other results from spectral geometry can be transferred from Laplacians to Dirac operators; we refer the reader to the monographs [31, 38, 46] for an introduction to and overview of this topic, but limit ourselves to the above-mentioned three examples.

The nonrelativistic limit provides a connection of the generalized MIT bag models with their nonrelativistic counterparts, i.e. Schrödinger operators, and is of independent interest, as it gives a physical interpretation of HκΩ. To find it one has to subtract the energy of the resting particle c22 and compute the limit of the resolvent of HκΩ-c22 as c→∞. Nonrelativistic limits of Dirac operators have been computed in many different settings. More information on three dimensional Dirac operators with regular potentials, for example, can be found in [50, Chapter 6] and the references therein. In [27] it was shown that the nonrelativistic limit of a family of one-dimensional Dirac operators with boundary conditions containing the counterpart of HκΩ is a Dirichlet or a Neumann Laplacian. Moreover, the nonrelativistic limit of one-dimensional Dirac operators with singular interactions supported on points, which are closely related to one-dimensional Dirac operators with boundary conditions, was studied extensively in [24, 25, 30, 32]. In higher dimensions, the nonrelativistic limit of Dirac operators with singular potentials supported on curves in R2 and surfaces in R3 was computed in various situations in [10, 11, 13, 18]. We also point out the paper [3], where it is shown that for bounded Ω the discrete eigenvalues of the MIT bag model, i.e. of HκΩ in (1.1) for κ=0, converge in the nonrelativistic limit to the eigenvalues of the Dirichlet Laplacian. However, in [3] only the convergence of the eigenvalues and not of the operator itself was studied.

In order to state our main result on the nonrelativistic limit of the operators HκΩ we make the following assumption on Ω, where we use the definition of a C2-domain as, e.g., in [41].

Hypothesis 1.1

Let Ω⊂R3 be a (bounded or unbounded) C2-domain, not necessarily connected, with a compact boundary and unit normal vector field ν pointing outwards of Ω. The bounded element in {Ω,R3\Ω¯} is denoted by Ω+, the unbounded element in {Ω,R3\Ω¯} is denoted by Ω-, and ν+ is the unit normal vector field pointing outwards of Ω+, so that ν=ν+ if Ω=Ω+ and ν=-ν+ if Ω=Ω-. For the common boundary we write Σ:=∂Ω=∂Ω+=∂Ω-.

Then, the main result of the present paper reads as follows:

Theorem 1.2

Let κ∈R, Ω⊂R3 be as in Hypothesis 1.1, and z∈C\[0,∞). Then, there exists a constant K(z) such that for all c sufficiently large z+c22∈ρ(HκΩ)∩ρ(-ΔDΩ) and1.3 HκΩ-z+c22-1-(-ΔDΩ-z)-1I2000L2(Ω;C4)→L2(Ω;C4)≤K(z)c,

where -ΔDΩ denotes the self-adjoint Dirichlet Laplacian in L2(Ω;C).

The strategy to prove Theorem 1.2 is to consider the self-adjoint orthogonal sum Hκ+⊕Hκ- in L2(Ω+;C4)⊕L2(Ω-;C4), which can be identified with a self-adjoint Dirac operator AκΣ in L2(R3;C4) with a δ-shell potential supported on Σ, see [6, 11, 16, 21]. Such types of Dirac operators with singular interactions are well-studied, see the review article [17] and the references therein. We collect some properties of AκΣ in Sect. 2.2 and provide a Krein type formula in Proposition 2.2 for its resolvent, which is the key tool for the analysis of the nonrelativistic limit. Each of the terms appearing in the resolvent formula will be examined separately and the main technical difficulty is the limit behavior of the inverse of1.4 ϑc+McCz+c2/2Mc,

involving a strongly singular boundary integral operator Cz+c2/2 on Σ, a coefficient matrix ϑc modelling the boundary condition in domHκΩ, and a scaling matrix Mc (see (2.5), (2.9), and (2.10) for details). In fact, it turns out that the operator in (1.4) does not converge to a boundedly invertible operator in one Sobolev space on Σ, but instead it is necessary to study the convergence of the inverse of (1.4) as an operator acting between different fractional order Sobolev spaces on Σ. Here we argue via the Schur complement and rely on an advanced and deep analysis of various boundary integral operators appearing in this context. Eventually, it turns out that the limit of AκΣ in the norm resolvent sense is an orthogonal sum of Dirichlet Laplacians and compressing the resolvents onto the original domain leads to (1.3).

It is well-known that the operator norm convergence in (1.3) implies the convergence of the corresponding spectra (see, e.g. [35, 49, 52]) and, in particular, if Ω is bounded, the spectrum of HκΩ is discrete and we conclude convergence of eigenvalues. This leads to spectral inequalities for the positive eigenvalues of the Dirac operators HκΩ, κ∈R, for c>0 sufficiently large; cf. Remark 3.7 for analogous results for the negative eigenvalues.

Corollary 1.3

Let κ∈R, Ω⊂R3 be a bounded C2-domain, B⊂R3 be a ball such that |B|=|Ω| and B1,B2⊂R3 be identical and disjoint balls such that |B1|+|B2|=|Ω|. Then, the following assertions hold for c>0 sufficiently large: (i) λj+(HκB)≤λj+(HκΩ) for j∈{1,2} and equality holds if and only if Ω is a ball.

(ii) λj+(HκB1∪B2)≤λj+(HκΩ) for j∈{3,4} and equality holds if and only if Ω is the union of two identical disjoint balls.

(iii) If, in addition, Ω is connected, then λj+(HκB)λl+(HκB)≤λj+(HκΩ)λl+(HκΩ),j∈{1,2},l∈{3,4},

and equality holds if and only if Ω is a ball.

The article is organized as follows. In Sect. 2 we introduce the free Dirac operator in R3 and some associated integral operators, show the connection of HκΩ and Dirac operators AκΣ with singular interactions supported on Σ=∂Ω, and recall some properties of the Dirichlet Laplacian. In Sect. 3 we compute the nonrelativistic limit of AκΣ, which allows us to prove Theorem 1.2 and Corollary 1.3.

Notations

The Dirac matrices are denoted by1.5 αk=0σkσk0,k∈{1,2,3},β=I200-I2,

where In is the n×n identity matrix, n∈N, andσ1=0110,σ2=0-ii0,σ3=100-1,

are the Pauli matrices. The Dirac matrices satisfy1.6 αjαk+αkαj=2δjkI4,αjβ+βαj=0,j,k∈{1,2,3},

where δjk is the Kronecker symbol. Moreover, the notations1.7 α·∇=∑j=13αj∂jandα·x=∑j=13αjxj,x=(x1,x2,x3)∈C3,

will often be used.

If M⊂R3 and k,l∈N, then the set of all continuous and k times continuously differentiable functions f:M→Cl is denoted by C(M;Cl) and Ck(M;Cl), respectively. Next, denote by F the Fourier transform on the space S′(R3;C) of tempered distributions. For the Sobolev spaces Hs(R3;C), s∈R, we shall use the definition1.8 Hs(R3;C)=f∈S′(R3;C):∫R3(1+|x|2)s|Ff(x)|2dx<∞,

with Hilbert space norm1.9 ‖f‖Hs(R3;C)2:=∫R3(1+|x|2)s|Ff(x)|2dx,f∈Hs(R3;C).

For Ω as in Hypothesis 1.1 the Sobolev spaces Hs(Ω;C), s>0, are defined via restrictions of functions from Hs(R3;C) onto Ω, and the spaces Ht(Σ;C), t∈[-2,2], on the boundary Σ of Ω are defined by using an open cover of Σ and a corresponding partition of unity, reducing it to Sobolev spaces on hypographs; see, e.g., [41, Chapter 3] for more details. We denote by γD:H1(Ω;C)→H1/2(Σ;C) the bounded Dirichlet trace operator and we shall use the same symbol for the trace operator γD:H1(R3;C)→H1/2(Σ;C). Sobolev spaces of vector valued functions are defined component-wise and in this context the action of the Dirichlet trace operator is also understood component-wise.

If A is a linear operator acting between two Hilbert spaces H and G, then its domain, range, and kernel are denoted by domA, ranA, and kerA, respectively. Whenever A is bounded and everywhere defined, then ‖A‖H→G is the operator norm of A. If A is self-adjoint in H, then the symbols ρ(A), σ(A), σess(A), and σdisc(A) are used for the resolvent set, spectrum, essential spectrum, and discrete spectrum of A, respectively.

Preliminaries

In this preliminary section we first collect several results about the free Dirac operator in R3 and associated integral operators. Afterwards, we show how the operators HκΩ in (1.1) are related to Dirac operators with δ-shell potentials, and we recall some useful properties of the single layer potential, single layer boundary integral operator, and the Dirichlet Laplacian that are needed to prove Theorem 1.2. Throughout this section we assume that Ω, Ω±, and Σ are as in Hypothesis 1.1.

The Free Dirac Operator and Associated Integral Operators

It is well-known that the free Dirac operator2.1 A0f=-ic(α·∇)f+c22βf,domA0=H1(R3;C4),

in R3 is self-adjoint in L2(R3;C4) and its spectrum is σ(A0)=(-∞,-c22]∪[c22,∞). For z∈ρ(A0)=C\((-∞,-c22]∪[c22,∞)) and f∈L2(R3;C4), the resolvent of A0 is given by(A0-z)-1f(x)=∫R3Gz(x-y)f(y)dy,x∈R3,

where the function Gz:R3\{0}→C4×4 is defined by2.2 Gz(x)=zc2I4+12β+1-iz2c2-c24|x|i(α·x)c|x|2eiz2/c2-c2/4|x|4π|x|

and the square root is chosen such that Imz2/c2-c2/4>0; cf. [50, Section 1.E].

Next, we introduce several integral operators and summarize some of their properties that are necessary to prove Theorem 1.2; we refer to [11, 15, 17] for more details. In the following γD:H1(R3;C4)→H1/2(Σ;C4) denotes the Dirichlet trace operator. For z∈ρ(A0) the map2.3 Φz∗:=γD(A0-z¯)-1:L2(R3;C4)→H1/2(Σ;C4)

is well-defined and bounded. It is not difficult to see that Φz∗ acts on f∈L2(R3;C4) asΦz∗f(x)=∫R3Gz¯(x-y)f(y)dy,x∈Σ.

The definition of Φz∗ in (2.3) allows to define the bounded anti-dual map2.4 Φz:=(Φz∗)′:H-1/2(Σ;C4)→L2(R3;C4).

With the help of Fubini’s theorem and (Gz¯(x))∗=Gz(-x) one shows that Φz acts on φ∈L2(Σ;C4) asΦzφ(x):=∫ΣGz(x-y)φ(y)dσ(y),x∈R3\Σ,

where dσ denotes the surface measure on Σ. We will also make use of the strongly singular boundary integral operator Cz:L2(Σ;C4)→L2(Σ;C4), z∈ρ(A0), acting via2.5 Czφ(x):=limε→0+∫Σ\B(x,ε)Gz(x-y)φ(y)dσ(y),x∈Σ,φ∈L2(Σ;C4),

where B(x,ε) is the ball of radius ε centered at x. For s∈[0,12] the map Cz gives rise to a bounded operator2.6 Cz:Hs(Σ;C4)→Hs(Σ;C4).

The adjoint of the realization of Cz in L2(Σ;C4) satisfies Cz∗=Cz¯ and it follows from (2.6) that Cz admits a bounded extension to Hs(Σ;C4), s∈[-12,0], such that2.7 Cz=(Cz¯)′:Hs(Σ;C4)→Hs(Σ;C4),s∈[-12,0],

where (Cz¯)′ denotes the anti-dual of Cz¯.

HκΩ and Dirac Operators with δ-Shell Potentials

In this subsection we show how the operators HκΩ defined in (1.1) are related to Dirac operators AκΣ with δ-shell potentials supported on Σ; the latter operators are well-studied, see, e.g., [6, 11, 17] and the references therein. Recall the notation Ω± and the unit outward normal vector field ν+ from Hypothesis 1.1. For a function f:R3→C4 we write f±:=f↾Ω±. Define the operator2.8 AκΣ=(-ic(α·∇)+c22β)f+⊕(-ic(α·∇)+c22β)f-,domAκΣ={f=f+⊕f-∈H1(Ω+;C4)⊕H1(Ω-;C4):-i(α·ν+)(γDf+-γDf-)=(sinh(κ)I4+cosh(κ)β)(γDf++γDf-)},

in L2(R3;C4). We note that AκΣ is the rigorously defined operator associated with the formal differential expression -ic(α·∇)+c22β+2c(sinh(κ)I4+cosh(κ)β)δΣ.

Our first observation is an immediate consequence from [11, Lemma 3.1 (ii)], which says that the operator formally given by -ic(α·∇)+c22β+(ηI4+τβ)δΣ decouples to the orthogonal sum of two Dirac operators with boundary conditions acting on functions in Ω± if and only if η2-τ2=-4c2; in the present setting the strength η of the electrostatic interaction in [11] is 2csinh(κ), the strength τ of the Lorentz scalar interaction is 2ccosh(κ), and the normal vector in the definition of Hκ- in (1.1) is -ν+. Note that this choice of η and τ is a natural parametrization of the arm of the hyperbola η2-τ2=-4c2 that contains the MIT bag boundary conditions. We also refer the reader to [6, Section 5], [16, Section 5.3], or [17, Section 5.2] for similar statements.

Lemma 2.1

The equality AκΣ=Hκ+⊕Hκ- holds.

In the next proposition we summarize some properties of the operator AκΣ that will be particularly useful for our analysis. Recall that A0 is the free Dirac operator defined in (2.1) and that Φz and Cz are the operators defined in (2.4) and (2.5), respectively. Moreover, define the two numbers2.9 a+:=12(cosh(κ)-sinh(κ))>0,a-:=-12(cosh(κ)+sinh(κ))<0,

and for c>0 the coefficient matrix ϑc and the scaling matrix Mc2.10 ϑc:=1ca+I200a-I2∈C4×4,Mc:=I200cI2∈C4×4.

Proposition 2.2

Let κ∈R and c>0. Then, the operator AκΣ in (2.8) is self-adjoint in L2(R3;C4), σ(AκΣ)=(-∞,-c22]∪[c22,∞), for z∈ρ(AκΣ) the linear operator ϑc+McCzMc admits a bounded inverse in H1/2(Σ;C4), and the formula(AκΣ-z)-1=(A0-z)-1-ΦzMc(ϑc+McCzMc)-1McΦz¯∗

holds.

Proof

It follows from [11, Lemma 3.3 and Theorems 3.4 & 4.1] or [17, Theorem 5.6] (in the case c=1) that AκΣ is self-adjoint in L2(R3;C4), that σess(AκΣ)=(-∞,-c22]∪[c22,∞), that I4+2c(sinh(κ)I4+cosh(κ)β)Cz is bijective in H1/2(Σ;C4) for z∈ρ(AκΣ)∩ρ(A0) and that the resolvent formula(AκΣ-z)-1=(A0-z)-1-Φz(I4+2c(sinh(κ)I4+cosh(κ)β)Cz)-1·2c(sinh(κ)I4+cosh(κ)β)Φz¯∗

holds. Note that the matrix 2c(sinh(κ)I4+cosh(κ)β) is invertible with inverse(2c(sinh(κ)I4+cosh(κ)β))-1=12c(-sinh(κ)I4+cosh(κ)β)=Mc-1ϑcMc-1.

Hence, also ϑc+McCzMc=Mc(12c(-sinh(κ)I4+cosh(κ)β)+Cz)Mc is bijective in H1/2(Σ;C4) and the claimed resolvent formula is true.

It remains to show that (-c22,c22)∩σ(AκΣ)=(-c22,c22)∩σp(AκΣ)=∅. For this, we use the Birman–Schwinger principle for AκΣ from [11, Lemma 3.3], which states that2.11 z∈-c22,c22∩σp(AκΣ)if and only if0∈σp(I4+2c(sinh(κ)I4+cosh(κ)β)Cz).

Let z∈(-c22,c22) and assume that φ∈ker(I4+2c(sinh(κ)I4+cosh(κ)β)Cz). Then,2.12 0=((I4+2cCz(-sinh(κ)I4+cosh(κ)β))·(I4+2c(sinh(κ)I4+cosh(κ)β)Cz)φ,φ)L2(Σ;C4)=(I4+4c2Cz2+2ccosh(κ)(Czβ+βCz))φ,φL2(Σ;C4).

With (1.6) and (2.2) one finds thatCzβ+βCz=212I4+zc2βSz2/c2-c2/4,

where Sz2/c2-c2/4 is the single layer boundary integral operator defined below in (2.14). In the present situation we have z2/c2-c2/4<0 and hence it follows that Sz2/c2-c2/4 is a non-negative operator in L2(Σ;C); cf. the text below (2.15) in the next subsection. Therefore, I4+4c2Cz2+2ccosh(κ)(Czβ+βCz) is a strictly positive operator in L2(Σ;C4) and we obtain φ=0 from (2.12). Therefore, by (2.11) we have z∉σp(AκΣ). □

From the properties of AκΣ one can now easily deduce the properties of HκΩ stated in the following corollary, when Ω coincides either with Ω+ or Ω-. The claims follow immediately from Lemma 2.1 and Proposition 2.2; for (i) one additionally uses that domHκΩ⊂H1(Ω;C4) is compactly embedded in L2(Ω;C4) if Ω is bounded, see also [7, Lemma 1.2], and that HκΩ commutes with the anti-linear time reversal operator Tf=-i(0I2I20)α2f¯, see the proof of [11, Proposition 4.2 (ii)] for details.

Corollary 2.3

Let κ∈R and c>0. Then, the operator HκΩ in (1.1) is self-adjoint in L2(Ω;C4) and the following holds: (i) If Ω is bounded, then σ(HκΩ)=σdisc(HκΩ)⊂(-∞,-c22]∪[c22,∞) and all eigenvalues of HκΩ have even multiplicity.

(ii) If Ω is unbounded, then σ(HκΩ)=(-∞,-c22]∪[c22,∞).

Moreover, for z∈C\((-∞,-c22]∪[c22,∞)) the resolvent formula(HκΩ-z)-1=PΩ(A0-z)-1PΩ∗-PΩΦzMc(ϑc+McCzMc)-1McΦz¯∗PΩ∗

holds, where PΩ:L2(R3;C4)→L2(Ω;C4) is the projection operator acting as f↦f↾Ω and its adjoint PΩ∗:L2(Ω;C4)→L2(R3;C4) is the embedding operator which extends a function g∈L2(Ω;C4) by zero.

The Dirichlet Laplacian and Associated Integral Operators

We begin by briefly recalling some properties of the single layer potential and single layer boundary integral operator associated with -Δ-μ, where -Δ is the self-adjoint Laplacian in L2(R3;C) defined on H2(R3;C) and μ∈ρ(-Δ)=C\[0,∞).

For φ∈L2(Σ;C) the single layer potential SLμ is the formal integral operator that acts as2.13 SLμφ(x)=∫Σeiμ|x-y|4π|x-y|φ(y)dσ(y),x∈R3\Σ,

and the single layer boundary integral operator Sμ is the mapping defined by2.14 Sμφ(x)=∫Σeiμ|x-y|4π|x-y|φ(y)dσ(y),x∈Σ,

where μ is again the complex square root satisfying Imμ>0 for μ∈C\[0,∞). It is well-known that for any s∈[-12,12] the map Sμ gives rise to a bounded and bijective operator2.15 Sμ:Hs(Σ;C)→Hs+1(Σ;C).

Moreover, we will use that for μ<0 the realization of Sμ in L2(Σ;C) is self-adjoint and non-negative. These claims can be shown in the same way as in [12, Lemma 2.6], where the two-dimensional case and μ=-1 is treated. Furthermore, by [41, Corollary 6.14] the mapping SLμ gives for any s∈(-12,1] rise to a bounded operatorSLμ:Hs-1/2(Σ;C)→Hs+1(Ω+;C)⊕Hs+1(Ω-;C).

Moreover, the representationsSLμ=(-Δ-μ)-1γD′:H-1/2(Σ;C)→H1(R3;C)

and2.16 Sμ=γD(-Δ-μ)-1γD′:H-1/2(Σ;C)→H1/2(Σ;C)

hold, where γD:H1(R3;C)→H1/2(Σ;C) is the bounded Dirichlet trace operator and γD′:H-1/2(Σ;C)→H-1(R3;C) its anti-dual map. We will also use that the L2-adjoint of SLμ is given by2.17 SLμ∗=γD(-Δ-μ¯)-1:L2(R3;C)→H3/2(Σ;C),

which is bounded, as the restriction γD:H2(R3;C)→H3/2(Σ;C) is bounded. Next, we state a useful continuity property of the map μ↦Sμ.

Lemma 2.4

Let M⊂C\[0,∞) be compact. Then, for all μ1,μ2∈M the operator Sμ1-Sμ2 has a bounded extension from H-3/2(Σ;C) to H3/2(Σ;C) and there exists a constant K(M)>0 such that the estimate2.18 ‖Sμ1-Sμ2‖H-3/2(Σ;C)→H3/2(Σ;C)≤K(M)|μ1-μ2|

holds. In particular, for any s∈[-32,32] the operator Sμ:Hs(Σ;C)→Hs(Σ;C) is uniformly bounded in μ∈M.

Proof

It suffices to show that(-Δ-μ1)-1-(-Δ-μ2)-1=(μ1-μ2)(-Δ-μ1)-1(-Δ-μ2)-1

gives rise to a bounded operator from H-2(R2;C) to H2(R2;C) that satisfies2.19 ‖(-Δ-μ1)-1-(-Δ-μ2)-1‖H-2(R3;C)→H2(R3;C)≤K(M)|μ1-μ2|,

as then (2.18) follows from (2.16) and the fact that γD has a continuous restriction γD:H2(R3;C)→H3/2(Σ;C) and γD′ a continuous extension γD′:H-3/2(Σ;C)→H-2(R3;C). To show (2.19), we compute for f∈H-2(R3;C), taking (1.9) into account,‖((-Δ-μ1)-1-(-Δ-μ2)-1)f‖H2(R3;C)2=∫R3(1+|x|2)2|F((-Δ-μ1)-1-(-Δ-μ2)-1)f(x)|2dx=∫R3(1+|x|2)21|x|2-μ1-1|x|2-μ22|Ff(x)|2dx=∫R3(1+|x|2)4|μ1-μ2|2|(|x|2-μ1)(|x|2-μ2)|2|Ff(x)|2(1+|x|2)2dx≤supx∈R3(1+|x|2)4|μ1-μ2|2|(|x|2-μ1)(|x|2-μ2)|2·‖f‖H-2(R3;C)2.

This shows (2.19) with K(M):=supx∈R3,μ1,μ2∈M(1+|x|2)2|(|x|2-μ1)(|x|2-μ2)|.

Eventually, it follows from (2.18) that Sμ:H-3/2(Σ;C)→H3/2(Σ;C) is uniformly bounded in μ∈M. Since Hs1(Σ;C) is continuously embedded in Hs2(Σ;C) for s1>s2, we conclude that Sμ:Hs(Σ;C)→Hs(Σ;C) is also uniformly bounded in μ∈M for any s∈[-32,32]. □

Let again Ω be a C2-domain as in Hypothesis 1.1. In the next lemma we express the resolvent of the self-adjoint Dirichlet Laplacian2.20 -ΔDΩf=-Δf,dom(-ΔDΩ)={f∈H2(Ω;C):γDf=0},

in L2(Ω;C) as the compression of the resolvent of the self-adjoint Laplacian -Δ in L2(R3;C) and a perturbation term. The statement follows from, e.g., [1, Theorem 4.4], [9, Theorem 3.2] or [14, Theorem 8.6.3], where instead of the single layer potential (2.13) and the single layer boundary integral operator (2.14) the terminology of γ-fields, Weyl functions or Q-functions, and Dirichlet-to-Neumann maps is used.

Lemma 2.5

Let Ω and Ω± be as in Hypothesis 1.1 and -ΔDΩ and -ΔD± be the corresponding Dirichlet Laplacians defined as in (2.20). Then, for the orthogonal sum -ΔD:=(-ΔD+)⊕(-ΔD-) and any z∈ρ(-ΔD)=C\[0,∞) the resolvent formula(-ΔD-z)-1=(-Δ-z)-1-SLzSz-1SLz¯∗

holds. In particular, one has(-ΔDΩ-z)-1=PΩ(-Δ-z)-1PΩ∗-PΩSLzSz-1SLz¯∗PΩ∗

with the projection and embedding operators PΩ and PΩ∗ from Corollary 2.3.

The Nonrelativistic Limit

In this section we compute the nonrelativistic limit of the operator AκΣ defined in (2.8) and use this to show Theorem 1.2 and Corollary 1.3. Again, we will always assume that Ω± is as in Hypothesis 1.1 and Σ=∂Ω±. Furthermore, we will often assume that z∈C\[0,∞) and c>|z|, as then z+c22∈ρ(A0)=ρ(AκΣ); cf. Proposition 2.2. In the following, the Krein type resolvent formula3.1 AκΣ-z+c22-1=A0-z+c22-1-Φz+c2/2Mc(ϑc+McCz+c2/2Mc)-1McΦz¯+c2/2∗

from Proposition 2.2 will play an important role. We will compute the limit of each of the terms on the right hand side separately. The convergence of (A0-(z+c2/2))-1, Φz+c2/2Mc, and McΦz¯+c2/2∗ is investigated in Sect. 3.1, the convergence of the map (ϑc+McCz+c2/2Mc)-1 is treated in Sect. 3.2, and the nonrelativistic limit of AκΣ is computed in Sect. 3.3.

Convergence of (A0-(z+c2/2))-1, Φz+c2/2Mc, and McΦz¯+c2/2∗

First, the nonrelativistic limit of the free Dirac operator A0 defined in (2.1) is discussed. This result is well-known, it follows, e.g., as a special case of the results in [50, Section 6]. However, since the result and the topology, in which the convergence takes place, are of importance in the analysis of Φz+c2/2Mc and McΦz¯+c2/2∗, we give a direct simple proof here to keep the presentation self-contained.

Proposition 3.1

Let z∈C\[0,∞) and c>|z|. Then, there exists a constant K(z) such that‖A0-z+c22-1-(-Δ-z)-1I2000‖L2(R3;C4)→H1(R3;C4)≤K(z)c.

Proof

We shall use (1.9) and compute for f∈L2(R3;C4) and c>|z|∫R3(1+|x|2)FA0-z+c22-1-(-Δ-z)-1I2000f(x)2dx=∫R3(1+|x|2)α·x+c2β+zc+c2I4c(|x|2-(z2c2+z))-1|x|2-zI2000Ff(x)2dx.

Next, we decompose the part of the integrand that does not depend on Ff in the last line of the equation. Using 12(β+I4)=(I2000) we findsupx∈R3(1+|x|2)|α·x+zcI4c(|x|2-z2c2-z)+12(β+I4)|x|2-z2c2-z-1|x|2-zI2000|2=supx∈R31+|x|2c2|α·x+zcI4|x|2-z2c2-z+z2c(|x|2-z2c2-z)(|x|2-z)I2000|2≤K(z)2c2

for some constant K(z) that does not depend on c, since the assumptions z∈C\[0,∞) and c>|z| ensure that there is no singularity in the last x-dependent expression. As ‖Ff‖L2(R3;C4)=‖f‖L2(R3;C4) we conclude∫R3(1+|x|2)FA0-z+c22-1-(-Δ-z)-1I2000f(x)2dx≤K(z)2c2‖f‖L2(R3;C4)2,

which shows the desired result. □

By using the convergence result from Proposition 3.1 and the definition (2.3) of Φz∗, it is not difficult to obtain the convergence of Φz+c2/2 and Φz+c2/2∗. Recall that SLμ, μ∈C\[0,∞), is the single layer potential defined in (2.13) and that Mc is the scaling matrix given by (2.10). Since there is a multiplication by c involved, the rate of convergence in the following proposition reduces to O(c-1/2). This is the main reason why we get this rate of convergence in Theorem 1.2.

Proposition 3.2

Let z∈C\[0,∞) and c>|z|. Then, there exists a constant K(z) such that3.2 Φz+c2/2Mc-SLzI2000H-1/2(Σ;C4)→L2(R3;C4)≤K(z)c

and3.3 McΦz+c2/2∗-SLz∗I2000L2(R3;C4)→H1/2(Σ;C4)≤K(z)c.

In particular, the operators Φz+c2/2Mc:H-1/2(Σ;C4)→L2(R3;C4) and the mappings McΦz+c2/2∗:L2(R3;C4)→H1/2(Σ;C4) are uniformly bounded in c.

Proof

Recall from (2.3) and (2.17) thatΦz+c2/2∗=γDA0-z¯+c22-1andSLz∗=γD(-Δ-z¯)-1.

Hence, (3.3) follows from Proposition 3.1 and the mapping properties of the trace operator; the stated rates of convergence are obtained by accounting for the matrix terms in equation (3.3). The claim in (3.2) follows from (3.3) by duality. The uniform boundedness of Φz+c2/2Mc and McΦz+c2/2∗ is clear as these operators converge. □

Convergence of (ϑc+McCz+c2/2Mc)-1

The more difficult part in the analysis of (3.1) is (ϑc+McCz+c2/2Mc)-1. To handle it in the computation of the nonrelativistic limit, first a more detailed consideration of Cz+c2/2 is provided. Define for z∈ρ(A0) the auxiliary operator Tz that formally acts on a sufficiently smooth function φ:Σ→C2 viaTzφ(x):=limε→0+∫Σ\B(x,ε)tz(x-y)φ(y)dσ(y),x∈Σ,

withtz(x):=1-iz2c2-c24|x|i(σ·x)4π|x|3eiz2/c2-c2/4|x|,x≠0.

Next, the definition of Gz in (2.2) impliesGz+c2/2(x)=zc2I4+I2000+1-iz+z2c2|x|i(α·x)c|x|2eiz+z2/c2|x|4π|x|.

This and the definitions of Cz and Sz in Eqs. (2.5) and (2.14) lead to3.4 Cz+c2/2=(zc2+1)Sz+z2/c2I21cTz+c2/21cTz+c2/2zc2Sz+z2/c2I2.

It follows from the latter representation and (2.6)–(2.7) that Tz+c2/2 gives rise to a bounded operator3.5 Tz+c2/2:Hs(Σ;C2)→Hs(Σ;C2),s∈[-12,12],

and that the anti-dual of Tz+c2/2 satisfies Tz+c2/2′=Tz¯+c2/2. In the next proposition we show that these operators are even uniformly bounded in c.

Proposition 3.3

Let z∈C\[0,∞) and c>|z|. Then, for any s∈[-12,12] the operators Tz+c2/2:Hs(Σ;C2)→Hs(Σ;C2) are uniformly bounded in c.

Proof

The proof of this proposition is split into three steps. In Step 1 the integral kernel tz+c2/2 of Tz+c2/2 is decomposed into a singular part d, which is independent of c, and a remainder term t~z,c which is easier to analyze. In Step 2 it is shown that the integral operator with kernel t~z,c gives rise to a bounded operator from L2(Σ;C2) to H1(Ω+;C2) that is uniformly bounded in c. By combining the results from Step 1 & 2, the proof of the proposition is completed in Step 3.

Step 1 Rewriting the exponential in the kerneltz+c2/2(x)=1-iz+z2c2|x|i(σ·x)4π|x|3eiz+z2/c2|x|,x≠0,

as a power series shows that the terms with |x|-2 cancel out. After combining and rearranging the coefficients of the remaining terms we obtain3.6 tz+c2/2(x)=d(x)+t~z,c(x),

where3.7 d(x)=i(σ·x)4π|x|3,x≠0,

andt~z,c(x)=∑k=0∞ik+3(k+2)!+ik+1(k+1)!z+z2c2k+2|x|k-1σ·x4π,x≠0.

Step 2 Now we consider t~z,c(x-y) for x∈Ω+ and y∈Σ and define the integral operator T~z,c for sufficiently smooth functions φ:Σ→C2 as3.8 T~z,cφ(x):=∫Σt~z,c(x-y)φ(y)dσ(y),x∈Ω+.

We will show that3.9 T~z,c:L2(Σ;C2)→H1(Ω+;C2)is uniformly bounded inc.

For this, we first establish some simple bounds on t~z,c and its first order derivatives that are independent of c. Observe that for a constant K1=K1(z) one has for all x∈Ω+ and y∈Σ3.10 |t~z,c(x-y)|≤∑k=0∞2k!(2|z|)k+2|x-y|k4π≤K1,

as the latter series is absolutely converging and defines a continuous function on the compact set Ω+¯×Σ. Likewise, there exists a constant K2=K2(z) such that for the partial derivatives of t~z,c and all x∈Ω+ and y∈Σ one has3.11 |∂xjt~z,c(x-y)|≤∑k=0∞2k!(2|z|)k+24π|∂xj((σ·(x-y))|x-y|k-1)|=∑k=0∞(2|z|)k+22πk!|σj+(k-1)(xj-yj)(σ·(x-y))|x-y|2||x-y|k-1≤K2|x-y|.

Since Ω+ is bounded, (3.10) and (3.11) imply(x,y)↦t~z,c(x-y)∈L2(Ω+×Σ;C2×2),(x,y)↦∂xjt~z,c(x-y)∈L2(Ω+×Σ;C2×2)

and there exists a constant K3=K3(z) such that3.12 ∫Ω+∫Σ|t~z,c(x-y)|2dσ(y)dx≤K3,∫Ω+∫Σ|∂xjt~z,c(x-y)|2dσ(y)dx≤K3.

Furthermore, using that for any y∈Σ one has x↦t~z,c(x-y)∈C∞(Ω+;C2×2), (3.11), and the dominated convergence theorem, it is not difficult to see that for any φ∈L2(Σ;C2) one has T~z,cφ∈C1(Ω+;C2) and3.13 ∂xjT~z,cφ(x)=∫Σ∂xjt~z,c(x-y)φ(y)dσ(y),x∈Ω+.

Combining (3.8) and (3.13) with (3.12) shows that T~z,c,∂xjT~z,c:L2(Σ;C2)→L2(Ω+;C2) are Hilbert–Schmidt operators that are uniformly bounded in c, and hence (3.9) is true.

Step 3 We verify that Tz+c2/2:Hs(Σ;C2)→Hs(Σ;C2) is uniformly bounded in c for any s∈[-12,12]. First, we do this for s=12. For that purpose, consider the operator γDT~z,c:L2(Σ;C2)→H1/2(Σ;C2), which is uniformly bounded in c by the results in Step 2, and hence also the restriction3.14 γDT~z,c:H1/2(Σ;C2)→H1/2(Σ;C2)

is uniformly bounded in c. Furthermore, we shall use that3.15 γDT~z,cφ(x)=∫Σt~z,c(x-y)φ(y)dσ(y),x∈Σ,

holds for all φ∈L2(Σ;C2) (and, in particular, for all φ∈H1/2(Σ;C2)). In fact, the estimate (3.10) extends to Ω+¯×Σ and this implies that the function T~z,cφ:Ω+→C2 admits a continuous extension onto Ω+¯, which shows (3.15).

Next, recall that the function d is defined by (3.7). For φ∈H1/2(Σ;C2) consider the integral operatorDφ(x):=limε→0+∫Σ\B(x,ε)d(x-y)φ(y)dσ(y),x∈Σ,

which is bounded in H1/2(Σ;C2) by [43, Theorem 4.3.1] as d is a homogeneous kernel of order 0 in the sense of [43, Section 4.3.2], see also [43, Example 4.2] (the boundedness of D would also follow from (3.16) and the reasoning below, as Tz+c2/2 is bounded in H1/2(Σ;C2) by (3.5)). From (3.6) and (3.15) we obtain3.16 Tz+c2/2φ=Dφ+γDT~z,cφ,φ∈H1/2(Σ;C2),

and now it follows from the uniform boundedness of the operator γDT~z,c in (3.14) that also Tz+c2/2:H1/2(Σ;C2)→H1/2(Σ;C2) is uniformly bounded in c.

To show the claim for s=-12, recall that Tz+c2/2 has a bounded extension in H-1/2(Σ;C2) given by Tz+c2/2=(Tz¯+c2/2)′. Hence, by the already shown uniform boundedness in c of Tz¯+c2/2 in H1/2(Σ;C2) also(Tz¯+c2/2)′=Tz+c2/2:H-1/2(Σ;C2)→H-1/2(Σ;C2)

is uniformly bounded in c. Finally, as Tz+c2/2 is uniformly bounded in H-1/2(Σ;C2) and H1/2(Σ;C2) in c, it follows with an interpolation argument using [41, Theorems B.2 and B.11] that Tz+c2/2:Hs(Σ;C2)→Hs(Σ;C2) is also uniformly bounded in c for any s∈(-12,12). This finishes the proof. □

Next, the convergence of (ϑc+McCz+c2/2Mc)-1 is analyzed. Recall that a± is defined by (2.9). By (3.4) one has the block structureϑc+McCz+c2/2Mc=(1ca++zc2+1Sz+z2/c2)I21cTz+c2/21cTz+c2/2(a-+zcSz+z2/c2)I2.

To proceed, note that for z∈C\[0,∞) and c>0 sufficiently large the operator a-+zcSz+z2/c2 is boundedly invertible in Hs(Σ;C), s∈[-12,12], with inverse given by3.17 (a-+zcSz+z2/c2)-1=1a-∑n=0∞-za-cSz+z2/c2n,

as a-<0 and by Lemma 2.4 the operator Sz+z2/c2 is (uniformly) bounded in Hs(Σ;C) in c. Thus, one can write3.18 ϑc+McCz+c2/2Mc=I21cTz+c2/2a-+zcSz+z2/c2-10I2·S~z,c00(a-+zcSz+z2/c2)I2I201ca-+zcSz+z2/c2-1Tz+c2/2I2,

where the Schur complement S~z,c is given by3.19 S~z,c=1ca+I2+zc2+1Sz+z2/c2I2-1cTz+c2/2a-+zcSz+z2/c2-1Tz+c2/2.

The first and the third factor in (3.18) are bijective in H1/2(Σ;C4). Since the map ϑc+McCz+c2/2Mc has this property as well by Proposition 2.2, we conclude that also S~z,c is bijective in H1/2(Σ;C2). In the following proposition, the convergence of S~z,c-1 is analyzed.

Proposition 3.4

Let z<0 and c>|z|. Then, there exists a constant K(z) such that for all c sufficiently large3.20 ‖S~z,c-1-Sz-1I2‖H3/2(Σ;C2)→H-1/2(Σ;C2)≤K(z)c.

Moreover, S~z,c-1:H1/2(Σ;C2)→H-1/2(Σ;C2) is uniformly bounded in c.

Proof

The proof of this proposition is split into four steps. In Step 1 we show that for s∈[-12,12] there exists a constant K1=K1(z,s) such that3.21 ‖S~z,c-SzI2‖Hs(Σ;C2)→Hs(Σ;C2)≤K1c

for c>0 sufficiently large. In Step 2 we verify that the realization of S~z,c in L2(Σ;C2) is bijective and there exists a constant K2 such that for c>0 sufficiently large3.22 ‖S~z,c-1‖L2(Σ;C2)→L2(Σ;C2)≤K2c.

Using this, we show in Step 3 our claim that S~z,c-1:H1/2(Σ;C2)→H-1/2(Σ;C2) is uniformly bounded in c, while in Step 4 we prove (3.20).

Step 1 For s∈[-12,12] fixed and c>0 sufficiently large we obtain the estimate3.23 ‖S~z,c-SzI2‖Hs(Σ;C2)→Hs(Σ;C2)≤(Sz+z2/c2-Sz)I2Hs(Σ;C2)→Hs(Σ;C2)+1c‖a+I2+zcSz+z2/c2I2-Tz+c2/2(a-+zcSz+z2/c2)-1Tz+c2/2‖Hs(Σ;C2)→Hs(Σ;C2)

from (3.19). For the first term on the right-hand side of (3.23) one has by Lemma 2.4‖(Sz+z2/c2-Sz)I2‖Hs(Σ;C2)→Hs(Σ;C2)≤‖(Sz+z2/c2-Sz)I2‖H-3/2(Σ;C2)→H3/2(Σ;C2)≤K1′z2c2

with some constant K1′=K1′(z). Note also that Sz+z2/c2 is uniformly bounded in Hs(Σ;C) for c>0 sufficiently large by Lemma 2.4. Therefore, since a-<0 we conclude from (3.17) and the estimate(a-+zcSz+z2/c2)-1Hs(Σ;C)→Hs(Σ;C)≤1-a-(1-za-c‖Sz+z2/c2‖Hs(Σ;C)→Hs(Σ;C))-1

that (a-+zcSz+z2/c2)-1 is also uniformly bounded in Hs(Σ;C) for c>0 sufficiently large. Combining this with Proposition 3.3 it follows that the second term on the right-hand side of (3.23) is bounded by K1′′c with some constant K1′′=K1′′(z,s); thus we conclude (3.21).

Step 2 For z<0 and c>0 sufficiently large we shall now consider the operatorS~z,c=1ca+I2+(zc2+1)Sz+z2/c2I2-1cTz+c2/2(a-+zcSz+z2/c2)-1Tz+c2/2

in L2(Σ;C2). Note that for c>0 sufficiently large Sz+z2/c2 is bounded, self-adjoint and nonnegative in L2(Σ;C2) (see the discussion after (2.15)) and hence the same holds for the operator (zc2+1)Sz+z2/c2. Furthermore, for c>0 sufficiently large Tz+c2/2 is bounded and self-adjoint in L2(Σ;C2) (see (3.5)), and together with (3.17) we conclude that also S~z,c is bounded and self-adjoint. As a-<0 and Sz+z2/c2 is uniformly bounded in c it is clear that a-+zcSz+z2/c2 is a negative operator in L2(Σ;C2) for c>0 sufficiently large, and the same is true for its inverse. Therefore,-1cTz+c2/2(a-+zcSz+z2/c2)-1Tz+c2/2

is a nonnegative operator in L2(Σ;C2) for c>0 sufficiently large. This implies S~z,c≥a+c for c>0 sufficiently large, which in turn yields (3.22) with K2=a+-1.

Step 3 We claim that S~z,c-1:H1/2(Σ;C2)→H-1/2(Σ;C2) is uniformly bounded in c. For this it suffices to prove that for c>0 sufficiently large there exists a constant K3=K3(z) such that3.24 ‖S~z,c-1‖H1(Σ;C2)→L2(Σ;C2)≤K3,

as then by duality and formal symmetry one also has‖S~z,c-1‖L2(Σ;C2)→H-1(Σ;C2)≤K3,

and an interpolation argument (see [41, Theorems B.2 and B.11]) leads to the assertion.

To show (3.24), we use3.25 S~z,c-1=Sz-1I2-S~z,c-1S~z,c-SzI2Sz-1I2

and the fact that Sz-1:H1(Σ;C)→L2(Σ;C) is bounded; cf. (2.15). Using (3.21) for s=0 with K1=K1(z,0) and (3.22) we obtain‖S~z,c-1‖H1(Σ;C2)→L2(Σ;C2)≤‖Sz-1‖H1(Σ;C)→L2(Σ;C)+‖S~z,c-1‖L2(Σ;C2)→L2(Σ;C2)‖SzI2-S~z,c‖L2(Σ;C2)→L2(Σ;C2)‖Sz-1‖H1(Σ;C)→L2(Σ;C)≤‖Sz-1‖H1(Σ;C)→L2(Σ;C)1+K2·c·K1c,

and hence (3.24) holds.

Step 4   Finally, we show (3.20). Using again (3.25), the fact that Sz:H1/2(Σ;C)→H3/2(Σ;C) is boundedly invertible, and the results from Step 1 and Step 3 we obtain‖S~z,c-1-Sz-1I2‖H3/2(Σ;C2)→H-1/2(Σ;C2)≤‖S~z,c-1‖H1/2(Σ;C2)→H-1/2(Σ;C2)‖SzI2-S~z,c‖H1/2(Σ;C2)→H1/2(Σ;C2)‖Sz-1‖H3/2(Σ;C)→H1/2(Σ;C)≤K(z)c.

This completes the proof of Proposition 3.4. □

Now we are ready to study the convergence of the inverse of ϑc+McCz+c2/2Mc.

Proposition 3.5

Let z<0 and c>|z|. Then, there exists a constant K(z) such that for all c sufficiently large(ϑc+McCz+c2/2Mc)-1-Sz-1I200a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)≤K(z)c.

Moreover, (ϑc+McCz+c2/2Mc)-1:H1/2(Σ;C4)→H-1/2(Σ;C4) is uniformly bounded in c.

Proof

It follows from (3.18) that3.26 (ϑc+McCz+c2/2Mc)-1=F1(c)F2(c)F3(c),

whereF1(c):=I20-1ca-+zcSz+c2/2-1Tz+c2/2I2,F2(c):=S~z,c-100a-+zcSz+z2/c2-1I2,F3(c):=I2-1cTz+c2/2a-+zcSz+z2/c2-10I2.

For c>0 sufficiently large we use the uniform boundedness of Sz+z2/c2 in Hs(Σ;C), s∈[-12,12], from Lemma 2.4 to estimate(a-+zcSz+z2/c2)-1-a--1Hs(Σ;C)→Hs(Σ;C)=-1a-∑n=1∞-za-cSz+z2/c2nHs(Σ;C)→Hs(Σ;C)≤-1a-za-c‖Sz+z2/c2‖Hs(Σ;C)→Hs(Σ;C)1-za-c‖Sz+z2/c2‖Hs(Σ;C)→Hs(Σ;C)≤K1c,

where K1=K1(z,s) is a constant; for the restriction onto H3/2(Σ;C) viewed as a mapping into H-1/2(Σ;C) this estimate yields3.27 (a-+zcSz+z2/c2)-1-a--1H3/2(Σ;C)→H-1/2(Σ;C)≤K1c,

and also shows that (a-+zcSz+z2/c2)-1 is uniformly bounded in Hs(Σ;C2), s∈[-12,12], for c>0 sufficiently large; cf. Step 1 in the proof of Proposition 3.4. Together with Proposition 3.3 this implies with a constant K2=K2(z) that3.28 ‖F1(c)-I4‖H-1/2(Σ;C4)→H-1/2(Σ;C4)≤K2c,‖F3(c)-I4‖H1/2(Σ;C4)→H1/2(Σ;C4)≤K2c.

In particular, F1(c) is uniformly bounded in H-1/2(Σ;C4) and F3(c) is uniformly bounded in H1/2(Σ;C4) in c, and the restrictions onto H1/2(Σ;C4) and H3/2(Σ;C4) satisfy the same bounds3.29 ‖F1(c)-I4‖H1/2(Σ;C4)→H-1/2(Σ;C4)≤K2c,‖F3(c)-I4‖H3/2(Σ;C4)→H1/2(Σ;C4)≤K2c.

Moreover, Proposition 3.4 and (3.27) implyF2(c)-Sz-1I200a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)≤K3c

with some constant K3=K3(z). Eventually, it follows from Proposition 3.4 and the uniform boundedness of (a-+zcSz+z2/c2)-1 as a mapping from H1/2(Σ;C) to H-1/2(Σ;C) that F2(c):H1/2(Σ;C4)→H-1/2(Σ;C4) is uniformly bounded. Combining this with (3.28) and (3.29) gives(ϑc+McCz+c2/2Mc)-1-Sz-100a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)≤‖F1(c)F2(c)(F3(c)-I4)‖H3/2(Σ;C4)→H-1/2(Σ;C4)+F1(c)F2(c)-Sz-1I200a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)+(F1(c)-I4)Sz-1I200a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)≤‖F1(c)F2(c)‖H1/2(Σ;C4)→H-1/2(Σ;C4)‖F3(c)-I4‖H3/2(Σ;C4)→H1/2(Σ;C4)+F1(c)H-1/2(Σ;C4)→H-1/2(Σ;C4)F2(c)-Sz-1I200a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)+F1(c)-I4H1/2(Σ;C4)→H-1/2(Σ;C4)Sz-1I200a--1I2H3/2(Σ;C4)→H1/2(Σ;C4)≤K(z)c,

which is exactly the claimed convergence result.

Finally, the claim about the uniform boundedness of the operator(ϑc+McCz+c2/2Mc)-1:H1/2(Σ;C4)→H-1/2(Σ;C4)

follows from (3.26) and the above observations on the uniform boundedness of F1(c) in H-1/2(Σ;C4), F2(c) from H1/2(Σ;C4) to H-1/2(Σ;C4), and F3(c) in H1/2(Σ;C4). □

Nonrelativistic Limit of AκΣ

With the preparations from the previous sections we are now ready to discuss the nonrelativistic limit of the Dirac operators AκΣ.

Proposition 3.6

Let AκΣ, κ∈R, be as in (2.8) and -ΔD:=(-ΔD+)⊕(-ΔD-), where -ΔD± is the Dirichlet Laplacian in Ω± from (2.20). Let z<0 and c>|z|. Then, there exists a constant K(z) such that for all c sufficiently largeAκΣ-z+c22-1-(-ΔD-z)-1I2000L2(R3;C4)→L2(R3;C4)≤K(z)c.

Proof

Let Mc and ϑc be defined by (2.10). As -c2<z<0, one has z∈ρ(-ΔD) and z+c22∈(-c22,c22)⊂ρ(AκΣ); cf Proposition 2.2. Furthermore, from Proposition 2.2 and Lemma 2.5 we obtain3.30 AκΣ-z+c22-1-(-ΔD-z)-1I2000=A0-z+c22-1-Φz+c2/2Mc(ϑc+McCz+c2/2Mc)-1McΦz+c2/2∗-(-Δ-z)-1-SLzSz-1SLz∗I2000=D1(c)+D2(c)+D3(c)+D4(c)

withD1(c):=A0-z+c22-1-(-Δ-z)-1I2000,D2(c):=-Φz+c2/2Mc(ϑc+McCz+c2/2Mc)-1McΦz+c2/2∗-SLz∗I2000,D3(c):=-Φz+c2/2Mc(ϑc+McCz+c2/2Mc)-1-Sz-100a--1I2SLz∗I2000,D4(c):=-Φz+c2/2Mc-SLzI2000Sz-100a--1I2SLz∗I2000.

First, it follows from Proposition 3.1 that ‖D1(c)‖L2(R3;C4)→L2(R3;C4)≤K1c for a constant K1=K1(z). To discuss D2(c) recall that Φz+c2/2Mc:H-1/2(Σ;C4)→L2(R3;C4) is uniformly bounded in c by Proposition 3.2 and (ϑc+McCz+c2/2Mc)-1:H1/2(Σ;C4)→H-1/2(Σ;C4) is uniformly bounded in c by Proposition 3.5. Hence, we find with Proposition 3.2 that there exists a constant K2=K2(z) such that‖D2(c)‖L2(R3;C4)→L2(R3;C4)≤‖Φz+c2/2Mc‖H-1/2(Σ;C4)→L2(R3;C4)·‖(ϑc+McCz+c2/2Mc)-1‖H1/2(Σ;C4)→H-1/2(Σ;C4)·McΦz+c2/2∗-SLz∗I2000L2(R3;C4)→H1/2(Σ;C4)≤K2c.

Next, as SLz∗:L2(R3;C)→H3/2(Σ;C) is bounded (see (2.17)), Proposition 3.5 implies that there exists a constant K3=K3(z) such that‖D3(c)‖L2(R3;C4)→L2(R3;C4)≤‖Φz+c2/2Mc‖H-1/2(Σ;C4)→L2(R3;C4)·(ϑc+McCz+c2/2Mc)-1-Sz-100a--1I2H3/2(Σ;C4)→H-1/2(Σ;C4)·SLz∗I2000L2(R3;C4)→H3/2(Σ;C4)≤K3c.

In a similar way, as Sz-1:H1/2(Σ;C)→H-1/2(Σ;C) is bounded (see (2.15)), we find with Proposition 3.2 that there exists a constant K4=K4(z) such that‖D4(c)‖L2(R3;C4)→L2(R3;C4)≤Φz+c2/2Mc-SLzI2000H-1/2(Σ;C4)→L2(R3;C4)·Sz-100a--1I2H1/2(Σ;C4)→H-1/2(Σ;C4)·SLz∗I2000L2(R3;C4)→H1/2(Σ;C4)≤K4c.

Now the statement of the proposition follows by combining the above estimates for the operators D1(c),D2(c),D3(c), and D4(c) with (3.30). □

Proof of Theorem 1.2 and Corollary 1.3

In this section we complete the proof of our main result by combining Lemma 2.1 and Proposition 3.6. For this let κ∈R, Ω,Ω± be as in Hypothesis 1.1, and Σ=∂Ω. Let the Dirac operator AκΣ be defined as in (2.8) and denote by PΩ:L2(R3;C4)→L2(Ω;C4) the operator PΩf=f↾Ω; cf. Corollary 2.3. Then, the self-adjoint Dirac operator HκΩ in L2(Ω;C4) satisfiesHκΩ=PΩAκΣPΩ∗

by Lemma 2.1 and since σ(HκΩ)⊂(-∞,-c22]∪[c22,∞) by Corollary 2.3 it is clear that z+c2/2 with z∈C\[0,∞) and c>|z| belongs to ρ(HκΩ)∩ρ(AκΣ), so thatHκΩ-z+c22-1=PΩAκΣ-z+c22-1PΩ∗.

Therefore, if z<0, then (1.3) follows from Proposition 3.6. In the general case z∈C\[0,∞) we obtain (1.3) by assuming that c>1 and using the identityHκΩ-z+c22-1-(-ΔDΩ-z)-1I2000=I4+(z+1)(-ΔDΩ-z)-1I2000·HκΩ--1+c22-1-(-ΔDΩ+1)-1I2000·I4+(z+1)HκΩ-z+c22-1.

This completes the proof of Theorem 1.2 and now we turn our attention to Corollary 1.3, which can be viewed as an immediate consequence of classical results on eigenvalues of Dirichlet Laplacians and convergence of spectra under operator norm convergence of resolvents. For the convenience of the reader and to keep the presentation self-contained we briefly provide the details of the arguments. In the present situation, it is convenient to apply [52, Satz 3.17 d)] or [31, Theorem 2.3.1] about the convergence of eigenvalues of nonnegative compact operators. More precisely, let Bc, c∈(c0,∞] for a suitable c0∈R, be a family of compact, self-adjoint, and nonnegative operators with eigenvalues μ1(Bc)≥μ2(Bc)≥⋯≥0 taking multiplicities into account. If Bc converges to B∞ in the operator norm, as c→∞, then by [52, Satz 3.17 d)] for all j∈N also μj(Bc) converges to μj(B∞), as c→∞. We apply this result toBc:=fHκΩ-(-1+c22-1)andB∞:=f(-ΔDΩ+1)-1I2000,

where f∈C(R) is a nonnegative function such that f(x)=0 for x∈(-∞,0]∪[2,∞), and f(x)=x for x∈(0,1]. It is clear that Bc and B∞ are nonnegative bounded self-adjoint operators, and from the compactness of the resolvents of HκΩ and -ΔDΩ, which holds as domHκΩ and dom(-ΔDΩ) are compactly embedded in L2(Ω;C4) and L2(Ω;C), respectively, it follows that Bc and B∞ are both compact.

In the following let c>1. For the eigenvalues λj±(HκΩ) of HκΩ ordered as in (1.2) we haveλj-(HκΩ)+1-c22-1<0andλj+(HκΩ)+1-c22-1≤1,j∈N;

cf. Corollary 2.3. From the choice of f it is clear that the positive eigenvalues of Bc are given by μj(Bc)=(λj+(HκΩ)+1-c22)-1. Similarly, σ(-ΔDΩ)⊂[0,∞) and the fact that two copies of (-ΔDΩ+1)-1 appear in the definition of B∞ leads toμ2j-1(B∞)=μ2j(B∞)=(λj(-ΔDΩ)+1)-1

for j∈N, where 0<λ1(-ΔDΩ)≤λ2(-ΔDΩ)≤⋯ denote the discrete eigenvalues of -ΔDΩ taking multiplicities into account. Now it follows from Theorem 1.2 and [49, Theorem VIII.20] that Bc converges to B∞ in the operator norm, as c→∞. Using that all eigenvalues of HκΩ have even multiplicity, see Corollary 2.3 (i), we conclude with [52, Satz 3.17 d)] from the above considerations that for any j∈Nμ2j-1(Bc)=μ2j(Bc)=λ2j-1+(HκΩ)-c22+1-1=λ2j+(HκΩ)-c22+1-1

tends toμ2j-1(B∞)=μ2j(B∞)=(λj(-ΔDΩ)+1)-1,asc→∞,

which is equivalent to3.31 λ2j-1+(HκΩ)-c22=λ2j+(HκΩ)-c22→λj(-ΔDΩ),asc→∞.

Eventually, to conclude Corollary 1.3 we note that (3.31) remains true if Ω is replaced by a ball B or the disjoint union of two balls B1∪B2. Thus, the claims follow immediately from the classical results for the Dirichlet Laplacian, which under the assumptions of Corollary 1.3 read as follows: (i) Faber–Krahn inequality: λ1(-ΔDB)≤λ1(-ΔDΩ) and equality holds if and only if Ω is a ball, see [29, 36] and also [31, Theorem 3.2.1 and Remark 3.2.2].

(ii) Hong–Krahn–Szegö inequality: λ2(-ΔDB1∪B2)≤λ2(-ΔDΩ) and equality holds if and only if Ω is the union of two identical disjoint balls, see [33, 37] and also [31, Theorem 4.1.1 and Remark 4.1.2].

(iii) Payne–Pólya–Weinberger inequality: If Ω is connected, then λ1(-ΔDB)λ2(-ΔDB)≤λ1(-ΔDΩ)λ2(-ΔDΩ)

and equality holds if and only if Ω is a ball, see [8, 45].

Remark 3.7

Finally, let us remark that from Theorem 1.2 and Corollary 1.3 one gets information about the positive part of the spectrum of HκΩ. Similar statements are also true for the negative part of the spectrum of HκΩ. Indeed, consider the self-adjoint unitary matrix U=(0-iI2iI20). Then, as Uαj+αjU=0, j∈{1,2,3}, and Uβ+βU=0, it is not difficult to see that HκΩ=-UH-κΩU, i.e. HκΩ and -H-κΩ are unitarily equivalent. Hence, it follows from Theorem 1.2 that for z∈C\(-∞,0] alsoHκΩ-z-c22-1=-UH-κΩU-z-c22-1=-UH-κΩ--z+c22-1U→-U(-ΔDΩ+z)-1I2000U=(-(-ΔDΩ)-z)-1000I2

in the operator norm, as c→∞. This convergence is of interest by its own, but similarly as in the proof of Corollary 1.3 one can conclude spectral inequalities for the negative eigenvalues λj-(HκΩ) of HκΩ; alternatively one can argue via the unitary equivalence HκΩ=-UH-κΩU. More precisely, for a bounded C2-domain Ω⊂R3, a ball B⊂R3 with |B|=|Ω|, and two identical and disjoint balls B1,B2⊂R3 with |B1|+|B2|=|Ω| the following assertions follow for sufficiently large c>0: (i) λj-(HκB)≥λj-(HκΩ) for j∈{1,2} and equality holds if and only if Ω is a ball.

(ii) λj-(HκB1∪B2)≥λj-(HκΩ) for j∈{3,4} and equality holds if and only if Ω is the union of two identical disjoint balls.

(iii) If, in addition, Ω is connected, then λj-(HκB)λl-(HκB)≤λj-(HκΩ)λl-(HκΩ),j∈{1,2},l∈{3,4},

and equality holds if and only if Ω is a ball.

Acknowledgements

This research was funded in part by the Austrian Science Fund  (FWF) 10.55776/P33568-N. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. This publication is based upon work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), https://www.cost.eu. The authors thank the referee for helpful comments and remarks that led to an improvement of the manuscript.

Funding

Open access funding provided by Graz University of Technology.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors have no conflict of interest to declare that are relevant to the content of this article.

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References

1. Alpay D Behrndt J Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators J. Funct. Anal. 2009 257 1666 1694
Alpay, D., Behrndt, J.: Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators. J. Funct. Anal. 257, 1666–1694 (2009)
2. Antunes P Benguria R Lotoreichik V Ourmières-Bonafos T A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities Commun. Math. Phys. 2021 386 2 781 818
Antunes, P., Benguria, R., Lotoreichik, V., Ourmières-Bonafos, T.: A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities. Commun. Math. Phys. 386(2), 781–818 (2021)
3. Arrizabalaga N Le Treust L Raymond N On the MIT bag model in the non-relativistic limit Commun. Math. Phys. 2017 354 2 641 669
Arrizabalaga, N., Le Treust, L., Raymond, N.: On the MIT bag model in the non-relativistic limit. Commun. Math. Phys. 354(2), 641–669 (2017)
4. Arrizabalaga N Le Treust L Raymond N Extension operator for the MIT bag model Ann. Fac. Sci. Toulouse Math. 2020 29 1 135 147
Arrizabalaga, N., Le Treust, L., Raymond, N.: Extension operator for the MIT bag model. Ann. Fac. Sci. Toulouse Math. 29(1), 135–147 (2020)
5. Arrizabalaga N Le Treust L Mas A Raymond N The MIT bag model as an infinite mass limit J. Éc. Polytech. Math. 2019 6 329 365
Arrizabalaga, N., Le Treust, L., Mas, A., Raymond, N.: The MIT bag model as an infinite mass limit. J. Éc. Polytech. Math. 6, 329–365 (2019)
6. Arrizabalaga N Mas A Vega L Shell interactions for Dirac operators: on the point spectrum and the confinement SIAM J. Math. Anal. 2015 47 2 1044 1069
Arrizabalaga, N., Mas, A., Vega, L.: Shell interactions for Dirac operators: on the point spectrum and the confinement. SIAM J. Math. Anal. 47(2), 1044–1069 (2015)
7. Arrizabalaga N Mas A Sanz-Perela T Vega L Eigenvalue curves for generalized MIT bag models Commum. Math. Phys. 2023 397 1 337 392
Arrizabalaga, N., Mas, A., Sanz-Perela, T., Vega, L.: Eigenvalue curves for generalized MIT bag models. Commum. Math. Phys. 397(1), 337–392 (2023)
8. Ashbaugh MS Benguria RD A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions Ann. Math. 1992 135 601 628
Ashbaugh, M.S., Benguria, R.D.: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. Math. 135, 601–628 (1992)
9. Behrndt J On compressed resolvents of Schrödinger operators with complex potentials Complex Anal. Oper. Theory 2021 15 12 (9 pages)
Behrndt, J.: On compressed resolvents of Schrödinger operators with complex potentials. Complex Anal. Oper. Theory 15, 12 (9 pages) (2021)
10. Behrndt J Exner P Holzmann M Lotoreichik V On the spectral properties of Dirac operators with electrostatic δ-shell interactions J. Math. Pures Appl. 2018 111 47 78
Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: On the spectral properties of Dirac operators with electrostatic -shell interactions. J. Math. Pures Appl. 111, 47–78 (2018)
11. Behrndt J Exner P Holzmann M Lotoreichik V On Dirac operators in R3 with electrostatic and Lorentz scalar δ-shell interactions Quantum Stud. 2019 6 295 314
Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: On Dirac operators in with electrostatic and Lorentz scalar -shell interactions. Quantum Stud. 6, 295–314 (2019)
12. Behrndt J Exner P Holzmann M Lotoreichik V The Landau Hamiltonian with δ-potentials supported on curves Rev. Math. Phys. 2020 32 2050010 (51 pages)
Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: The Landau Hamiltonian with -potentials supported on curves. Rev. Math. Phys. 32, 2050010 (51 pages) (2020)
13. Behrndt, J., Exner, P., Holzmann, M., Tušek, M.: On two-dimensional Dirac operators with -shell interactions supported on unbounded curves with straight ends. To appear in Singularities, Asymptotics, and Limiting Models, Springer INdAM Series (2024)
14. Behrndt, J., Hassi, S., de Snoo, H.: Boundary Value Problems, Weyl Functions, and Differential Operators, Volume 108 of Monographs in Mathematics. Birkhäuser/Springer, Cham (2020)
15. Behrndt J Holzmann M On Dirac operators with electrostatic δ-shell interactions of critical strength J. Spectr. Theory 2020 10 147 184
Behrndt, J., Holzmann, M.: On Dirac operators with electrostatic -shell interactions of critical strength. J. Spectr. Theory 10, 147–184 (2020)
16. Behrndt J Holzmann M Mas A Self-adjoint Dirac operators on domains in R3 Ann. Henri Poincaré 2020 21 2681 2735 32765187
Behrndt, J., Holzmann, M., Mas, A.: Self-adjoint Dirac operators on domains in . Ann. Henri Poincaré 21, 2681–2735 (2020)32765187
17. Behrndt J Holzmann M Stelzer C Stenzel G Boundary triples and Weyl functions for Dirac operators with singular interactions Rev. Math. Phys. 2024 36 2 2350036 (65 pages)
Behrndt, J., Holzmann, M., Stelzer, C., Stenzel, G.: Boundary triples and Weyl functions for Dirac operators with singular interactions. Rev. Math. Phys. 36(2), 2350036 (65 pages) (2024)
18. Behrndt J Holzmann M Stenzel G Schrödinger operators with oblique transmission conditions in R2 Commun. Math. Phys. 2023 401 3149 3167 37476817
Behrndt, J., Holzmann, M., Stenzel, G.: Schrödinger operators with oblique transmission conditions in . Commun. Math. Phys. 401, 3149–3167 (2023)37476817
19. Benguria RD Fournais S Stockmeyer E Van Den Bosch H Self-adjointness of two-dimensional Dirac operators on domains Ann. Henri Poincaré 2017 18 4 1371 1383
Benguria, R.D., Fournais, S., Stockmeyer, E., Van Den Bosch, H.: Self-adjointness of two-dimensional Dirac operators on domains. Ann. Henri Poincaré 18(4), 1371–1383 (2017)
20. Benguria RD Fournais S Stockmeyer E Van Den Bosch H Spectral gaps of Dirac operators describing graphene quantum dots Math. Phys. Anal. Geom. 2017 20 2 11 (12 pages)
Benguria, R.D., Fournais, S., Stockmeyer, E., Van Den Bosch, H.: Spectral gaps of Dirac operators describing graphene quantum dots. Math. Phys. Anal. Geom. 20(2), 11 (12 pages) (2017)
21. Benhellal B Spectral analysis of Dirac operators with delta interactions supported on the boundaries of rough domains J. Math. Phys. 2022 63 1 011507 (34 pages)
Benhellal, B.: Spectral analysis of Dirac operators with delta interactions supported on the boundaries of rough domains. J. Math. Phys. 63(1), 011507 (34 pages) (2022)
22. Bogolioubov PN Sur un modèle à quarks quasi-indépendants Ann. Inst. Henri Poincaré. Sect. A 1968 8 2 163 189
Bogolioubov, P.N.: Sur un modèle à quarks quasi-indépendants. Ann. Inst. Henri Poincaré. Sect. A 8(2), 163–189 (1968)
23. Briet P Krejčiřík D Spectral optimization of Dirac rectangles J. Math. Phys. 2022 63 1 013502 (11 pages)
Briet, P., Krejčiřík, D.: Spectral optimization of Dirac rectangles. J. Math. Phys. 63(1), 013502 (11 pages) (2022)
24. Budyika V Malamud M Posilicano A Nonrelativistic limit for 2p×2p-Dirac operators with point interactions on a discrete set Russ. J. Math. Phys. 2017 24 4 426 435
Budyika, V., Malamud, M., Posilicano, A.: Nonrelativistic limit for -Dirac operators with point interactions on a discrete set. Russ. J. Math. Phys. 24(4), 426–435 (2017)
25. Carlone R Malamud M Posilicano A On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set J. Differ. Equ. 2013 254 9 3835 3902
Carlone, R., Malamud, M., Posilicano, A.: On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set. J. Differ. Equ. 254(9), 3835–3902 (2013)
26. Chodos A Jaffe RL Johnson K Thorn CB Weisskopf VF New extended model of hadrons Phys. Rev. D 1974 9 12 3471 3495
Chodos, A., Jaffe, R.L., Johnson, K., Thorn, C.B., Weisskopf, V.F.: New extended model of hadrons. Phys. Rev. D 9(12), 3471–3495 (1974)
27. Cuenin J-C Estimates on complex eigenvalues for Dirac operators on the half-line Integr. Equ. Oper. Theory 2014 79 3 377 388
Cuenin, J.-C.: Estimates on complex eigenvalues for Dirac operators on the half-line. Integr. Equ. Oper. Theory 79(3), 377–388 (2014)
28. DeGrand T Jaffe RL Johnson K Kiskis J Masses and other parameters of the light hadrons Phys. Rev. D 1975 12 7 2060 2076
DeGrand, T., Jaffe, R.L., Johnson, K., Kiskis, J.: Masses and other parameters of the light hadrons. Phys. Rev. D 12(7), 2060–2076 (1975)
29. Faber, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitzungsber. Bayer. Akad. Wiss. München Math.-Phys. Kl. 169–172 (1923)
30. Gesztesy F Šeba P New analytically solvable models of relativistic point interactions Lett. Math. Phys. 1987 13 4 345 358
Gesztesy, F., Šeba, P.: New analytically solvable models of relativistic point interactions. Lett. Math. Phys. 13(4), 345–358 (1987)
31. Henrot A Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics 2006 Basel Birkhäuser Verlag
Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006)
32. Heriban L Tušek M Non-self-adjoint relativistic point interaction in one dimension J. Math. Anal. Appl. 2022 516 2 126536 (28 pages)
Heriban, L., Tušek, M.: Non-self-adjoint relativistic point interaction in one dimension. J. Math. Anal. Appl. 516(2), 126536 (28 pages) (2022)
33. Hong I On an inequality concerning the eigenvalue problem of membrane Kodai Math. Sem. Rep. 1954 6 113 114
Hong, I.: On an inequality concerning the eigenvalue problem of membrane. Kodai Math. Sem. Rep. 6, 113–114 (1954)
34. Johnson K The MIT bag model Acta Physica Pol. B 1975 6 865 892
Johnson, K.: The MIT bag model. Acta Physica Pol. B 6, 865–892 (1975)
35. Kato T Perturbation Theory for Linear Operators. Classics in Mathematics 1995 Berlin Springer-Verlag
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer-Verlag, Berlin (1995). (Reprint of the 1980 edition)
36. Krahn E Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises Math. Ann. 1925 94 97 100
Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)
37. Krahn E Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen Acta Univ. Dorpat. A 1926 9 1 44
Krahn, E.: Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta Univ. Dorpat. A 9, 1–44 (1926)
38. Levitin M Mangoubi D Polterovich I Topics in Spectral Geometry. Graduate Studies in Mathematics 2023 Providence American Mathematical Society
Levitin, M., Mangoubi, D., Polterovich, I.: Topics in Spectral Geometry. Graduate Studies in Mathematics, vol. 237. American Mathematical Society, Providence (2023)
39. Lotoreichik V Ourmières-Bonafos T A sharp upper bound on the spectral gap for graphene quantum dots Math. Phys. Anal. Geom. 2019 22 13 (30 pages)
Lotoreichik, V., Ourmières-Bonafos, T.: A sharp upper bound on the spectral gap for graphene quantum dots. Math. Phys. Anal. Geom. 22, 13 (30 pages) (2019)
40. Lotoreichik, V., Ourmières-Bonafos, T.: Spectral asymptotics of the Dirac operator in a thin shell. arXiv:2307.09033
41. McLean W Strongly Elliptic Systems and Boundary Integral Equations 2000 Cambridge Cambridge University Press
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
42. Moroianu A Ourmières-Bonafos T Pankrashkin K Dirac operators on hypersurfaces as large mass limits Commun. Math. Phys. 2020 374 3 1963 2013
Moroianu, A., Ourmières-Bonafos, T., Pankrashkin, K.: Dirac operators on hypersurfaces as large mass limits. Commun. Math. Phys. 374(3), 1963–2013 (2020)
43. Nédélec JC Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems 2001 New York Springer-Verlag
Nédélec, J.C.: Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Springer-Verlag, New York (2001)
44. Ourmières-Bonafos T Vega L A strategy for self-adjointness of Dirac operators: application to the MIT bag model and δ-shell interactions Publ. Mat. 2018 62 397 437
Ourmières-Bonafos, T., Vega, L.: A strategy for self-adjointness of Dirac operators: application to the MIT bag model and -shell interactions. Publ. Mat. 62, 397–437 (2018)
45. Pólya G On the characteristic frequencies of a symmetric membrane Math. Z. 1955 63 331 337
Pólya, G.: On the characteristic frequencies of a symmetric membrane. Math. Z. 63, 331–337 (1955)
46. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, vol 27. Princeton University Press, Princeton (1951)
47. Rabinovich VS Boundary problems for three-dimensional Dirac operators and generalized MIT bag models for unbounded domains Russ. J. Math. Phys. 2020 27 4 500 516
Rabinovich, V.S.: Boundary problems for three-dimensional Dirac operators and generalized MIT bag models for unbounded domains. Russ. J. Math. Phys. 27(4), 500–516 (2020)
48. Rabinovich VS Boundary value problems for 3D-Dirac operators and MIT bag model Springer Proc. Math. Stat. 2021 357 479 495
Rabinovich, V.S.: Boundary value problems for 3D-Dirac operators and MIT bag model. Springer Proc. Math. Stat. 357, 479–495 (2021)
49. Reed M Simon B Methods of Modern Mathematical Physics I. Functional Analysis 1972 Cambridge Academic Press
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, Cambridge (1972)
50. Thaller B The Dirac Equation. Texts and Monographs in Physics 1992 Berlin Springer-Verlag
Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Springer-Verlag, Berlin (1992)
51. Vu, T.: Spectral inequality for Dirac right triangles. J. Math. Phys. 64(4), 041502 (18 pages) (2023)
52. Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil I. Grundlagen. Mathematische Leitfäden. B. G. Teubner, Stuttgart (2000)
