Spectral Transition for Dirac Operators with Electrostatic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}δ-Shell Potentials Supported on the Straight Line

In this note the two dimensional Dirac operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\eta $$\end{document}Aη with an electrostatic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}δ-shell interaction of strength \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in {\mathbb {R}}$$\end{document}η∈R supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\pm 2$$\end{document}η=±2 the continuous spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\eta $$\end{document}Aη inside the spectral gap of the free Dirac operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0$$\end{document}A0 collapses abruptly to a single point.


Introduction
Differential operators that admit a spectral transition are of particular interest in mathematical analysis and its applications. Typically, one expects that the properties of a model described by a differential operator depend continuously on the parameters. However, in some cases it turns out that there is an abrupt change in the spectral properties-in other words, a spectral transition. A well known example in this regard is the Smilansky model [29,30] (see also [4][5][6]) or the indefinite Laplacian studied in [13,20]. A spectral transition was also observed for Dirac operators with singular potentials supported on bounded curves in R 2 and surfaces in R 3 . More precisely, when studying perturbations A η of the free Dirac operator by an electrostatic δ-shell potential of strength η ∈ R, it turned out that there is an abrupt change in the spectral properties for η = ±2. While for η = ±2-which is referred to as the non-critical case-it is shown in the three dimensional situation in [2,3,7,8] that the essential spectrum consists of two unbounded rays and finitely many eigenvalues between these rays, the critical case η = ±2 remained initially open. In the critical case it was then proved in [10,18,27] that there is a loss of smoothness in the operator domain and that there may be one additional point in the essential spectrum; for general combinations of interaction strengths in the two dimensional setting see [12]. We note that similar effects also appear in the study of Dirac operators on bounded domains with suitable boundary conditions; cf. [11,17,18,21,26].
In this note we study the spectrum of a Dirac operator in R 2 with an electrostatic δ-shell potential of strength η ∈ R supported on the straight line Σ ∼ = R which is formally given by here and in the following σ 1 , σ 2 , σ 3 are the C 2×2 -valued Pauli spin matrices defined in (1.3), σ 0 is the 2 × 2-identity matrix, and m ∈ R \ {0}. In order to define this expression rigorously, we denote by R 2 + and R 2 − the upper and lower half plane, respectively, and we use the notation f ± := f R 2 ± for the restriction of a function f defined on R 2 . Next, let : . This operator is the rigorous mathematical definition of a Dirac operator with an electrostatic δ-shell interaction of strength η and models, for m > 0, the propagation of a particle with mass m and spin 1/2 under the influence of such a potential; cf. [2,8,12,21]. Observe that for η = 0 the operator in (1.1) coincides with the free Dirac operator and recall that A 0 is self-adjoint in L 2 (R 2 ; C 2 ) with purely (absolutely) continuous spectrum The main result of this paper is the following theorem on the self-adjointness and the spectra of the operators A η . It turns out that the continuous spectrum grows under the influence of the δ-shell interaction and also half of the gap (−|m|, |m|) is filled when η approaches the critical values ±2 from above or below. In the critical case η = ±2 a spectral transition appears: the continuous spectrum inside (−|m|, |m|) vanishes abruptly and 0 becomes an infinite dimensional eigenvalue.
, +∞ . For η = ±2 the spectrum of A η is purely continuous and for η = ±2 the point 0 is an isolated eigenvalue of A η with infinite multiplicity and the remaining spectrum is purely continuous.
Note that the spectrum of A η is invariant under the transformation η → − 4 η . This symmetry would also follow from the stronger fact that A η and A −4/η are unitary equivalent; this can be shown in the same way as in [12,Proposition 4.8 (i) and Proposition 4.15 (i)].
The proof of Theorem 1.1 is based on an efficient abstract technique that was applied in a similar form also in [12,21]: We use a so-called boundary triple and its Weyl function to reduce the spectral analysis of A η in the gap (−|m|, |m|) of A 0 to a certain boundary operator in L 2 (Σ; C 2 ) L 2 (R; C 2 ); cf. [9,19,22,23] for details on boundary triples and Weyl functions in the extension theory of symmetric operators. Since Σ is a straight line the spectral properties of this boundary operator can be studied with the help of the Fourier transform. From the limit behaviour of the Weyl function and the boundary operator towards the real line we also conclude that the set (−∞, −|m|] ∪ [|m|, +∞) consists of purely continuous spectrum of A η for all η ∈ R. For m = 0 this method is also applicable, then it follows for all η ∈ R that σ(A η ) = R, i.e. there is no spectral transition. We note that the method of direct integrals decomposing A η in its fibers would yield a similar result; cf. [24] for a related problem on Dirac operators with Robin type boundary conditions. In this note we prefer to work with the boundary triple technique, since it allows generalizations for more complicated curves Σ in a natural way, which we plan to study in the near future.

Notations Let
be the Pauli spin matrices and denote by σ 0 the 2 × 2-identity matrix. Note that the Pauli matrices satisfy

Proof of Theorem 1.1
Let A 0 be the free Dirac operator in (1.2) and recall that for z ∈ ρ(A 0 ) the resolvent of A 0 is given by and K j is the modified Bessel functions of the second kind and order j; cf.
[1] and [31]. Here and in the following the complex square root is chosen such that it is holomorphic in C \ (−∞, 0] and Re √ z > 0. An important object in our analysis is the mapping C z which is defined for z ∈ ρ(A 0 ) by where S(Σ; C 2 ) S(R; C 2 ) denotes the Schwartz space and the convolution is understood in the sense of distributions. The action of C z is In the following we will denote by F the Fourier transform on Σ R.
In particular, C z gives rise to a bounded operator in H s (Σ; C 2 ) for any s ∈ R.
Since the Fourier transform takes K 0 (κ|x|) to π/2(p 2 + κ 2 ) −1/2 we obtain and hence the calculation rules for the Fourier transform of distributions [28, Chapter IX] lead to for ϕ ∈ S(Σ; R 2 ), which yields the claimed result about the representation of FC z F −1 . Finally, taking the definition of the norm in H s (Σ; C 2 ) H s (R; C 2 ) with the help of the Fourier transform into account, one sees that C z gives rise to a bounded operator in H s (Σ; C 2 ) for all s ∈ R. In the following we shall make use of the closed symmetric restriction of the free Dirac operator A 0 , and its adjoint ; the above mentioned properties of S and S * can be shown in the same way as in [10, Proposition 3.1], where similar operators in R 3 with compact surfaces Σ have been studied.

Lemma 2.2. For η ∈ R \ {0} the operator
is self-adjoint in L 2 (Σ; C 2 ) and we have In particular, the operator A η is self-adjoint in L 2 (R 2 ; C 2 ).
Proof. Define the C 2×2 -valued function Then, by Lemma 2.1 we have Since θ is a symmetric matrix and F is unitary, we conclude that Θ is selfadjoint in L 2 (Σ; C 2 ) L 2 (R; C 2 ). Using (2.2) it is not difficult to verify that (2.3) holds and hence the self-adjointness of A η follows, see, e.g., [9,Corollary 2.1.4 (v)].
In the next lemma we analyze, when zero belongs to the point spectrum or continuous spectrum of Θ − M (z) for z ∈ (−|m|, |m|).
For z ∈ (−|m|, |m|) the following holds: Proof. Let z ∈ (−|m|, |m|) and observe that Using Lemma 2.1 we conclude and hence it suffices to consider the self-adjoint multiplication operator with the function θ z in L 2 (R; C 2 ). In the following we discuss for which η ∈ R \{0} and z ∈ (−|m|, |m|) the multiplication operator θ z has 0 as an eigenvalue or as a point in the continuous spectrum. Note first that det θ z (p) = (p 2 + 1) 1 We verify assertion (ii). In the case η = ±2 we have det θ z (p) = 0 if and only if z = 0, and in this situation det θ z (p) = 0 for all p ∈ R. Therefore, 0 is an eigenvalue of infinite multiplicity of the multiplication operator θ z . Observe that for z ∈ (−|m|, |m|) \ {0} we have det θ z (p) = 0 and hence one finds that 0 is in the resolvent set of θ z . Now we prove (i). If η = ±2, then det θ z (p) = 0 is equivalent to When p ∈ R varies the left hand side can be any non-negative number, and hence (2.6) has a solution, whenever the right hand side is non-negative, i.e. if and only if z ≤ z − or z ≥ z + , where z ± are the zeros of the polynomial on the right hand side of the last equation given by It is also clear that for a fixed z ∈ (−|m|, |m|) with z ≤ z − or z ≥ z + there are only two values p ∈ R with (2.7) holds, and hence 0 is in the continuous spectrum of the multiplication operator associated with θ z .
In order to prove Theorem 1.1, we also investigate the limiting behaviour of (Θ−M (z)) −1 , when z ∈ C\R approaches σ( Proof. In order to show the claims, we note first with the help of (2.5) that the operator F(Θ − M (z)) −1 F −1 , z ∈ C \ R, is the maximal multiplication operator associated with the matrix-valued function By the continuity of the complex square root one sees for a fixed p ∈ R that the limit c x+i0 (p) := lim y 0 c x+iy (p) exists. One verifies for any number x ∈ (−∞, −|m|]∪[|m|, +∞) in a similar way as in (2.6) and (2.7) that c x+i0 (p) has no zero, if η = ±2 or if xη η 2 −4 < 0, and for xη η 2 −4 > 0 the term c x+i0 (p) has two zeros at (2.10) Let us prove (i). Let x ∈ (−∞, −|m|] ∪ [|m|, +∞) be fixed. Since the map z → M (z) is the Weyl function of a boundary triple it is a Nevanlinna function and the values M (z) are bounded and everywhere defined operators in L 2 (Σ; C 2 ); cf. [9,Corollary 2.3.7]. It follows that z → (Θ − M (z)) −1 is also a Nevanlinna function and for z ∈ ρ(A η ) ∩ ρ(A 0 ) the values are also bounded and everywhere defined operators in L 2 (Σ; C 2 ). From the operator representation of Nevanlinna functions (see, e.g., [25,Theorem 4.2]) we then conclude that there exists a constant C 1 > 0 such that for y > 0 sufficiently small. Since (Θ − M (x + iy)) −1 is unitarily equivalent to the multiplication operator with the function θ −1 x+iy , we conclude that for all y > 0 sufficiently small. Next,we define for y > 0 the set

R,
if c x+i0 has no zero, (2.12) and prove for all sufficiently small y > 0 and p ∈ I x (y) that for some C 2 > 0, where the absolute value is understood elementwise. In order to show (2.13), we note first that for some C 3 > 0 independent of p ∈ R and y ∈ [0, 1] one has the estimate c x+iy (p)θ −1 x+iy (p) ≤ C 3 . (2.14) Consider first the case when c x+i0 has no zero. From the definition of c x+iy we conclude that there exists P 0 > 0 such that c x+iy (p) > 1 holds for all |p| > P 0 and all sufficiently small y > 0. Moreover, the map [−P 0 , P 0 ] × [0, 1] (p, y) → c x+iy (p) is uniformly continuous and hence, there exists a constant C 4 > 0 such that for all sufficiently small y > 0 and p ∈ [−P 0 , P 0 ] the relation |c x+iy (p)| > C 4 holds, as c x+i0 has no zero. Thus, for y > 0 sufficiently small |c x+iy | is uniformly bounded from below by a positive constant, which together with (2.14) leads to the estimate in (2.13).
Assume now that c x+i0 has the zeros p ± , let y > 0 be sufficiently small, and fix p ∈ R with |p − p ± | ≥ √ y. From (2.10) we get p 2 ± + m 2 − x 2 > 0 and thus,