Spectral Shift Function for SchrÃ¶dinger Operators in Constant ...

Lemma 2.3. [38, Theorem 2.1, Proposition 4.1] Let 0 ≤ U ∈ L ∞ (R 2 ). Assume thatln U(X ⊥ ) = −µ|X ⊥ | 2β (1 + o(1)), |X ⊥ | → ∞,**for** some β ∈ (0, ∞), µ ∈ (0, ∞). Then **for** each q ∈ Z + we havewhere⎧⎪⎨ϕ β (s) :=⎪⎩n + (s; p q Up q ) = ϕ β (s)(1 + o(1)), s ↓ 0,b| ln s| 1/β if 0 < β < 1,2µ 1/β 1ln (1+2µ/b)| ln s| if β = 1,ββ−1 (ln | ln s|)−1 | ln s| if 1 < β < ∞.s ∈ (0, e −1 ). (2.12)Lemma 2.4. [38, Theorem 2.2, Proposition 4.1] Let 0 ≤ U ∈ L ∞ (R 2 ). Assume that the support of Uis compact, and that there exists a constant C > 0 such that U ≥ C on an open non-empty subset ofR 2 . Then **for** each q ∈ Z + we havewheren + (s; p q Up q ) = ϕ ∞ (s) (1 + o(1)), s ↓ 0,ϕ ∞ (s) := (ln | ln s|) −1 | ln s|, s ∈ (0, e −1 ). (2.13)Employ**in**g Lemmas 2.2, 2.3, 2.4, we easily f**in**d that asymptotic estimates (2.8) and (2.10) entail thefollow**in**gCorollary 2.1. [10, Corollaries 3.1 – 3.2] Let (1.2) hold with m 0 > 3.i) Assume that the hypotheses of Lemma 2.2 hold with U = W and α > 2. Then we haveξ(2bq − λ; H − , H 0 ) = − b2π{∣ X ⊥ ∈ R 2 |W (X ⊥ ) > 2 √ ∣∣λ}∣(1 + o(1)) =−ψ α (2 √ λ) (1 + o(1)), λ ↓ 0, (2.14)ξ(2bq + λ; H ± , H 0 ) = ± b ∫2π 2 arctan ((2 √ λ) −1 W (X ⊥ ))dX ⊥ (1 + o(1)) =R 2±12 cos (π/α) ψ α(2 √ λ) (1 + o(1)), λ ↓ 0,the function ψ α be**in**g def**in**ed **in** (2.11).ii) Assume that the hypotheses of Lemma 2.3 hold with U = W . Then we haveξ(2bq − λ; H − , H 0 ) = −ϕ β (2 √ λ) (1 + o(1)), λ ↓ 0, β ∈ (0, ∞),the functions ϕ β be**in**g def**in**ed **in** (2.12). If, **in** addition, V satisfies (1.1) **for** some m ⊥ > 2 and m 3 > 2,we haveξ(2bq + λ; H ± , H 0 ) = ± 1 2 ϕ β(2 √ λ) (1 + o(1)), λ ↓ 0, β ∈ (0, ∞).iii) Assume that the hypotheses of Lemma 2.4 hold with U = W . Then we haveξ(2bq − λ; H − , H 0 ) = −ϕ ∞ (2 √ λ) (1 + o(1)), λ ↓ 0,the function ϕ ∞ be**in**g def**in**ed **in** (2.13). If, **in** addition, V satisfies (1.1) **for** some m ⊥ > 2 and m 3 > 2,we haveξ(2bq + λ; H ± , H 0 ) = ± 1 2 ϕ ∞(2 √ λ) (1 + o(1)), λ ↓ 0,the function ϕ ∞ be**in**g def**in**ed **in** (2.13).5

In particular, we f**in**d thatξ(2bq + λ; H − , H 0 )limλ↓0 ξ(2bq − λ; H − , H 0 ) = 12 cos π αif W has a power-like decay at **in**f**in**ity (i.e. if the assumptions of Corollary 2.1 i) hold), orξ(2bq + λ; H − , H 0 )limλ↓0 ξ(2bq − λ; H − , H 0 ) = 1 2(2.15)(2.16)if W decays exponentially or has a compact support (i.e. if the assumptions of Corollary 2.1 ii) - iii)are fulfilled). Relations (2.15) and (2.16) could be **in**terpreted as analogues of the classical Lev**in**son**for**mulae (see e.g. [40]).Remarks: i) S**in**ce the ranks of p q W p q and p q W λ p q are **in**f**in**ite, the quantities n + (s2 √ λ; p q W p q ) andTr arctan ((s2 √ λ) −1 p q W λ p q ) tend to **in**f**in**ity as λ ↓ 0 **for** every s > 0. There**for**e, Theorems 2.1 and 2.2imply that the SSF ξ(·; H ± , H 0 ) has a s**in**gularity at each Landau level. The existence of s**in**gularitiesof the SSF at strictly positive energies is **in** sharp contrast with the non-magnetic case b = 0 where theSSF ξ(E; −∆ + V, −∆) is cont**in**uous **for** E > 0 (see e.g. [40]). The ma**in** reason **for** this phenomenonis the fact that the Landau levels play the role of thresholds **in** σ(H 0 ) while the free Laplacian −∆ hasno strictly positive thresholds **in** its spectrum.It is conjectured that the s**in**gularity of the SSF ξ(·; H ± (b), H 0 (b)), b > 0, at a given Landau level2bq, q ∈ Z + , could be related to a possible accumulation of resonances and/or eigenvalues of H at2bq. Here it should be recalled that **in** the case b = 0 the high energy asymptotics (see [27]) and thesemi-classical asymptotics (see [28]) of the derivative of the SSF **for** appropriate compactly supportedperturbations of the Laplacian, are related by the Breit-Wigner **for**mula to the asymptotic distributionnear the real axis of the resonances def**in**ed as poles of the meromorphic cont**in**uation of the resolventof the perturbed operator.ii) In the case q = 0, when by (1.3) we have ξ(−λ; H − , H 0 ) = −N(−λ; H − ) **for** λ > 0, asymptoticrelations of the type of (2.14) have been known s**in**ce long ago (see [43], [42], [44], [31], [17]). Animportant characteristic feature of the methods used **in** [31], and later **in** [38], is the systematic use,explicit or implicit, of the connection between the spectral theory of the Schröd**in**ger operator withconstant magnetic field, and the theory of Toeplitz operators act**in**g **in** holomorphic spaces of Fock-Segal-Bargmann type, and the related pseudodifferential operators with generalised anti-Wick symbols(see [12], [3], [41], [15]). Various important aspects of the **in**teraction between these two theories havebeen discussed **in** [37] and [7, Section 9]). The Toeplitz-operator approach turned out to be especiallyfruitful **in** [38] where electric potentials decay**in**g rapidly at **in**f**in**ity (i.e. decay**in**g exponentially, orhav**in**g compact support) were considered (see Lemmas 2.3 - 2.4). It is shown **in** [11] that the precisespectral asymptotics **for** the Landau Hamiltonian perturbed by a compactly supported electric potentialU of fixed sign recovers the logarithmic capacity of the support of U.iii) Let us mention several other exist**in**g extensions of Lemmas 2.2 – 2.4. Lemmas 2.2 and 2.4 have beengeneralised to the multidimensional case where p q is the orthogonal projection onto a given eigenspaceof the Schröd**in**ger operator with constant magnetic field of full rank, act**in**g **in** L 2 (R 2d ), d > 1 (see [31]and [25] respectively). Moreover, Lemma 2.4 has been generalised **in** [25] to a relativistic sett**in**g wherep q is an eigenprojection of the Dirac operator. F**in**ally, **in** [36] Lemmas 2.2 – 2.4 have been extendedto the case of the 2D Pauli operator with variable magnetic field from a certa**in** class **in**clud**in**g thealmost periodic fields with non-zero mean value (**in** this case the role of the Landau levels is played bythe orig**in**), and electric potentials U satisfy**in**g the assumptions of Lemmas 2.2 – 2.4. In the case ofcompactly supported U of def**in**ite sign, [11] conta**in**s a more precise version of the correspond**in**g resultof [36], **in**volv**in**g aga**in** the logarithmic capacity of the support of U.iv) To the author’s best knowledge, the s**in**gularities at the Landau levels of the SSF **for** the 3DSchröd**in**ger operator **in** constant magnetic field has been **in**vestigated **for** the first time **in** [10]. However,it is appropriate to mention here the article [19] where an axisymmetric potential V = V (|X ⊥ |, x 3 ) has6

Remark: The third part of Proposition 2.1 was not **in**cluded **in** [7, Proposition 2.2]. However, it followseasily from the representation of the SSF described **in** Subsection 3.1 below, and the hypotheses V ≢ 0and V ∈ C(R 3 ).Theorem 2.4. (cf. [7, Theorem 2.3]) Assume that (1.1) holds. Let q ∈ Z + , λ ∈ R \ {0}. If λ < 0,suppose also that (2.20) holds. Then we havelimb→∞ b−1 ξ(2bq + λ; H(b), H 0 (b)) = 1 ∫ξ(λ; χ(X ⊥ ), χ 0 ) dX ⊥ . (2.21)2π R 2The proofs of Theorems 2.3 and 2.4 are conta**in**ed **in** Subsection 4.2. We present these proofs underthe additional assumption that V has a def**in**ite sign, and refer the reader to the orig**in**al paper [7] **for**the proofs **in** the general case.By Proposition 2.1 iii), if ±V ≥ 0, then the r.h.s. of (2.21) is different from zero if λ > Λ, λ ≠ 0.Un**for**tunately, we cannot prove that the same is true **for** general non-sign-def**in**ite electric potentialsV . On the other hand, it is obvious that **for** arbitrary V we have ∫ R 2 ξ(λ; χ(X ⊥ ), χ 0 )dX ⊥ = 0 if λ < Λ.The last theorem of this subsection conta**in**s a more precise version of (2.21) **for** the case λ < Λ.Theorem 2.5. (cf. [7, Theorem 2.4]) Let (1.1) hold.i) Let λ < Λ. Then **for** sufficiently large b > 0 we have ξ(λ; H(b), H 0 (b)) = 0.ii) Let q ∈ Z + , q ≥ 1, λ < Λ. Assume that the partial derivatives of 〈x 3 〉 m3 V with respect to thevariables X ⊥ ∈ R 2 exist, and are uni**for**mly bounded on R 3 . Then we havelimb→∞ b−1/2 ξ(2bq + λ; H(b), H 0 (b)) = 14π 2 q−1∑(2(q − l))∫R −1/2 V (x)dx. (2.22)3The first part of the theorem is trivial, and follows immediately from (2.19). We omit the proof ofTheorem 2.5 ii) and refer the reader to the orig**in**al work [7].Remarks: i) Relations (2.18), (2.21), and (2.22) can be unified **in**to a s**in**gle asymptotic **for**mula. In orderto see this, notice that a general result on the high-energy asymptotics of the SSF **for** 1D Schröd**in**geroperators (see e.g. [40]) implies, **in** particular, thatlimE→∞ E1/2 ξ(E; χ(X ⊥ ), χ 0 ) = 12π∫Rl=0V (X ⊥ , x 3 ) dx 3 , X ⊥ ∈ R 2 .Then relation (2.18) with 0 < E/2 ∉ Z + , or relations (2.21) and (2.22) with E = 2q, q ∈ Z + , entailξ(Eb + λ; H(b), H 0 (b)) = b2π[E/2]∑l=0∫R 2 ξ(b(E − 2l) + λ; χ(X ⊥ ), χ 0 )dX ⊥ (1 + o(1)), b → ∞. (2.23)On its turn, (2.23) can be re-written as∫ ∫ξ(Eb + λ; H(b), H 0 (b)) = ξ(Eb + λ − s; χ(X ⊥ ), χ 0 )dX ⊥ dν b (s) (1 + o(1)), b → ∞,R R 2{where ν b (s) := b ∑ ∞ 0 if s ≤ 0,2π l=0 Θ(s − 2bl), s ∈ R, and Θ(s) := is the Heaviside function. It1 if s > 0,is well-known that ν is the **in**tegrated density of states **for** the 2D Landau Hamiltonian (see (2.6)).ii) By (1.3) **for** λ < 0 we have ξ(λ; H(b), H 0 ) = −N(λ; H(b)). The asymptotics as b → ∞ of thecount**in**g function N(λ; H 0 (b)) with λ < 0 fixed, has been **in**vestigated **in** [32] under considerably lessrestrictive assumptions on V than **in** Theorems 2.3 ∫ – 2.5. The asymptotic properties as λ ↑ 0, and asλ ↓ Λ if Λ < 0, of the asymptotic coefficient − 12πN(λ; χ(XR 2 ⊥ )dX ⊥ which appears at the r.h.s. of(2.21) **in** the case of a negative perturbation, have been studied **in** [33]. The asymptotic distribution8

of the discrete spectrum **for** the 3D magnetic Pauli and Dirac operators **in** strong magnetic fields hasbeen considered **in** [35] and [34] respectively. The ma**in** purpose **in** [32], [34], and [35] was to obta**in** thema**in** asymptotic term (without any rema**in**der estimates) of the correspond**in**g count**in**g function of thediscrete spectrum under assumptions close to the m**in**imal ones which guarantee that the Hamiltoniansare self-adjo**in**t, and the asymptotic coefficient is well-def**in**ed. Other results which aga**in** describethe asymptotic distribution of the discrete spectrum of the Schröd**in**ger and Dirac operator **in** strongmagnetic fields, but conta**in** also sharp rema**in**der estimates, have been obta**in**ed [17], [9], and [18] underassumptions on V which, naturally, are considerably more restrictive than those **in** [32], [34], and [35].iii) Generalisations of asymptotic relation (2.18) **in** several directions can be found **in** [26]. In particular,[26, Theorem 4] implies that if V ∈ S(R 3 ), then the SSF ξ(Eb+λ; H(b), H 0 (b)), E ∈ (0, ∞)\2Z + , λ ∈ R,admits an asymptotic expansion of the **for**mξ(Eb + λ; H(b), H 0 (b)) ∼∞∑j=0c j b 1−2j2 , b → ∞.iv) Together with the po**in**twise asymptotics as b → ∞ of the SSF **for** the pair (H 0 (b), H(b)) (see(2.18), (2.21), or (2.22)), it also is possible to consider its weak asymptotics, i.e. the asymptotics of theconvolution of the SSF with an arbitrary ϕ ∈ C ∞ 0 (R). Results of this type are conta**in**ed **in** [6].2.4 High energy asymptotics of the SSFTheorem 2.6. [8, Theorem 2.1] Assume that V satisfies (1.1). Then we havelim E −1/2 ξ(E; H, H 0 ) = 1 ∫E→∞,E∈O r 4π 2 V (x)dx, r ∈ (0, b), (2.24)R 3where O r := {E ∈ (0, ∞)|dist(E, 2bZ + )}.We omit the proof of Theorem 2.6 which is quite similar to that of Theorem 2.3, and refer the readerto the orig**in**al paper [8].Remarks: i) It is essential to avoid the Landau levels **in** (2.24), i.e. to suppose that E ∈ O r , r ∈ (0, b),as E → ∞, s**in**ce by Theorems 2.1 - 2.2, the SSF has s**in**gularities at the Landau levels, at least **in** thecase ±V ≥ 0.ii) For E ∈ R set∫(ξ cl (E) := Θ(E − |p + A(x)| 2 ) − Θ(E − |p + A(x)| 2 − V (x)) ) dxdp =T ∗ R 34π3()E∫R 3/2+ − (E − V (x)) 3/2+ dx3where Θ, as above, is the Heaviside function. Note that ξ cl (E) is **in**dependent of the magnetic fieldb ≥ 0. Evidently, under the assumptions of Theorem 2.6 we have lim E→∞ E −1/2 ξ cl (E) = 2π ∫ R 3 V (x)dx.Hence, if ∫ R 3 V (x)dx ≠ 0, then (2.24) is equivalent toξ(E; H, H 0 ) = (2π) −3 ξ cl (E)(1 + o(1)), E → ∞, E ∈ O r , r ∈ (0, b).iii) As far as the author is **in****for**med, the high-energy asymptotics of the SSF **for** 3D Schröd**in**geroperators **in** constant magnetic fields was **in**vestigated **for** the first time **in** [8]. Nonetheless, **in** [19]the asymptotic behaviour as E → ∞, E ∈ O r , of the SSF ξ(E; H (m) , H (m)0 ) **for** the operator pair(H (m) , H (m)0 ) (see (2.17)) with fixed m ∈ Z has been been **in**vestigated. It does not seem possible todeduce (2.24) from the results of [19] even **in** the case of axial symmetry of V .9

3 Auxiliary Results3.1 A. Pushnitski’s representation of the SSFIn the first part of this subsection we summarise several results due to A. Pushnitski on the representationof the SSF **for** a pair of lower-bounded self-adjo**in**t ∫ operators (see [29]).Let I ∈ R be a Lebesgue measurable set. Set µ(I) := 1 dtπ 1+t. Note that µ(R) = 1.2Lemma 3.1. [29, Lemma 2.1] Let T 1 = T1 ∗ ∈ S ∞ and T 2 = T2 ∗ ∈ S 1 . Then∫n ± (s 1 + s 2 ; T 1 + t T 2 ) dµ(t) ≤ n ± (s 1 ; T 1 ) + 1 ‖T 2 ‖ 1 , s 1 , s 2 > 0. (3.1)πs 2RLet H ± and H 0 be two lower-bounded self-adjo**in**t operators. Assume thatLet λ 0 < **in**f σ(H ± ) ∪ σ(H 0 ). Suppose thatFor z ∈ C with Im z > 0 set T (z): = V 1/2 (H 0 − z) −1 V 1/2 .IV := ±(H ± − H 0 ) ≥ 0. (3.2)(H − λ 0 ) −γ − (H 0 − λ 0 ) −γ ∈ S 1 , γ > 0, (3.3)V 1/2 (H 0 − λ 0 ) −1/2 ∈ S ∞ , (3.4)V 1/2 (H 0 − λ 0 ) −γ′ ∈ S 2 , γ ′ > 0. (3.5)Lemma 3.2. [29, Lemma 4.1] Let (3.2), (3.4), and (3.5) hold. Then **for** almost every E ∈ R theoperator-norm limit T (E + i0) := n − lim δ↓0 T (E + iδ) exists, and by (3.4) we have T (E + i0) ∈ S ∞ .Moreover, 0 ≤ Im T (E + i0) ∈ S 1 .Theorem 3.1. [29, Theorem 1.2] Let (3.2) – (3.5) hold. Then the SSF ξ(·; H ± , H 0 ) **for** the operatorpair (H ± , H 0 ) is well-def**in**ed, and **for** almost every E ∈ R we have∫ξ(E; H ± , H 0 ) = ± n ∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t).RRemark: The representation of the SSF described **in** the above theorem was generalised to non-signdef**in**iteperturbations **in** [14] **in** the case of trace-class perturbations, and **in** [30] **in** the case of relativelytrace-class perturbations. These generalisations are based on the concept of **in**dex of orthogonal projections(see [2]).Suppose now that V satisfies (1.1), and ±V ≥ 0. Then relations (3.2) – (3.5) hold with V = |V |,H 0 = H 0 , and γ = γ ′ = 1. For z ∈ C, Im z > 0, set T (z) := |V | 1/2 (H 0 − z) −1 |V | 1/2 . By Lemma 3.2,**for** almost every E ∈ R the operator-norm limitexists, andT (E + i0) := n − limδ↓0T (E + iδ) (3.6)0 ≤ Im T (E + i0) ∈ S 1 . (3.7)The follow**in**g proposition conta**in**s a more precise version of the above statement, and provides estimatesof the norm of T (E + i0), and the trace-class norm of Im T (E + i0).10

Proposition 3.1. [7, Lemma 4.2] Assume that (1.1) holds, and E ∈ R \ 2bZ + . Then the operator limit(3.6) exists, and we have‖T (E + i0)‖ ≤ C 1 (dist (E, 2bZ + )) −1/2 (3.8)with C 1 **in**dependent of E and b.Moreover, (3.7) holds, and if E < 0 then Im T (E + i0) = 0, while **for** E ∈ (0, ∞) \ 2bZ + we have‖Im T (E + i0)‖ 1 = Tr Im T (E + i0) = b[∑E 2b ](E − 2bl) −1/2 |V (x)|dx. (3.9)4π∫R 3By Lemma 3.1 and Proposition 3.1, the quantity∫˜ξ(E; H ± , H 0 ) = ± n ∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t), E ∈ R \ 2bZ + , (3.10)Ris well-def**in**ed **for** every E ∈ R \ 2bZ + , and bounded on every compact subset of R \ 2bZ + . Moreover,by [7, Proposition 2.5], ˜ξ(·; H ± , H 0 ) is cont**in**uous on R \ {2bZ + ∪ σ pp (H ± )}. On the other hand, byTheorem 3.1 we have˜ξ(E; H ± , H 0 ) = ξ(E; H ± , H 0 ) (3.11)**for** almost every E ∈ R. As expla**in**ed **in** the **in**troduction, **in** the case of sign-def**in**ite perturbations wewill identify the SSF ξ(E; H ± , H 0 ) with ˜ξ(E; H ± , H 0 ), while **in** the case of non-sign-def**in**ite perturbations,we will identify it with the generalisation of ˜ξ(E; H ± , H 0 ) described **in** [7, Section 3] on the basisof the general results of [14] and [30].Here it should be underl**in**ed that **in** contrast to the case b = 0, we cannot rule out the possibility thatthe operator H has **in**f**in**ite discrete spectrum, or eigenvalues embedded **in** the cont**in**uous spectrum byimpos**in**g conditions about the fast decay of the potential V at **in**f**in**ity. First, it is well-known that ifV satisfiesV (x) ≤ −Cχ(x), x ∈ R 3 , (3.12)where C > 0, and χ is the characteristic function of a non-empty open subset of R 3 , then the discretespectrum of H is **in**f**in**ite (see [1, Theorem 5.1], [38, Theorem 2.4]). Further, if V is axisymmetric andsatisfies (3.12), then the operator H (q) def**in**ed **in** (2.17) with q ≥ 0 has at least one eigenvalue **in** the**in**terval (2bq − ‖V ‖ L ∞ (R 3 ), 2bq), and hence the operator H has **in**f**in**itely many eigenvalues embedded**in** its cont**in**uous spectrum (see [1, Theorem 5.1]). Assume now that V is axisymmetric and satisfiesthe estimateV (X ⊥ , x 3 ) ≤ −Cχ ⊥ (X ⊥ )〈x 3 〉 −m3 , (X ⊥ , x 3 ) ∈ R 3 , (3.13)where C > 0, χ ⊥ is the characteristic function of a non-empty open subset of R 2 , and m 3 ∈ (0, 2) whichis compatible with (1.1) if m 3 ∈ (1, 2). Then, us**in**g the argument of the proof of [1, Theorem 5.1]and the variational pr**in**ciple, we can easily check that **for** each q ≥ 0 the operator H (q) has **in**f**in**itelymany discrete eigenvalues which accumulate to the **in**fimum 2bq of its essential spectrum. Hence, if Vis axisymmetric and satisfies (3.13), then below each Landau level 2bq, q ∈ Z + , there exists an **in**f**in**itesequence of f**in**ite-multiplicity eigenvalues of H, which converges to 2bq. Note however that the claims**in** [10, p. 385] and [8, p. 3457] that [1, Theorem 5.1]) implies the same phenomenon **for** axisymmetricnon-positive potentials compactly supported **in** R 3 , are not justified. The challeng**in**g and **in**terest**in**gproblem about the accumulation at a given Landau level of embedded eigenvalues and/or resonancesof H will be considered **in** a future work.F**in**ally, we note that generically the only possible accumulation po**in**ts of the eigenvalues of H are theLandau levels (see [1, Theorem 4.7], [13, Theorem 3.5.3 (iii)]). Further **in****for**mation on the location ofthe eigenvalues of H can be found **in** [7, Proposition 2.6].l=011

3.2 Estimates **for** Birman-Schw**in**ger operatorsFor x, x ′ ∈ R 2 denote by P q,b (x, x ′ ) the **in**tegral kernel of the orthogonal projection p q (b) onto thesubspace Ker (h(b) − 2bq), q ∈ Z + , the Landau Hamiltonian h(b) be**in**g def**in**ed **in** (2.6). It is wellknownthatP q,b (x, x ′ ) = b2π L q( b|x − x ′ | 2(see [21] or [37, Subsection 2.3.2]) where L q (t) := 1 q! et d q (t q e −t )dt qLaguerre polynomials. Note that2) (exp − b )4 (|x − x′ | 2 + 2i(x 1 x ′ 2 − x ′ 1x 2 ))(3.14)= ∑ q( q) (−t)kk=0 k k!, t ∈ R, q ∈ Z + , are theP q,b (x, x) = b2π , q ∈ Z +, x ∈ R 2 . (3.15)Def**in**e the orthogonal projections P q : L 2 (R 3 ) → L 2 (R 3 ), q ∈ Z + , by P q := p q ⊗ I 3 where I 3 is theidentity operator **in** L 2 (R x3 ).(−1For z ∈ C with Im z > 0, def**in**e the operator R(z) := − d2 − z)dx bounded **in** L 2 (R). Note that the23operator R(z) admits the **in**tegral kernel R z (x 3 − x ′ 3) where R z (x) = ie i√z|x| /(2 √ z), x ∈ R, the branchof √ z be**in**g chosen so that Im √ z > 0.Def**in**e that the operatorsT q (z) := |V | 1/2 P q (H 0 − z) −1 |V | 1/2 , q ∈ Z + ,)bounded **in** L 2 (R 3 ). We have T q (z) = |V |(p 1/2 q (b) ⊗ R(z − 2bq) |V | 1/2 .For λ ∈ R, λ ≠ 0, def**in**e R(λ) as the operator with **in**tegral kernel R λ (x 3 − x ′ 3) where⎧⎨R λ (x) := lim R λ+iδ (x) =δ↓0 ⎩e −√ −λ|x|2 √ −λie i√ λ|x|2 √ λif λ < 0,if λ > 0,x ∈ R. (3.16)Evidently, if w 1 , w 2 ∈ L 2 (R) and λ ≠ 0, then w 1 R(λ)w 2 ∈ S 2 . For E ∈ R, E ≠ 2bq, q ∈ Z + , set( )T q (E) := |V | 1/2 p q (b) ⊗ R(E − 2bq) |V | 1/2 .Then lim δ↓0 ‖T q (E + iδ) − T q (E)‖ 2 = 0 (see [10, Proposition 4.1]).Proposition 3.2. Let E ∈ R, q ∈ Z + , E ≠ 2bq. Let (1.1) hold. Thenwith C 2 **in**dependent of E, b, and q.Proof. We have‖T q (E)‖ ≤ C 2 |E − 2bq| −1/2 , (3.17)‖T q (E)‖ 2 2 ≤ C 2 b|E − 2bq| −1 , (3.18)T q (E) = MG q,m⊥ ⊗ t(E − 2bq)M (3.19)where M is the multiplier by the bounded function |V (X ⊥ , x 3 )| 1/2 〈X ⊥ 〉 m⊥/2 〈x 3 〉 m3/2 , (X ⊥ , x 3 ) ∈ R 3 ,G q,m⊥ : L 2 (R 2 ) → L 2 (R 2 ) is the operator with **in**tegral kernel〈X ⊥ 〉 −m ⊥/2 P b,q (X ⊥ , X ′ ⊥)〈X ′ ⊥〉 −m ⊥/2 , X ⊥ , X ′ ⊥ ∈ R 2 ,and t(λ) : L 2 (R) → L 2 (R), λ ∈ R \ {0}, is the operator with **in**tegral kernel〈x 3 〉 −m3/2 R λ (x 3 − x ′ 3)〈x ′ 3〉 −m3/2 , x 3 , x ′ 3 ∈ R.12

Then we have‖T q (E)‖ ≤ ‖M‖ 2 ∞‖G q,m⊥ ‖ ‖t(E − 2bq)‖ ≤ ‖M‖ 2 ∞‖G q,m⊥ ‖ ‖t(E − 2bq)‖ 2 , (3.20)where ‖M‖ ∞ := ‖M‖ L∞ (R 2 ). Evidently,(see (3.15)), and‖T q (E)‖ 2 ≤ ‖M‖ 2 ∞‖G q,m⊥ ‖ 2 ‖t(E − 2bq)‖ 2 , (3.21)‖G q,m⊥ ‖ ≤ 1, (3.22)‖G q,m⊥ ‖ 2 2 ≤ Tr p q 〈X ⊥ 〉 −m ⊥p q = b ∫〈X ⊥ 〉 −m ⊥dX ⊥2π R 2 (3.23)‖t(E − 2bq)‖ 2 2 ≤∫1〈x 3 〉 −m3 dx 3 (3.24)4|E − 2bq| R(see (3.16)). Now the comb**in**ation of (3.20), (3.22), and (3.24) yields (3.17), while the comb**in**ation of(3.21), (3.23), and (3.24) yields (3.18).Remark: Us**in**g more sophisticated tools than those of the proof of Proposition 3.2, it is shown **in** [7]that **for** E ≠ 2bq we have not only T q (E) ∈ S 2 , but also T q (E) ∈ S 1 . We will not use this fact here.Proposition 3.3. Assume that V satisfies (1.1). Let E ∈ R \ 2bZ + , q ∈ Z + . Then we have 0 ≤Im T q (E) ∈ S p with any p > 2/m ⊥ . If E < 2bq, then Im T q (E) = 0. If E > 2bq, then the estimate)n + (s; Im T q (E)) ≤ C 3(1 + b (E − 2bq) −1/m ⊥s −2/m ⊥(3.25)holds **for** each s > 0 with C 3 **in**dependent of s, b, and E. Moreover, if E > 2bq, then we have‖Im T q (E)‖ 1 = Tr Im T q (E) = b∫4π (E − 2bq)−1/2 |V (x)|dx. (3.26)R 3Proof. By (3.19), we haveIm T q (E) = MG p,m⊥ ⊗ Im t(E − 2bq)M.If E < 2bq, then Im t(E − 2bq) = 0. If E > 2bq, then Im t(E − 2bq) admits the **in**tegral kernel12 √ E − 2bq 〈x 3〉 −m3/2 cos ( √ E − 2bq (x 3 − x ′ 3))〈x ′ 3〉 −m3/2 , x 3 , x ′ 3 ∈ R.S**in**ce the function 〈X ⊥ 〉 −m⊥/2 is radially symmetric, the eigenvalues ν k , k ∈ N, of the operator G p,m⊥ ≥0 can be computed explicitly, and **for** k ≥ k 0 we have ν k ≤ C 3b ′ m⊥/2 k −m⊥/2 with k 0 ∈ N and C 3′**in**dependent of b and E (see the proof of [7, Lemma 9.4]). Further, if E > 2bq, ∫ we have rank Im t(E −12bq) = 2, and the eigenvalues of Im t(E −2bq) are upper-bounded by2 √ E−2bq R 〈x 3〉 −m3 dx 3 . There**for**e,( C′n + (s; Im T q (E)) ≤ k 0 + 2 3 ‖M‖ 2 ∞b m⊥/2 s −1 ∫) 2/m⊥2 √ 〈x 3 〉 −m3 dx 3 , s > 0,E − 2bq Rwhich entails immediately (3.25). F**in**ally, if we write the trace of the operator Im T q (E) as the **in**tegralof the diagonal value of its kernel, and take **in**to account (3.15) and (3.16), we get (3.26).13

Proposition 3.4. [10, Proposition 4.2] Let q ∈ Z + , λ ∈ R, |λ| ∈ (0, b), and δ > 0. Assume that Vsatisfies (1.1). Then the operator series T + q (2bq + λ + iδ) := ∑ ∞l=q+1 T l(2bq + λ + iδ), andconverge **in** S 2 . Moreover,T + q (2bq + λ) :=‖T + q (2bq + λ)‖ 2 2 ≤ C 0b8π∞∑l=q+1T l (2bq + λ) (3.27)∞∑(2b(l − q) − λ)∫R −3/2 V (x)dx. (3.28)3l=q+1F**in**ally, lim δ↓0 ‖T + q (2bq + λ + iδ) − T + q (2bq + λ)‖ 2 = 0.4 Proofs of the Ma**in** Results4.1 Proofs of the results on the s**in**gularities of the SSF at the Landau levelsThe first step **in** the proofs of both Theorems 2.1 and 2.2 is to show that we can replace the operatorT (E + i0) by T q (E) **in** the r.h.s of (3.10) when we deal with the first asymptotic term of ˜ξ(E; H ± , H 0 )as the energy E approaches a given Landau level 2bq, q ∈ Z + . More precisely, we pick q ∈ Z + , λ ∈ Rwith |λ| ∈ (0, b), and set Tq − (2bq + λ) := ∑ q−1l=0 T l(2bq + λ); if q = 0 the sum should be set equal to zero.Evidently,T (2bq + λ + i0) = Tq − (2bq + λ) + T q (2bq + λ) + T q + (2bq + λ),Re T (2bq + λ + i0) = Re T − q (2bq + λ) + Re T q (2bq + λ) + T + q (2bq + λ),Im T (2bq + λ + i0) = Im T − q (2bq + λ) + Im T q (2bq + λ),the operator T q + (2bq + λ) be**in**g def**in**ed **in** (3.27). Comb**in****in**g the Weyl **in**equalities (2.3), Lemma 3.1,(3.26), the Chebyshev-type estimates (2.5) with p = 2, (3.18), and (3.28), we easily f**in**d that theasymptotic estimates∫n ± (1 + ε; Re T q (2bq + λ) + t Im T q (2bq + λ)) dµ(t) + O(1) ≤R∫R∫Rn ± (1; Re T (2bq + λ + i0) + t Im T (E + i0)) dµ(t) ≤n ± (1 − ε; Re T q (2bq + λ) + t Im T q (2bq + λ)) dµ(t) + O(1) (4.1)hold as λ → 0 **for** each ε ∈ (0, 1) (see [10, Proposition 5.1] **for** details).If λ > 0, then T q (2bq − λ) is a self-adjo**in**t operator with **in**tegral kernel12 √ λ1 √∫|V (X⊥ , x 3 )| P q,b (X ⊥ , X ′2π⊥)Re ip(x3−x′ 3 ) √p 2 + λdp |V (X⊥ ′ , x′ 3 )| =√|V (X⊥ , x 3 )| P q,b (X ⊥ , X ′ ⊥)e −√ λ|x 3−x ′ 3 | √|V (X ′ ⊥ , x′ 3 )|, (X ⊥, x 3 ), (X ′ ⊥, x ′ 3) ∈ R 3 .In particular, Im T q (2bq − λ) = 0, and Re T q (2bq − λ) = T q (2bq − λ) ≥ 0. There**for**e,∫n ± (s; Re T q (2bq − λ) + t Im T q (2bq − λ)) dµ(t) = n ± (s; T q (2bq − λ)), s > 0, λ > 0. (4.2)R14

S**in**ce T q (2bq − λ) ≥ 0, we have n − (s; T q (2bq − λ)) = 0 **for** all s > 0 and λ > 0, which comb**in**ed with(3.10), (4.1), and (4.2), implies (2.7). In order to prove (2.8), we writewhere O q (λ) is an operator with **in**tegral kernel12 √ λT q (2bq − λ) = O q (λ) + ˜T q (λ)√|V (X⊥ , x 3 )| P q,b (X ⊥ , X ′ ⊥)√|V (X ′ ⊥ , x′ 3 )|, (X ⊥, x 3 ), (X ′ ⊥, x ′ 3) ∈ R 3 ,and ˜T q (λ) := T q (2bq − λ) − O q (λ). By (1.2) we have n − lim λ↓0 ˜Tq (λ) = ˜T q (0) where ˜T q (0) is a compactoperator with **in**tegral kernel−2√ 1 √|V (X⊥ , x 3 )| P q,b (X ⊥ , X ⊥)|x ′ 3 − x ′ 3| |V (X⊥ ′ , x′ 3 )|, (X ⊥, x 3 ), (X ⊥, ′ x ′ 3) ∈ R 3 .Hence, the Weyl **in**equalities easily imply that the asymptotic estimatesn + (s ′ ; O q (λ)) + O(1) ≤ n + (s; T q (2bq − λ)) ≤ n + (s ′′ ; O q (λ)) + O(1) (4.3)hold **for** every 0 < s ′ < s < s ′′ as λ ↓ 0. Further, def**in**e the operator K : L 2 (R 3 ) → L 2 (R 2 ) by∫√(Ku)(X ⊥ ) := P q,b (X ⊥ , X ⊥)∫R ′ |V (X⊥ ′ , x′ 3 )|u(X′ ⊥, x ′ 3) dx ′ 3 dX ⊥, ′ X ⊥ ∈ R 2 ,2 Rwhere u ∈ L 2 (R 3 ). The adjo**in**t operator K ∗ : L 2 (R 2 ) → L 2 (R 3 ) is given by(K ∗ v)(X ⊥ , x 3 ) := √ ∫|V (X ⊥ , x 3 )| P q,b (X ⊥ , X ⊥)v(X ′ ⊥) ′ dX ⊥, ′ (X ⊥ , x 3 ) ∈ R 3 ,R 2where v ∈ L 2 (R 2 ). Obviously, O q (λ) = 12 √ λ K∗ K, p q W p q = K K ∗ . There**for**e,n + (s; O q (λ)) = n + (s2 √ λ; p q W p q ), s > 0, λ > 0. (4.4)Now the comb**in**ation of (3.10) with (4.1) – (4.4) entails (2.8). Thus, we are done with the proof ofTheorem 2.1.In order to complete the proof of Theorem 2.2, we recall that if λ > 0, then the operator Re T q (2bq + λ)admits the **in**tegral kernel− 12 √ √ (√ √|V (X⊥ , x 3 )| s**in** λ|x3 − x 3|)′ P q,b (X ⊥ , X ⊥)′ |V (X⊥ ′ λ, x′ 3 )|, (X ⊥, x 3 ), (X ⊥, ′ x ′ 3) ∈ R 3 ,and hence n − lim λ↓0 Re T q (2bq + λ) = ˜T q (0). Apply**in**g the Weyl **in**equalities and the evident identities∫n ± (s; tT )dµ(t) = 1 π Tr arctan (s−1 T ), s > 0,Rwhere T = T ∗ ≥ 0, T ∈ S 1 , we f**in**d that asymptotic estimates∫1π Tr arctan (((1 + ε)s)−1 Im T q (2bq + λ)) + O(1) ≤ n ± (s; Re T q (2bq + λ) + t Im T q (2bq + λ))dµ(t) ≤1π Tr arctan (((1 − ε)s)−1 Im T q (2bq + λ)) + O(1) (4.5)are valid as λ ↓ 0 **for** each s > 0 and ε ∈ (0, 1). Def**in**e the operator K : L 2 (R 3 ) → L 2 (R 2 ) 2 byKu := v = (v 1 , v 2 ) ∈ L 2 (R 2 ) 2 , u ∈ L 2 (R 3 ),R15

wherev 1 (X ⊥ ) := P q,b (X ⊥ , X ⊥) ′ cos( √ √λx ′ 3) |V (X⊥ ′ , x′ 3 )|u(X′ ⊥, x ′ 3) dx ′ 3 dX ⊥,′R∫v 2 (X ⊥ ) := P q,b (X ⊥ , X ⊥)∫R ′ s**in**( √ √λx ′ 3) |V (X⊥ ′ , x′ 3 )|u(X′ ⊥, x ′ 3) dx ′ 3 dX ⊥, ′ X ⊥ ∈ R 2 .2 R∫R 2 ∫Evidently, the adjo**in**t operator K ∗ : L 2 (R 2 ) 2 → L 2 (R 3 ) is given by(K ∗ v)(X ⊥ , x 3 ) := cos( √ λx 3 ) √ ∫|V (X ⊥ , x 3 )| P q,b (X ⊥ , X ⊥)v ′ 1 (X ⊥) ′ dX ⊥+′R 2s**in**( √ λx 3 ) √ ∫|V (X ⊥ , x 3 )| P q,b (X ⊥ , X ⊥)v ′ 2 (X ⊥) ′ dX ⊥, ′ (X ⊥ , x 3 ) ∈ R 3 ,R 2where v = (v 1 , v 2 ) ∈ L 2 (R 2 ) 2 . Obviously,Im T q (2bq + λ) = 12 √ λ K∗ K, p q W λ p q = K K ∗ ,and, there**for**e,n + (s; Im T q (2bq + λ)) = n + (s2 √ λ; p q W λ p q ), s > 0, λ > 0,Tr arctan (s −1 Im T q (2bq + λ)) = Tr arctan ((s2 √ λ) −1 p q W λ p q ), s > 0, λ > 0. (4.6)Now the comb**in**ation of (3.10), (4.1), (4.5), and (4.6) yields (2.10).4.2 Proofs of the results on the strong-magnetic-field asymptotics of theSSFIn this subsection we prove Theorems 2.3 and 2.4 under the additional assumption that ±V ≥ 0. Asbe**for**e if V ≥ 0 (or if V ≤ 0), we will write H + and χ + (X ⊥ ), X ⊥ ∈ R 2 , (or H − and χ − (X ⊥ )) **in**steadof H and χ(X ⊥ ) respectively.First, we prove Theorem 2.3. For brevity setA = A(b) = Re T (Eb + λ),B = B(b) = Im T (Eb + λ).Note that if E ∈ (0, ∞) \ 2Z + , and λ ∈ R, then (3.8) and (3.9) imply‖A(b)‖ = O(b −1/2 ), ‖B(b)‖ = O(b −1/2 ), ‖B(b)‖ 1 = O(b 1/2 ), b → ∞. (4.7)Assume that b is so large that ‖A(b)‖ < 1. Then the operator I − A is boundedly **in**vertible, andlim b→∞ ‖(I − A(b)) −1 ‖ = 1. By the Birman-Schw**in**ger pr**in**ciple we have∫∫n ± (1; A + tB)dµ(t) = n ± (1; tB 1/2 (I ∓ A) −1 B 1/2 )dµ(t) =Further,Tr arctanTr arctan∫ ∞0RRn + (s; B 1/2 (I ∓ A) −1 B 1/2 )dµ(s) = 1 π Tr arctan (B 1/2 (I ∓ A) −1 B 1/2) . (4.8)(B 1/2 (I ± A) −1 B 1/2) (≤ Tr B 1/2 (I ± A) −1 B 1/2) = Tr B ∓ Tr ( (I ± A) −1 AB ) , (4.9)(B 1/2 (I ± A) −1 B 1/2) (≥ Tr B 1/2 (I ± A) −1 B 1/2) − 1 3 ‖B1/2 (I ± A) −1 B 1/2 ‖ 3 3 =16

Tr B ∓ Tr ( (I ± A) −1 AB ) − 1 3 ‖B1/2 (I ± A) −1 B 1/2 ‖ 3 3. (4.10)By (4.7) we have|Tr ( (I ± A) −1 AB ) | ≤ ‖(I ± A) −1 A‖‖B‖ 1 = O(1), b → ∞, (4.11)‖B 1/2 (I ± A) −1 B 1/2 ‖ 3 3 ≤ ‖B 1/2 (I ± A) −1 B 1/2 ‖ 2 ‖B 1/2 (I ± A) −1 B 1/2 ‖ 1 = O(b −1/2 ), b → ∞. (4.12)Putt**in**g together (4.8) – (4.12), and bear**in**g **in** m**in**d (3.10), we getξ(Eb + λ; H ± (b), H 0 (b)) = ± 1 Tr B(b) + O(1), b → ∞. (4.13)πRecall**in**g (3.9), we f**in**d that the asymptotic estimateTr B(b) = b1/24π[E/2]∑(E − 2l)∫R −1/2 |V (x)|dx + O(b −1/2 ) (4.14)3l=0holds as b → ∞. Now the comb**in**ation of (4.13) and (4.14) yields (2.18).Next, we pass to the proof of Theorem 2.4 under the additional assumption that ±V ≥ 0. To this endwe establish some auxiliary results. Introduce the operatorτ(X ⊥ ; z) := |V (X ⊥ , .)| 1/2 (χ 0 − z) −1 |V (X ⊥ , .)| 1/2 ,def**in**ed on L 2 (R), and depend**in**g on the parameters X ⊥ ∈ R 2 and z ∈ C with Im z > 0. The operatorτ(X ⊥ ; z) admits the **in**tegral kernel|V (X ⊥ , x 3 )| 1/2 R z (x 3 − x ′ 3)|V (X ⊥ , x ′ 3)| 1/2 , x 3 , x ′ 3 ∈ R.Evidently, τ(X ⊥ ; z) ∈ S 2 . For X ⊥ ∈ R 2 , λ ∈ R\{0}, def**in**e the operator τ(X ⊥ ; λ+i0) : L 2 (R) → L 2 (R)as the operator with **in**tegral kernel|V (X ⊥ , x 3 )| 1/2 R λ (x 3 − x ′ 3)|V (X ⊥ , x ′ 3)| 1/2 , x 3 , x ′ 3 ∈ R,the function R λ (x), x ∈ R, be**in**g def**in**ed **in** (3.16). Some explicit simple calculations with the kernelof the operator τ(X ⊥ ; λ + i0) yield the follow**in**gProposition 4.1. Let X ⊥ ∈ R 2 , λ ∈ R \ {0}. Assume that (1.1) holds.i) We have τ(X ⊥ ; λ + i0) ∈ S 2 ,‖τ(X ⊥ ; λ + i0)‖ 2 2 ≤ 1 (∫) 2|V (X ⊥ , x 3 )|dx 3 , (4.15)4|λ| Rand τ(X ⊥ ; λ + iδ) → τ(X ⊥ ; λ + i0) **in** S 2 as δ ↓ 0, uni**for**mly with respect to X ⊥ ∈ R 2 .ii) We have Im τ(X ⊥ ; λ+i0) ≥ 0, and Im τ(X ⊥ ; λ+i0) = 0 if λ < 0. If λ > 0, then rank Im τ(X ⊥ ; λ+i0) = 2, and( ∫)1n + (s; Im τ(X ⊥ ; λ + i0)) ≤ 2Θ2 √ |V (X ⊥ , x 3 )|dx 3 − s , s > 0. (4.16)λ RFor X ⊥ ∈ R 2 , λ ∈ R \ {0}, s > 0, set∫Ξ ± λ,s (X ⊥) := n ± (s; Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0) dµ(t). (4.17)R17

Corollary 4.1. Let (1.1) hold. Fix X ⊥ ∈ R 2 , λ ∈ R \ {0}, s > 0. Then we haveξ(λ; χ ± (X ⊥ ), χ 0 ) = ± Ξ ∓ λ,1 (X ⊥) (4.18)where ξ(·; χ ± (X ⊥ ), χ 0 ) is the representative of SSF **for** the operator pair (χ ± (X ⊥ ), χ 0 ) which is monotonousand left-cont**in**uous **for** λ < 0, and cont**in**uous **for** λ > 0.Proof. It suffices to apply Theorem 3.1 with H ± = χ ± (X ⊥ ) and H 0 = χ 0 .Corollary 4.2. Under the assumptions of Corollary 4.1 we haveΞ ± λ,s (·) ∈ L1 (R 2 ). (4.19)Proof. Comb**in**e Lemma 3.1 **for** T 1 = Re τ(X ⊥ ; λ + i0) and T 2 = Im τ(X ⊥ ; λ + i0), with Proposition4.1.Proposition 4.2. Let λ > 0. Assume that (1.1) holds. Then the function ∫ R 2 Ξ ± λ,s (X ⊥)dX ⊥ is cont**in**uouswith respect to s > 0.Proof. Fix s > 0. First of all we will show that **for** almost every (X ⊥ , t) ∈ R 2 × R the functionss ′ ↦→ n ± (s ′ ; Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0))are cont**in**uous at the po**in**t s ′ = s. Evidently, this is equivalent to±s ∉ σ(Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0)). (4.20)In order to prove (4.20), we will use an argument quite close to the one of the proof of [29, Lemma 4.1].Note that the compact operator Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0) depends l**in**early on t. By theFredholm alternative the setsΩ ± (s, X ⊥ , λ) := {z ∈ C | ± s ∈ σ(Re τ(X ⊥ ; λ + i0) + z Im τ(X ⊥ ; λ + i0))}either co**in**cide with C, or are discrete. However, i ∈ Ω ± (s, X ⊥ , λ) is equivalent to dim Ker (χ 0 ∓s −1 |V (X ⊥ , .)| − λ) ≥ 1. On the other hand, it is well-known that the operators χ 0 ∓ s −1 |V (X ⊥ , .)| haveno positive eigenvalues (see e.g. [39, Theorem XIII.58]) s**in**ce (1.1) implies lim |x3|→∞ |x 3 |V (X ⊥ , x 3 ) = 0.There**for**e, dim Ker (χ 0 ∓s −1 |V (X ⊥ , .)|−λ) = 0, i ∉ Ω ± (s, X ⊥ , λ), and the sets Ω ± (s, λ, X ⊥ ) are discrete.In particular, |R ∩ Ω ± (s, X ⊥ , λ)| = 0. Put˜Ω ± (s, λ) := {(X ⊥ , t) ∈ R 2 × R| ± s ∈ σ(Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0))}.The eigenvalues of the compact operator Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0) are cont**in**uous, andhence measurable with respect to (X ⊥ , t) ∈ R 2 × R. There**for**e, the sets ˜Ω ± (s, λ) are measurable, andby the Fub**in**i-Tonelli theorem∫∫|˜Ω ± (s, λ)| = 1˜Ω± (s,λ)∫R (X ⊥, t)dtdX ⊥ = |R ∩ Ω ± (s, X ⊥ , λ)|dX ⊥ = 02 RR 2where 1˜Ω± (s,λ) denotes the characteristic function of Ω± (s, λ). On the other hand,lim n ±(s ′ ; Re τ(X ⊥ ; λ+i0)+t Im τ(X ⊥ ; λ+i0)) = n ± (s; Re τ(X ⊥ ; λ+i0)+t Im τ(X ⊥ ; λ+i0)) (4.21)s ′ →sif (X ⊥ , t) ∉ ˜Ω ± (s, λ). The Weyl **in**equalities (2.3) and estimates (4.15) – (4.16) implyn ± (s ′ ; Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0)) ≤18

(∫) 2 ( ∫)1|t||V (Xs ′2 ⊥ , x 3 )|dx 3 + 2Θ √ |V (X ⊥ , x 3 )|dx 3 − s ′ . (4.22)λ Rλ RNote that the r.h.s. is **in** L 1 (R 2 × R; dX ⊥ dµ(t)) **for** each s ′ > 0, and is a sum of two monotonousfunctions of∫s ′ > 0. Bear**in**g **in** m**in**d (4.21) – (4.22), we apply the dom**in**ated convergence theorem, andget lim s ′ →s Ξ ± R 2 λ,s(X ′ ⊥ )dX ⊥ = ∫ Ξ ± R 2 λ,s (X ⊥)dX ⊥ .SetΦ ± λ,s (t) := ∫R 2 n ± (s; Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0)) dX ⊥ , t ∈ R.Corollary 4.3. Assume that (1.1) holds. Let λ > 0, s > 0. Then lim s ′ →s Φ ± λ,s(t) = Φ ± ′ λ,s(t) **for** almostevery t ∈ R.Proof. S**in**ce the functions Φ ± λ,s(t) are non-**in**creas**in**g with respect to s > 0, the one-sided limitsΦ ± λ,s−0 (t) ≥ Φ± λ,s+0 (t) exist. Next, Proposition 4.2 implies ∫ Ξ ± R 2 λ,s−0 (X ⊥)dX ⊥ = ∫ Ξ ± R 2 λ,s+0 (X ⊥)dX ⊥ .By the Fub**in**i theorem ∫ Ξ ± R 2 λ,s (X ⊥)dX ⊥ = ∫ R Φ± λ,s (t)dµ(t). Hence, ∫ )(Φ ± R λ,s−0 (t) − Φ± λ,s+0 (t) dµ(t) =0. S**in**ce the functions Φ ± λ,s−0 (t) − Φ± λ,s+0(t) are non-negative, we conclude that{}∣ ∣∣ t ∈ R ∣Φ ± ∣∣λ,s−0 (t) > Φ± λ,s+0 (t) = 0.The follow**in**g proposition conta**in**s key limit**in**g relations used **in** the proof of Theorem 2.4.Proposition 4.3. (cf. [7, Proposition 7.1]) Let (1.1) hold. Then we have**for** every t ∈ R and each **in**teger p ≥ 2.limb→∞ b−1 Tr (Re T q (2bq + λ) + t Im T q (2bq + λ)) p =∫1Tr (Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0)) p dX ⊥ (4.23)2π R 2Proof. Let t ∈ R, x ∈ R. If λ > 0, set ˜R λ,t (x) := − s**in**(√ λ|x|)2 √ λR λ (x) = e−√ −λ|x|2 √ −λ . We have Tr (Re T q (2bq + λ) + t Im T q (2bq + λ)) p =+ t cos(√ λ x)2 √ . If λ < 0, then ˜R λλ,t (x) =∫R 2p ∫R p Π p j=1 |V (X ⊥,j, x 3,j )|Π ′ pj=1P q,b (X ⊥,j , X ⊥,j+1 ) ˜R λ,t (x 3,j − x 3,j+1 )Π p j=1 dX ⊥,jdx 3,jwhere the notation Π ′ pj=1 means that **in** the product of p factors the variables X ⊥,p+1 and x 3,p+1 shouldbe set equal respectively to X ⊥,1 and x 3,1 . Change the variablesThus we obta**in**X ⊥,1 = X ⊥,1 ′ , X ⊥,j = X ⊥,1 ′ + b−1/2 X ⊥,j ′ , j = 2, . . . , p. (4.24)Tr (Re T q (2bq + λ) + t Im T q (2bq + λ)) p =∫b |V (X ⊥,1∫R ′ , x 3,1)|Π p j=2 |V (X′ ⊥,1 + b−1/2 X ⊥,j ′ , x 3,j)| ×2p R pP q,1 (0, X ′ ⊥,2 )Πp−1 j=2 P q,1(X ⊥′,j , X ⊥′,j+1 )P q,1 (X ′ ⊥,p , 0) Π′ pj=1 ˜R λ,t (x 3,j − x 3,j+1 )Π p j=1 dX′ ⊥,j dx 3,j. (4.25)19

Here and **in** the sequel, if p = 2, then the product Π p−1j=2 P q,b(X ⊥′,j , X ⊥′,j+1 ) should be set equal to one.Bear**in**g **in** m**in**d (1.1) and (3.14), and apply**in**g the dom**in**ated convergence theorem, we easily f**in**d that(4.25) entailslimb→∞ b−1 Tr (Re T q (2bq + λ + i0) + t Im T q (2bq + λ + i0)) p =∫ ∫R 2∫R p Π p j=1 |V (X ⊥,1, x 3,j )|Π ′ pj=1 ˜R λ,t (x 3,j − x 3,j+1 )dX ⊥,1 Π p j=1 dx 3,j ×P q,1 (0, X ⊥,2 )Π p−1j=2 P q,1(X ⊥,j , X ⊥,j+1 )P q,1 (X ⊥,p , 0)Π p j=2 dX ⊥,j =R 2(p−1) ∫Tr ( Re τ(X ⊥,1 ; λ + i0) + t Im τ(X ⊥,1 ; λ + i0) ) pdX⊥,1 ×R∫2 P q,1 (0, X ⊥,2 )Π p−1j=2 P q,1(X ⊥,j , X ⊥,j+1 )P q,1 (X ⊥,p , 0)Π p j=2 dX ⊥,j.R 2(p−1)In order to conclude that the above limit**in**g relation is equivalent to (4.23), it rema**in**s to recall (3.15),and note that∫P q,1 (0, X ⊥,2 )Π p−1j=2 P q,1(X ⊥,j , X ⊥,j+1 )P q,1 (X ⊥,p , 0)Π p j=2 dX ⊥,j = P q,1 (0, 0) = 1R 2π .2(p−1)Corollary 4.4. Assume that the assumptions of Theorem 2.4 hold. Then we havelimb→∞ b−1 n ± (s; Re T q (2bq + λ) + t Im T q (2bq + λ)) =∫1n ± (s; Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0))dX ⊥ ,2π R 2**for** each t ∈ R, provided that s > 0 is a cont**in**uity po**in**t of the r.h.s.Proof. It suffices to notice that the norm of the operator T q (2bq + λ) is uni**for**mly bounded with respectto b, and to apply a suitable version of the Kac-Murdock-Szegö theorem (see e.g. [32, Lemma 3.1])which tells us that under appropriate hypotheses the convergence of the moments of a given measureimplies the convergence of the measure itself, and to take **in**to account Proposition 4.3.Now we are **in** position to prove Theorem 2.4. By A. Pushnitski’s representation of the SSF (see (3.10)and (4.18)), **in** order to check the validity of (2.21), it suffices to show thatlimb→∞∫Rb−1 n ± (1; Re T (2bq + λ + i0) + tIm T (2bq + λ + i0))dµ(t) =∫ ∫1n ± (1; Re τ(X ⊥ ; λ + i0) + tIm τ(X ⊥ ; λ + i0))dµ(t)dX ⊥ . (4.26)2π R R 2Argu**in**g as **in** the derivation of (4.1), we easily f**in**d that the asymptotic estimates∫n ± (1 + ε; Re T q (2bq + λ) + t Im T q (2bq + λ)) dµ(t) + o(b) ≤R∫∫RRn ± (1; Re T (2bq + λ + i0) + t Im T (2bq + λ + i0)) dµ(t) ≤n ± (1 − ε; Re T q (2bq + λ) + tIm T q (2bq + λ)) dµ(t) + o(b), (4.27)20

hold as b → ∞ **for** each ε ∈ (0, 1). Assume λ > 0. Corollary 4.3 and Corollary 4.4 implylimb→∞ b−1 n ± (s; Re T q (2bq + λ) + t Im T q (2bq + λ)) =∫1n ± (s; Re τ(X ⊥ ; λ + i0) + tIm τ(X ⊥ ; λ + i0)) dX ⊥ (4.28)2π R 2**for** any fixed s > 0, and almost every t ∈ R. Further, by (2.3), (2.5) with p = 2, Proposition 3.2, andProposition 3.3 we haveb −1 n ± (s; Re T q (2bq + λ + i0) + t Im T q (2bq + λ + i0)) ≤ C 4 (1 + |t| 2/m ⊥), t ∈ R, (4.29)with C 4 which may depend on s > 0, λ ∈ R \ {0}, q and m ⊥ but is **in**dependent of b ≥ 1 and t. Notethat the function on the r.h.s of (4.29) is **in** L 1 (R; dµ). By (4.28) – (4.29), the dom**in**ated convergencetheorem and the Fub**in**i Theorem imply12πlimb→∞∫Rb−1 n ± (s; Re T q (2bq + λ + i0) + t Im T q (2bq + λ + i0) dµ(t) =∫R 2 ∫n ± (s; Re τ(X ⊥ ; λ + i0) + t Im τ(X ⊥ ; λ + i0)) dµ(t) dX ⊥ , s > 0. (4.30)RPutt**in**g together (4.27) and (4.30), we f**in**d that the follow**in**g estimates∫Ξ ± λ,1+ε (X ⊥)dX ⊥ ≤ lim **in**fR 2 b→∞∫Rb−1 n ± (1; Re T (2bq + λ + i0) + t Im T (2bq + λ + i0) dµ(t) ≤∫∫lim sup b −1 n ± (1; Re T (2bq + λ + i0) + t Im T (2bq + λ + i0) dµ(t) ≤ Ξ ± λ,1−ε (X ⊥)dX ⊥b→∞ RR 2are valid **for** each ε ∈ (0, 1). Lett**in**g ε ↓ 0, and tak**in**g **in**to account Proposition 4.2, we obta**in** (4.26),and hence (2.21), **in** the case λ > 0. The modifications of the argument **for** λ < 0 are quite obvious; **in**this case we essentially use assumption (2.20) guarantee**in**g that λ is a cont**in**uity po**in**t of the r.h.s of(2.21).Acknowledgements. I would like to thank my co-authors V**in**cent Bruneau, Claudio Fernández,and Alexander Pushnitski **for** hav**in**g let me present our jo**in**t results **in** this survey article. The majorpart of this work has been done dur**in**g my visit to the Institute of Mathematics of the Czech Academyof Sciences **in** December 2004. I am s**in**cerely grateful to Miroslav Engliš **for** his warm hospitality.The partial support by the Chilean Science Foundation Fondecyt under Grant 1050716 is acknowledged.References[1] J.Avron, I.Herbst, B.Simon, Schröd**in**ger operators with magnetic fields. I. General **in**teractions,Duke Math. J. 45 (1978), 847-883.[2] J.Avron, R.Seiler, B.Simon, The **in**dex of a pair of projections, J. Funct. Anal. 120 (1994),220–237.[3] F.A.Berez**in**, M.A.Shub**in**, The Schröd**in**ger Equation, Kluwer Academic Publishers, Dordrecht,1991.[4]M.Š.Birman, M. G. Kreĭn, On the theory of wave operators and scatter**in**g operators, Dokl.Akad. Nauk SSSR 144 (1962), 475–478 (Russian); English translation **in** Soviet Math. Doklady3 (1962).21

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[42] A. V. Sobolev, Asymptotic behavior of energy levels of a quantum particle **in** a homogeneousmagnetic field, perturbed by an attenuat**in**g electric field. I, Probl. Mat. Anal. 9 (1984), 67-84(Russian); English translation **in**: J. Sov. Math. 35 (1986), 2201-2212.[43] S.N.Solnyshk**in**, Asymptotic behaviour of the energy of bound states of the Schröd**in**ger operator**in** the presence of electric and homogeneous magnetic fields, Probl. Mat. Fiziki, Len**in**gradUniversity 10 (1982), 266-278 (Russian); Engl. transl. **in** Selecta Math. Soviet. 5 (1986), 297-306.[44] H.Tamura, Asymptotic distribution of eigenvalues **for** Schröd**in**ger operators with homogeneousmagnetic fields, Osaka J. Math. 25 (1988), 633-647.[45] D. R. Yafaev, Mathematical scatter**in**g theory. General theory. Translations of MathematicalMonographs, 105 AMS, Providence, RI, 1992.Georgi D. RaikovDepartamento de MatemáticasFacultad de CienciasUniversidad de ChileLas Palmeras 3425, Santiago, ChileE-mail: graykov@uchile.cl24