Magnetic field influence on the discrete spectrum of locally deformed leaky wires
aa r X i v : . [ m a t h . SP ] J un MAGNETIC FIELD INFLUENCE ON THE DISCRETE SPECTRUM OF LOCALLYDEFORMED LEAKY WIRES
DIANA BARSEGHYAN AND PAVEL EXNERA bstract . We consider magnetic Schr¨odinger operator H = ( i ∇ + A ) − αδ Γ with an attractive singu-lar interaction supported by a piecewise smooth curve Γ being a local deformation of a straight line.The magnetic field B is supposed to be nonzero and local. We show that the essential spectrum is[ − α , ∞ ), as for the non-magnetic operator with a straight Γ , and demonstrate a su ffi cient conditionfor the discrete spectrum of H to be empty.
1. I ntroduction
It is well known that a magnetic field, even a local one, can influence substantially the behaviorof waveguide systems, in particular their geometrically induced discrete spectrum. While a particleconfined to a fixed-profile tube with Dirichlet boundary can exists in localized states wheneverthe tube is bent or locally deformed [8], the presence of a local magnetic field can destroy sucha discrete spectrum. This is a consequence of a Hardy-type inequality proved by Ekholm andKovaˇr´ık [5], for a more general form of this result see [12, Thm 9.9].The analogous e ff ect of bound state existence coming from the geometry of the interaction sup-port was observed is a class of singular Schr¨odinger operators, usually dubbed leaky quantum wires ,with attractive contact interaction supported by a curve [8, Chap. 10]. Here too, as observed firstin [7], perturbation of a straight line by a bend or local deformation induces, under suitable asymp-totic straightness conditions, the existence of a nonempty discrete spectrum. The above waveguideresult then leads to the natural question what would happen with such bound states in presence ofa magnetic field. The aim of this letter is to provide a partial answer.Let us mention that magnetic Schr¨odinger operators with such singular interactions have beenstudied recently, however, in a di ff erent situation where the field was homogeneous. In such a casethat spectral picture changes completely. The magnetic field itself turns the essential spectrum intothe family of infinitely degenerate Landau levels and the e ff ect of the singular interaction dependson its support: a finite curve splits the Landau levels into clusters of eigenvalues [1], a straight lineproduces an absolutely continuous spectrum analogous to the well-known edges states [3] or theIwatsuka model [10, 11], and a non-straight support may give rise to isolated eigenvalues [9].A local field is a much weaker perturbation which does not change the essential spectrum, thequestion is what has to happen so that the discrete spectrum would be destroyed. We are going to Mathematics Subject Classification.
Key words and phrases.
Singular Schr¨odinger operators, magnetic field, discrete spectrum. Note the the presence of a magnetic field is not the only e ff ect that may cause an operator subcriticality, for anotherexample see, e.g., [2]. prove one such su ffi cient condition referring to the situation when the curve supporting the singularinteraction is a local deformation of a straight line. To be more specific, we are going to consideroperator that can be formally written as(1.1a) H = ( i ∇ + A ) − αδ Γ , where A is a vector potential corresponding to magnetic field B , α >
0, and Γ is a curve of the form Γ = ( x , γ ( x )), where γ ( · ) a continuous, piecewise C smooth function. There are several ways todefine the operator properly, the simplest one is to identify it with the unique self-adjoint operatorassociated with the quadratic form(1.1b) q : q [ ψ ] = Z R | i ∇ + A | d x d y − α Z Γ | ψ | d µ, ψ ∈ H ( R ) , where µ is the measure on Γ referring to the arc length of the curve. In addition to the statedregularity of Γ , we will assume that(a) the magnetic field B = rot A is compactly supported, supp( B ) ⊂ Ω : = ( − a , a ) for some a > Ω the curve Γ coincides with the straight line Γ : = { x , x } x ∈ R and for x ∈ ( − a , a ) it stayswithin Ω , that is, | γ ( x ) | ≤ a .Let us note that if we rotate the axes passing to the coordinates x ± y , the segment of Γ in Ω mayor may not be graph of a function. It is straightforward to check that under the stated assumptionthe form is closed and bounded from below, hence the operator H is well defined. The operatordomain of H is the same as for the operator without the magnetic field consisting of functions ψ ∈ H ( R ) ∩ H ( R \ Γ ) with the property that the normal derivative of ψ at the point x ∈ Γ hasthe jump equal to − αψ ( x ). 2. M ain results It is expected that the two local perturbations, the magnetic field and the departure of Γ from thestraight line will not alter the essential spectrum. This is indeed the case: Theorem 1.
Under the stated assumptions the essential spectrum of operator H coincides with thehalf-line (cid:2) − α , ∞ (cid:1) . Our main interest concerns the possibility that the applied magnetic field destroys the discretespectrum. The main result of this letter is the following su ffi cient condition: Theorem 2.
Adopt the same assumptions and suppose that B is not identically zero. Then thereexists a constant α = α ( Ω , B ) > such that for α ∈ (0 , α ] the spectrum of operator (1.1a) below − α is empty provided that k γ ′ k ∞ ≤ α α − . AGNETIC FIELD INFLUENCE ON THE DISCRETE SPECTRUM OF LOCALLY DEFORMED LEAKY WIRES 3
3. P roofs
Let us begin with the proof of Theorem 1 . First we are going to show that the essential spectrumof H contains the half-line (cid:2) − α , ∞ (cid:1) . We employ the Weyl criterion [13, Thm VII.12]. Put λ = − α + p , p ,
0, and consider the sequence of vectors ψ k : ψ k ( x , y ) = √ k e − α √ | x − y | e ip ( x + y ) / χ (cid:18) xk (cid:19) , where χ is a smooth function with support in (1 ,
2) and k ∈ N .It is straightforward to check that ψ k ∈ Dom( H ); the idea is to use the fact that there is a part ofplane where the magnetic field has no influence. Indeed, choosing the Landau gauge for the vectorpotential A , that is, putting A ( x , y ) = (cid:16) − R y B ( x , t ) d t , (cid:17) , we get A ( x , y ) = | x | > a . For k largeenough we thus have ∂ψ k ( x , y ) ∂ x = √ k e − α √ | x − y | e ip ( x + y ) / (cid:20)(cid:18) − α √ x − y ) + ip (cid:19) χ (cid:18) xk (cid:19) + k χ ′ (cid:18) xk (cid:19) (cid:21) ,∂ψ k ( x , y ) ∂ y = √ k e − α √ | x − y | e ip ( x + y ) / (cid:18) α √ x − y ) + ip (cid:19) , hence using (1.1b) and d µ = √ x we get by a direct computation q [ ψ k ] − (cid:16) α + p (cid:17) k ψ k k = (cid:16) α + p (cid:17) √ α k χ k + k √ α k χ ′ k − α √ k χ k + (cid:16) α − p (cid:17) √ α k χ k = k √ α k χ ′ k = O ( k − ) , and since k ψ k k = √ α k χ k is independent of k , we infer that − α + p ∈ σ ( H ). Moreover, onecan choose a sequence { k n } ∞ n = such that k n → ∞ as n → ∞ and the supports of the functions ψ k n are mutually disjoint which means that − α + p ∈ σ ess ( H ).Next we have to establish that the spectrum of H below − α , if any, can be only discrete.Neumann bracketing yields the estimate(3.1) H ≥ H ⊕ H , where H is the Neumann restriction of H to L ( R \ Ω ) and H is the complementary Neumannrestriction to L ( Ω ). Denoting Γ Ω : = Γ ↾ Ω and using the conventional abbreviations for partial D. BARSEGHYAN AND P. EXNER derivatives, one can check easily that Z R \ Ω | i ∇ u + Au | d x d y − α Z Γ \ Γ Ω | u | d µ ≥ Z R \ ( − a , a ) Z R | u y | d y d x + Z R \ ( − a , a ) Z R (cid:12)(cid:12)(cid:12)(cid:12) iu x − u Z y B ( x , t ) d t (cid:12)(cid:12)(cid:12)(cid:12) d x d y − α √ Z R \ ( − a , a ) | u ( x , x ) | d x ≥ Z R \ ( − a , a ) Z R | u y | d y d x − α √ Z R \ ( − a , a ) | u ( x , x ) | d x + Z R \ ( − a , a ) Z R (cid:12)(cid:12)(cid:12)(cid:12) iu x − u Z y B ( x , t ) d t (cid:12)(cid:12)(cid:12)(cid:12) d x d y − α √ Z R \ ( − a , a ) | u ( x , x ) | d x . In the last term on the right-hand side we may replace R R \ ( − a , a ) | u ( x , x ) | d x by α √ R R \ ( − a , a ) | u ( y , y ) | d y .Noting then that for any y ∈ R \ [ − a , a ] the integral R y B ( x , t ) d t depends only on x and using thefact that the principal eigenvalue of operator − d d t − α √ δ ( t ) is − α , the above estimate implies Z R \ Ω | i ∇ u + Au | d x d y − α Z Γ \ Γ Ω | u | d µ ≥ − α (cid:18) Z R \ ( − a , a ) Z R + Z R Z R \ ( − a , a ) (cid:19) | u | d x d y ≥ − α Z R \ Ω | u | d x d y ;(3.2)this means that the spectrum of H below − α is empty.It remains to deal with operator H . Our task will be done if we check that its spectrum ispurely discrete; by minimax principle, it is su ffi cient to show that H is bounded from below byan operator with a purely discrete spectrum. To this aim we will estimate the integral R Γ Ω | u | d µ appearing in the singular perturbation using the magnetic Sobolev norm of u . To begin with, it iseasy to see that for any fixed x there is a y = y ( x ) ∈ ( − a , a ) such that | u ( x , y ) | ≤ a Z a − a | u ( x , z ) | dz . In view of the identity | u ( x , y ) | = Z yy ∂∂ z | u ( x , z ) | d z + | u ( x , y ) | , where the derivative in the integral equals 2 Re (cid:0) u ( x , z ) u z ( x , z ) (cid:1) , we have the estimate | u ( x , y ) | ≤ ε Z a − a | u z ( x , z ) | dz + ε Z a − a | u ( x , z ) | d z + √ a Z a − a | u ( x , z ) | d z ≤ ε Z a − a | u z ( x , z ) | dz + ε + √ a ! Z a − a | u ( x , z ) | d z . AGNETIC FIELD INFLUENCE ON THE DISCRETE SPECTRUM OF LOCALLY DEFORMED LEAKY WIRES 5 for any ε >
0. Choosing thus A = (cid:16) − R y B ( x , t ) dt , (cid:17) we infer that Z Γ Ω | u | d µ = Z a − a | u ( x , γ ( x )) | q + γ ′ ( x ) d x ≤ q + k γ ′ k ∞ ε Z Ω | i ∇ u + Au | d x d y + ε + √ a ! Z Ω | u | d x d y ! . This means that H is, in the sense of quadratic forms, estimated from below by the operator − αε q + k γ ′ k ∞ ! (cid:0) i ∇ u + A (cid:1) − α ε + √ a ! which for small enough values of ε has a purely discrete spectrum. This establishes the claim ofTheorem 1.In order to prove Theorem 2 we need the following two lemmata: Lemma 1.
Assume that B ∈ L loc ( R ) is not identically zero on a bounded domain ω ⊂ R , then theprincipal eigenvalue of the magnetic Neumann Laplacian on ω with the field B is positive.Proof. We have to demonstrate thatinf H ( ω ) , k u k = Z ω | i ∇ u + Au | d x d y ≥ c holds for some positive c .Let B = B ( p , R ) ⊂ ω be a ball such that R B B ( x , y ) d x d y is not zero. We are going to use thetechnics developed in [6]. By diamagnetic inequality we get Z ω | i ∇ u + Au | d x d y ≥ Z ω | i ∇ u + Au | d x d y + Z B | i ∇ u + Au | d x d y ≥ Z ω |∇| u || d x d y + Z B | i ∇ u + Au | d x d y . (3.3)According to Lemma 3.1 in [5] there is a constant c = c ( B ) > Z B | i ∇ u + Au | d x d y ≥ c Z B | u | d x d y . The estimate (3.3) implies that for any u satisfying k u k = Z ω | i ∇ u + Au | d x d y ≥ Z ω |∇| u || d x d y + c Z B | u | d x d y ≥ min (cid:8) , c (cid:9) inf w ∈H ( ω ) , w ≥ , k w k = (cid:16)R ω |∇ w | d x d y + R B | w | d x d y (cid:17) (3.4)Let us estimate the right-hand side of (3.4). For any w ∈ H ( ω ) with the unit L norm we use therepresentation w = βϕ + f , D. BARSEGHYAN AND P. EXNER where ϕ = √ vol( ω ) is the positive normalized ground state eigenfunction of the Neumann Laplacianon ω , f is orthogonal to ϕ , and(3.5) | β | + k f k L ( ω ) = . This choice means that Z ω |∇ f | d x d y ≥ λ Z ω | f | d x d y , where λ is the second eigenvalue of the Neumann Lapacian on ω , and consequently Z ω |∇ w | d x d y + Z B | w | d x d y = Z ω |∇ f | d x d y + Z B | βϕ + f | d x d y ≥ λ Z ω | f | d x d y + Z B | βϕ + f | d x d y . (3.6)The functions ϕ and f are orthogonal on ω but not on B , hence to estimate further the right-handside of (3.6) we have to consider separately three cases. The second integral is certainly not smallerthan (cid:0) k βϕ k L ( B ) − k f k L ( B ) (cid:1) , hence if | β | Z B | ϕ | d x d y ≥ Z B | f | d x d y the expression in question can be estimated as Z B | βϕ + f | d x d y ≥ λ Z ω | f | d x d y + | β | Z B | ϕ | d x d y , and similarly for Z B | f | d x d y ≥ | β | Z B | ϕ | d x d y the lower bound is Z B | βϕ + f | d x d y ≥ λ Z ω | f | d x d y + | β | Z B | ϕ | d x d y . In the remaining case when14 Z B | f | d x d y < | β | Z B | ϕ | d x d y < Z B | f | d x d y we just know that the squared norm di ff erence is nonnegative. Instead, we then split the first termon the right-hand side of (3.6) and estimate one half of it from below from the last inequalityobtaining λ Z ω | f | d x d y + | β | λ Z B | ϕ | d x d y . Putting these bounds together we get Z ω |∇ w | d x d y + Z B | w | d x d y ≥ λ Z ω | f | d x d y + | β | (cid:26) , λ (cid:27) vol( B )vol( ω ) . AGNETIC FIELD INFLUENCE ON THE DISCRETE SPECTRUM OF LOCALLY DEFORMED LEAKY WIRES 7 If β ≥ / min n , λ o vol( B )vol( ω ) , in theopposite case relation (3.5) shows that the lower bound is λ > min n , λ o vol( B )vol( ω ) . Combiningfinally this conclusion with relation (3.4) we arrive at the bound(3.7) inf σ ( − ∆ ω N ) ≥ vol( B )16vol( ω ) min (cid:26) , λ (cid:27) min (cid:26) , c (cid:27) , which proves the lemma. (cid:3) We also need the following simple estimate:
Lemma 2.
Let I ⊂ R be an interval and denote by | I | its length, then for any g ∈ H ( I ) and allx ∈ I we have | g ( x ) | ≤ | I | Z I | g ′ ( t ) | d t + | I | Z I | g ( t ) | d tProof. One can check easily that there exists a point x ∈ I such that | g ( x ) | ≤ √| I | sZ I | g ( t ) | dt ;this fact in combination with Schwarz inequality yields | g ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z xx g ′ ( t ) d − g ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | I | Z I | g ′ ( t ) | d + | I | Z I | g ( t ) | d . which is what we have set up to prove. (cid:3) After this preliminary let us return to proof of the theorem. Applying Lemma 2 to u ( x , y ) on Ω for a fixed x one gets(3.8) | u ( x , y ) | ≤ C Z a − a ( | u t ( x , t ) | + | u ( x , t ) | ) dt , y ∈ ( − a , a ) , where(3.9) C : = max n a , a o . In what follows we use again the notation Γ Ω = { x , γ ( x ) } x ∈ ( − a , a ) . We have Z Γ Ω | u | d µ = Z a − a | u ( x , γ ( x )) | q + γ ′ ( x ) d x and relation(3.8) allows to estimate the above expression as follows, Z Γ Ω | u | d µ ≤ C Z a − a q + γ ′ ( x ) Z a − a (cid:16) | u y ( x , y ) | + | u ( x , y ) | (cid:17) d y d x , with the constant C defined in (3.9); in this way we arrive to the bound(3.10) Z Γ Ω | u | d µ ≤ C q + k γ ′ k ∞ Z Ω (cid:16) | u y ( x , y ) | + | u ( x , y ) | (cid:17) d x d y . D. BARSEGHYAN AND P. EXNER
Using once more Landau gauge for the vector potential, A = (cid:16) − R y B ( x , t ) d t , (cid:17) , we infer from(3.10) that(3.11) Z Γ Ω | u | d µ ≤ C q + k γ ′ k ∞ Z Ω | i ∇ u + Au | d x d y + Z Ω | u | d x d y ! . Next we use Lemma 1 by which the ground state eigenvalue of the magnetic Neumann Laplacianon Ω is not smaller than some κ = κ Ω , B >
0. The inequality (3.11) implies Z Ω | i ∇ u + Au | d x d y − α Z Γ Ω | u | d µ ≥ − C α q + k γ ′ k ∞ ! Z Ω | i ∇ u + Au | d x d y − C α q + k γ ′ k ∞ Z Ω | u | d x d y ≥ κ − C α q + k γ ′ k ∞ ! Z Ω | u | d x d y − C α q + k γ ′ k ∞ Z Ω | u | d x d y = κ − C α (1 + κ ) q + k γ ′ k ∞ ! Z Ω | u | d x d y , (3.12)hence by choosing(3.13) α q + k γ ′ k ∞ ≤ κ ( κ + C we achieve that the right-hand side of (3.12) becomes non negative. In combination with (3.2) thismeans that the spectrum below − α is empty.To finish the proof we have to specify the range of the values of α compatible with (3.13). Wenote that k γ ′ k ∞ ≥
1. Indeed, was it not the case we get a contradiction because the identity γ ( a ) = Z a − a γ ′ ( t ) d t + γ ( − a )would lead to a contradiction with γ ( ± a ) = ± a . Hence p + k γ ′ k ∞ ≥ √ α ≤ α : = κ √ κ + C , where C is the number given by (3.9). For a fixed α ∈ (0 , α ] the condition (3.13) then reads p + k γ ′ k ∞ ≤ √ α α which shows how much the curve Γ may depart from the straight line withoutgiving rise to bound states, k γ ′ k ∞ ≤ α α − . This completes the proof of Theorem 2.
AGNETIC FIELD INFLUENCE ON THE DISCRETE SPECTRUM OF LOCALLY DEFORMED LEAKY WIRES 9
4. C oncluding remarks
Let us add a few comments. First of all, for weak magnetic fields the value α of the criticalcoupling is expected to be small. By (3.14) its behavior depends on the principal eigenvalue ofthe magnetic Neumann Laplacian on Ω which is expected to go zero as B → Ω ′ ⊃ Ω witha smooth boundary. Furthermore, for a fixed γ one can always achieve that the discrete spectrumis empty by choosing α small enough. In this respect it is useful to mention that the analogy withDirichlet waveguides that we mentioned in the introduction as an inspiration is far from complete:a counterpart to the coupling strength α could be, in a sense, the inverse of the waveguide widthbut the latter cannot take arbitrary values, for smooth waveguides at least.On the other hand, our geometric assumptions are rather restrictive and it would be useful tofind an extension of the present result, at least to curves Γ straight outside a compact region but notnecessarily coming from a local deformation of a straight line. Acknowledgment.
The authors are obliged to Hynek Kovaˇr´ık for useful comments. The workof P.E. was in part supported by the European Union within the project CZ.02.1.01 / / / / eferences [1] J. Behrndt, P. Exner, M. Holzmann, V. Lotoreichik: The Landau Hamiltonian with δ -potentials supported on curves, Rev. Math. Phys. (2020), 2050010.[2] D. Borisov, T. Ekholm, H. Kovaˇr´ık: Spectrum of the magnetic Schr¨odinger operator in a waveguide with combinedboundary conditions, Ann. Henri Poincar´e (2005), 327–342.[3] S. de Bi´evre, J.V. Pul´e: Propagating edge states for a magnetic Hamiltonian, Math. Phys. El. J. (1999), 3.[4] B. Colbois, A. El Soufi, S. Ilias, A. Savo: Eigenvalues upper bounds for the magnetic Schr¨odinger operator, arXiv:1709.09482 .[5] T. Ekholm, H. Kovaˇr´ık: Stability of the magnetic Schr¨odinger operator in a waveguide, Comm. PDE (2005),539–565.[6] T. Ekholm, H. Kovaˇr´ık, F. Portmann: Estimates for the lowest eigenvalue of magnetic Laplacians, J. Math. Anal.Appl. (2016), 330–346.[7] P. Exner, T. Ichinose: Geometrically induced spectrum in curved leaky wires,
J. Phys. A: Math. Gen. (2001),1439–1450.[8] P. Exner, H. Kovaˇr´ık: Quantum Waveguides , Springer International, Heidelberg 2015.[9] P. Exner, V. Lotoreichik, A. P´erez-Obiol: On the bound states of magnetic Laplacians on wedges,
Rep. Math. Phys. (2018), 161–185.[10] A. Iwatsuka: Examples of absolutely continuous Schr¨odinger operators in magnetic fields, Publ. RIMS (1985),385-401.[11] M. Mantoiu, R. Purice: Some propagation properties of the Iwatsuka model, Commun. Math. Phys. (1997),691-708.[12] N. Raymond:
Bound States of the Magnetic Schr¨odinger Operator , EMS Publ., Z¨urich 2017.[13] M. Reed, B. Simon,
Methods of Modern Mathematical Physics, I. Functional Analysis , Academic Press, New York1981. D epartment of M athematics , F aculty of S cience , U niversity of O strava , 30. dubna
22, 70103 O strava , C zech R epublic , E- mail : diana . barseghyan @ osu . cz D oppler I nstitute for M athematical P hysics and A pplied M athematics , C zech T echnical U niversity in P rague ,Bˇ rehov ´ a
7, 11519 P rague , C zech R epublic , and D epartment of T heoretical P hysics , N uclear P hysics I nstitute , C zech A cademy of S ciences , 25068 ˇR e ˇ z , C zech R epublic , E- mail : exner @ ujf . cas ..