Stability of the Magnetic Schrödinger Operator in a Waveguide

Communications in Partial Differential Equations, 3: 539 565, 5 Copyright Taylor & Francis, Inc. ISSN 36-53 print/153-4133 online DOI: 1.181/PDE-5113 Stability of the Magnetic Schrödinger Operator in a Waveguide TOMAS EKHOLM 1 AND HYNEK KOVAŘÍK 1 Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden Faculty of Mathematics and Physics, Stuttgart University, Stuttgart, Germany The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right. Keywords Hardy inequality; Magnetic field; Schrödinger operator. Mathematics Subject Classification 35P5; 81Q1. 1. Introduction It has been known for a long time that a bending of a two-dimensional quantum waveguide induces the existence of bound states Exner and Šeba (1989, Goldstone and Jaffe (199, Duclos and Exner (1995. From the mathematical point of view, this means that the Dirichlet Laplacian on a smooth asymptotical straight planar waveguide has at least one isolated eigenvalue below the threshold of the essential spectrum. Similar results have been obtained for a locally deformed waveguide, which corresponds to adding a small bump to the straight waveguide; see Bulla et al. (1997 and Borisov et al. (1. In both cases an appropriate transformation Received May 6, 4; Accepted October 11, 4 Also on leave of absence from Nuclear Physics Institute, Academy of Sciences, Řež, near Prague, Czech Republic. Address correspondence to Tomas Ekholm, Department of Mathematics, Royal Institute of Technology, S-141 Stockholm, Sweden; E-mail: tomase@math.kth.se or Hynek Kova rík, Stuttgart University, Faculty of Mathematics and Physics, Stuttgart University, D-75 69 Stuttgart, Germany, Fax: 49-711-6855594; E-mail: kovarik@mathematik.unistuttgart.de 539

54 Ekholm and Kovařík is used to pass to a unitary equivalent operator on the straight waveguide with an additional potential, which is for small perturbations of the described type proved to be dominated by an attractive term. As a result, at least one isolated eigenvalue appears below the essential spectrum for any bending, satisfying certain regularity properties, respectively for an arbitrarily small bump. The crucial point is that for low energies the Dirichlet Laplacian in a planar waveguide in behaves effectively as a one-dimensional system, in which the Schrödinger operators with attractive potentials have a negative discrete eigenvalue no matter how weak the potential is. This is related to the well-known fact that the Hardy inequality fails to hold in dimensions one and two. The purpose of this paper is to prove that in the presence of a suitable magnetic field some critical strength of the deformation is needed for these bound states to appear. The magnetic field is not supposed to affect the essential spectrum of the Dirichlet Laplacian. We will deal with two generic examples of a magnetic field, a locally bounded field and an Aharonov Bohm field. The crucial technical tool of the present work is a Hardy type inequality for magnetic Dirichlet forms in the waveguide. For d 3 the classical Hardy inequality states that d ux x dx 4 d d ux dx (1.1 for all u H 1 d. Hence if d 3 and V C d, V, the operator V does not have negative eigenvalues for small values of the parameter. However, if d = 1 and V C d is nonnegative, then Vxux dx d ux dx d (1. implies V =. It is known that if V is nonnegative and not identically zero, then the spectrum of V contains some negative eigenvalues for any >; see Blankenbelcer et al. (1977, (Weidl, 1999, Lemma 5.1. Let us consider the magnetic Schrödinger operator i + A, where A is a magnetic vector potential. Laptev and Weidl (1999 proved a modified version of the inequality (1.1 in for the quadratic form of a magnetic Schrödinger operator, Const ux 1 +x dx i + Aux dx (1.3 and gave a sharp result for the case of an Aharonov Bohm field. See Laptev and Weidl (1999 for details and precise assumptions on the magnetic field under which inequality (1.3 holds true. This work was later extended in Balinsky (3, to multiple Aharonov Bohm magnetic potentials; see also Evans and Lewis (4, Balinsky et al. (in press, and Melgaard et al. (4. Recently another generalization of the result by Laptev and Weidl was obtained in Balinsky et al. (4. In our model we study the spectrum of i + A in L with Dirichlet boundary conditions. An essential difference to the above mentioned cases is that due to the Dirichlet boundary conditions, the spectrum starts from 1. Consequently, inequality (1.3 becomes trivial. Therefore we shall subtract the

Stability of the Magnetic Schrödinger Operator in a Waveguide 541 threshold of the spectrum and prove a Hardy inequality in the form ux ( Const dx i + Aux ux dx (1.4 1 + x1 for all u in the magnetic Sobolev space HA 1. Inequality (1.4 is then used to prove the stability of the spectrum of the corresponding magnetic Schrödinger operator under local geometrical perturbations. Stability results in another form of the magnetic Schrödinger operator on d, d, was also studied in Mantoiu and Richard (. The text is organized in the following way. The main results are formulated in Section. In Section 3 we prove the Hardy inequalities for the magnetic Schrödinger operator with a locally bounded field and an Aharonov Bohm field. The main new ingredient of our result is that we subtract the threshold of the essential spectrum. We also prove the asymptotical behavior of the corresponding constant in the Hardy inequality in the limit of weak fields. In Section 4 we prove the stability of the essential spectrum of the operator i + A in locally deformed and curved waveguides for certain magnetic potentials; see Theorem 4.1. In Section 5 we use the Hardy inequalities to prove that the spectrum of i + A is stable under weak enlargement of the waveguide. We also give an asymptotical estimate on the critical strength of the deformation, for which the discrete spectrum of i + A will be empty. In particular, if the magnetic field equals B, then the critical strength of the deformation is proportional to as. Moreover, we prove by a trial function argument that the same behavior of with another constant, is sufficient also for the presence of eigenvalues below the essential spectrum; see Theorem 5.. The latter shows that the order of in our estimate is optimal. Locally curved waveguides are studied in Section 6. We consider a waveguide with the curvature, where is a positive parameter and is a fixed smooth function with compact support. Similarly as in Section 5, we show that there exists a such that for all < there will be no discrete spectrum of i + A. The behavior of in the limit of weak fields is at least proportional to,as.. The Main Results Here we formulate the main results of the paper without giving any explicit estimates of the involved constants. Explicit formulae for these constants are given in the respective sections..1. Hardy-Type Inequalities We state a Hardy inequality for magnetic Dirichlet forms separately for the case of an Aharonov Bohm field and for a locally bounded field. The following notation will be used. Let be the strip given by =, B r p be the open ball centred at p with radius r, and p r = 1 Bx ydx dy (.1 B r p

54 Ekholm and Kovařík be the flux of B through the ball B r p. We will say that A = a 1 a is a magnetic vector potential associated with a magnetic field B L loc if B = curl A in the distributional sense and a 1 a L loc. Let HA 1 denote the completion of in the norm C u HA 1 =u + i + L Au L (. Theorem.1. Let B L loc be a real-valued magnetic field. Assume that there is a ball B R p, with p = x y such that the function p r is not identically zero for r R. Then u c H 1 + x x dx dy ( i + Au u dx dy (.3 holds for all u HA 1, where A is a magnetic vector potential associated with B and c H is a positive constant given in (3.34. Theorem.. Assume there is a point p = x y and a radius R, such that B R p. Let A be a magnetic vector potential, such that A L loc \B Rp, and for x y B R p we have ( Ax y = y + y x x + y y x x x x + y y (.4 where \. Then u dx dy c AB x x + y y ( i + Au u dx dy (.5 holds for all u H 1 A \p, where c AB is a positive constant given in (3.66. Remark. With the translational invariance in mind, we will throughout the paper assume that x =... Geometrically Deformed Waveguides in a Magnetic Field As an application of Theorem.1 and Theorem. we prove stability results for the spectrum of the magnetic Schrödinger operator under geometrical perturbations of the waveguide. We will consider the following two cases...1. Local Enlargement. Let f be a nonnegative function in C 1,, and put = { s t <t<+ fs } (.6 Let M d be the self-adjoint operator associated with the closed quadratic form i + Au ds dt (.7

Stability of the Magnetic Schrödinger Operator in a Waveguide 543 on HA 1, where A is either the magnetic vector potential for the Aharonov Bohm field, given by ( y + y Ax y = x + y y x \ (.8 x + y y or a magnetic vector potential associated with a magnetic field B L, such that B is nontrivial in. Then the following statements hold: Theorem.3. Suppose that B L, such that B is nontrivial in. Then there is a positive number depending on f, f, a 1 and a such that for the discrete spectrum of M d is empty. Theorem.4. Let A be defined by (.8. Then there exists a positive number depending on f and f such that for the discrete spectrum of M d is empty.... Curved Waveguides. Let a and b be real-valued functions in C. Define the set = s t s = ax yb x t = bx + ya x where x y (.9 where is to be explained below and =. We assume the normalization a x + b x = 1 (.1 for all x. The boundary of for which y = is a curve given by and the signed curvature of is given by = ax bx x (.11 x = b xa x a xb x (.1 Assume that C 1 and let the natural condition x > 1 hold for all x. We prohibit to be self-intersecting. To be able to study mildly curved waveguides we consider the waveguide with curvature, where is a positive number and is a given curvature. For the sake of simplicity we will write instead of. Let M c be the self-adjoint operator associated with the closed quadratic form i + Au ds dt (.13 on H 1 A, where A is either the magnetic vector potential for the Aharonov Bohm field given in (.8 or a magnetic vector potential associated with a magnetic field B L, such that B is nontrivial in. Then the following statements hold: Theorem.5. Suppose that B L, such that B is nontrivial in. There exists a positive number depending on,, a 1, and a such that for the discrete spectrum of M c is empty.

544 Ekholm and Kovařík Theorem.6. Let A be as given in (.8. There exists a positive number depending on and such that for the discrete spectrum of M c is empty. 3. Hardy-Type Inequalities In this section we will prove Theorems.1 and.. Since one would like to know how the corresponding constants depend on the physical parameters of the model, we will keep the calculations as explicit as possible, although it makes the proofs rather technical. 3.1. Proof of Theorem.1 The substitution ux y = vx y sin y transforms inequality (.3 into v sin y c H 1 + x x dx dy i + Av sin ydxdy (3.1 for all v HA 1. This inequality and hence also Theorem.1 will be proved in three steps. First we prove an inequality similar to (3.1 in a ball surrounding a certain point of the strip, see Lemma 3.1. This result will then be extended to a box enclosing the given ball; see Lemma 3.; and finally to the whole strip. 3.1.1. Step 1. Hardy Inequality in a Circle. Lemma 3.1. Let the magnetic field B and the ball B R p be given as in Theorem.1. Then u dx dy c 1 i + Au dx dy (3. B R p B R p holds for all u H 1 A B Rp, where A is any magnetic vector potential associated with B and the constant c 1 is given by (3.15. Proof. We can construct a magnetic vector potential Ax y = a 1 x y a x y associated with B this way: a 1 x y = y y a x y = x 1 1 Btx ty y + y t dt (3.3 Btx ty y + y t dt (3.4 Then curl A = x a y a 1 = B in the distributional sense and Ax y x y y = for all x y. Owing to gauge invariance we can assume that the components of A are given by (3.3 and (3.4. Let r be polar coordinates centred at the point p. In these coordinates inequality (3. is equivalent to ( u rdrd c 1 ur + r iu + rar u rdrd (3.5 B R p B R p where ar = A sin cos.

Stability of the Magnetic Schrödinger Operator in a Waveguide 545 For fixed r we consider the operator K r = i + rar in L, which was studied in Laptev and Weidl (1999. The operator K r is self-adjoint on the domain H 1 with periodic boundary conditions. The spectrum of K r is discrete, and the eigenvalues { } and the orthonormal set of eigenfunctions k k= k k= are given by k = k r = k + r ar d = k + p r (3.6 and k r = 1 e i k ir ars ds (3.7 The quadratic form of K r satisfies the inequality r u d for all ur H 1, where r = dist p r. Thus B R p iu + rau d (3.8 r u rdrd r iu + rau rdrd (3.9 B R p holds for all u H 1 B R p. Define the function R 1 by ( r = r where r = max r R r 1 (3.1 r Since p is piecewise continuous differentiable and p = it follows that is well-defined. It is clear that r 1 and that there exists at least one r R such that r = 1. Let v H 1 R such that vr = ; then we have the inequalities R r vr rdr R3 3R r + r 3 6r R r v r rdr (3.11 and r vr rdr r r v r r dr (3.1 j1 where j 1 is the first zero of the Bessel function J. Using (3.11 and (3.1, we conclude that ( u rdrd u +1 u rdrd B R p B R p r iu + rau rdrd B R p

546 Ekholm and Kovařík where + + c c 1 B R p B R p B R p ( r j 1 r 1 u rdr + R3 3R r + r 3 6r R r iu + rau rdrd u +u r r dr d r 1 u rdr d ( ur + r iu + rau rdrd (3.13 c = 4 max { j 1 r 6r 1 R 3 3R r + r 3 } (3.14 c 1 = max { + 4c c 4 c } (3.15 c = max r r r r (3.16 r R Remark. Note that the constant c 1 1 gives a lower bound of the lowest eigenvalue of the magnetic Schrödinger operator on the disc of radius R with magnetic Neumann boundary conditions. 3.1.. Step. Extension to a Box. Lemma 3.. Let A, B, and B R p be given as in Theorem.1. Put R = R R ; then R ( R x u sin ydxdy c 4 R i + Au sin ydxdy (3.17 holds for all u H 1 R, where c 4 is a positive constant given in (3.6. Proof. The operator d 1 on the domain { u H 1 dy H uy = } is greater than or equal to c 3 = min { y y } 1 (3.18 In terms of quadratic forms this means that for v in H 1 satisfying vy = we have vy sin ydy c 1 3 v y sin y dy (3.19 Let u H 1 R and let 1 be defined by y y if h y = x<y<h + x R x 1 otherwise. (3.

Stability of the Magnetic Schrödinger Operator in a Waveguide 547 where h ± x = y ± R x. We write u = 1 u + u and use (3.19 to obtain u sin ydy h+ x h x + 4 c 3 ( 1 u sin ydy h+ u y sin x ydy+ h x u sin ydy (3.1 R x Hence we get R R x u sin ydydx R c 3 + 4 + 4R c 3 c 3 B R p u sin ydxdy R u y sin ydydx (3. for all u H 1 R. In particular it holds for u, where u C R, hence R R x u sin ydydx R c 3 + 4 + 4R c 3 c 3 B R p u sin ydxdy R u y sin ydydx (3.3 We can estimate the first term on the r.h.s. by Lemma 3.1 and the second term by the diamagnetic inequality (see, for instance, Kato, 1973; Simon, 1979; Avron et al., 1978 or Hundertmark and Simon, 4 saying that ux y i + Aux y (3.4 holds almost everywhere. We get R x u sin R c ydxdy 1 c 3 + 4c 1 R c 3 cos y +R i + Au B sin ydxdy R p + 4R i + Au sin ydxdy c 3 R = c 4 i + Au sin y dx dy (3.5 R for all u C R, with c 4 = R c 1 c 3 + 4c 1 + 4R (3.6 c 3 cos y +R The theorem now follows by continuity.

548 Ekholm and Kovařík 3.1.3. Step 3. Extension to the Strip. Proof of Theorem.1. saying that We need the classical one-dimensional Hardy inequality u t dt 4 u dt (3.7 holds for any u H 1, such that u = (see Hardy, 19. Let m = R and define the mapping 1 by 1 if x >m x = x (3.8 if x <m m Let u H 1. By writing u = u + u1 and using (3.7 we obtain where u sin y u dx dy sin ydxdy+ u1 sin ydxdy (3.9 1 + x x ( 16 ux +u sin ydxdy+ u sin ydxdy m 16 u x sin ydx dy+ c 5 R x u sin y dx dy R c 5 = (3.3 64 + 4R R 4 (3.31 Equation (3.9 holds in particular for u =v, where v is any function from C with bounded support. Thus by (3.4 and Lemma 3. we get where v sin y dx dy c 1 + x 6 i + Av sin y dx dy (3.3 c 6 = 16 + c 4 c 5 (3.33 Since the space of functions from C with bounded support is dense in HA 1, we conclude that inequality (3.3 holds for all v HA 1. In view of (3.1 the statement of the theorem follows with c H = c 1 6 (3.34

Stability of the Magnetic Schrödinger Operator in a Waveguide 549 3.. Weak Magnetic Fields Here we discuss the behaviour of the constant c H in (.3 for weak magnetic fields. Let pb r be the flux of the magnetic field B through the ball B r p defined in (.1 and define the following constants: ( 1 k 1 = max r 1 pb r (3.35 r R k = max r r pb r pbr (3.36 r R k 4 = R c 3 + 4k 1 + 4c k 4 1 k c 3 cos y +R (3.37 Theorem 3.3. If we replace the magnetic field B in Theorem.1 by B, where, then the constant c H in (.3 satisfies the estimate for. c H 1 k 4 c 5 + 4 (3.38 Proof. First note that the constants c, c 3 and c 5 are independent of. As we get c 1 = k1 + 4c k1 4k and c = k. This implies that c 4 = k 4 + 1 and therefore (3.38 holds. 3.3. Proof of Theorem. We will again make use of the substitution ux y = vx y sin y. This transforms inequality (.5 into u sin ydxdy c AB i + Au sin y dx dy (3.39 x + y y The proof of Theorem. will again be done in three steps. 3.3.1. Step 1. Lemma 3.4. Let the magnetic vector potential A and the ball B R p be given as in Theorem.. Then the inequality cos ( y iu + Au sin ydxdy + p x y u dx dy B R p B R p x + y y (3.4 holds true for all u C B R p such that u = in a neighbourhood of p, where = min k k (3.41 Proof. Let us introduce polar coordinates centred at the point p and put D n = { r n 1RN 1 <r<nrn 1}, where N is a natural number. Assume that

55 Ekholm and Kovařík u C B R p such that u = in a neighbourhood of p. In each D n we have iu + Au sin ydxdy D n ( = ur + r iu + u sin y + r sin r dr d D n ( cos y + nr dr d iu N + u (3.4 D n r To study the form on the r.h.s. of (3.4 we make use of the one-dimensional selfadjoint operator K on L given by defined on the set K = i + (3.43 K = { u H 1 u = u } (3.44 The spectrum of K is discrete and its eigenvalues k k orthonormal system of eigenfunctions k k are given by and the complete k = k + (3.45 and k = 1 e i k (3.46 We can write the function u in the Fourier expansion ur = k k r k (3.47 Then we have r 1 iu + u dr d r D n Dn 1 k k k d dr nrn 1 k r 1 k n 1RN 1 k dr k D n r 1 u dr d (3.48 Finally we sum up the inequality over the rings. For any N we have N ( iu + Au sin ydxdy cos y + nr dr d iu B R p N + u n=1 D n r N ( cos y + nr dr d u N n=1 D n r N ( cos y + r + R dr d u n=1 D n N r (3.49 Hence the desired result follows as N.

3.3.. Step. Stability of the Magnetic Schrödinger Operator in a Waveguide 551 Lemma 3.5. The inequality uy sin ydy y y c 7 u y sin ydy (3.5 holds true for all functions u H 1 such that uy =, where 4 c 7 = max { y y } (3.51 Proof. It is clear that From the Hardy inequality, 1 vy dy 4 y min { y y } d (3.5 dy v y dy (3.53 for all functions v H 1 such that v =, where is any positive number, it follows that 1 4y y d (3.54 dy for v H 1 satisfying vy =. The estimates (3.5 and (3.54 imply that 1 ( c y y 7 d dy 1 (3.55 which in terms of the quadratic form means that vy dy y y c 7 v y vy dy (3.56 holds for all v H 1 such that vy =. The substitution vy = uy sin y implies that u H 1 and that uy =. From (3.56 we get uy sin ydy y y c 7 u y sin ydy (3.57 for functions u H 1 such that uy =. 3.3.3. Step 3. Proof of Theorem.. Because of density arguments it will be enough to establish inequality (3.39 for u C with bounded support such that u = in a neighborhood of the point p.

55 Ekholm and Kovařík Assume x R R and put h ± x = y ± R x. Let be defined by (3.. Since ux H 1, we have u sin ydy x + y y c 7 4c 7 by Lemma 3.5. Thus the inequality u y + u sin ydy+ h+ x h x u sin ydy x + y y u y sin ydy+ ( + 4c 7R h+ x u sin ydy R x h x x + y y (3.58 R x u sin ydy x + y y 4c 7 R u y sin ydy h+ + R x u sin ydy 1 + c 7 (3.59 h x x + y y holds true. Let R = R R. We integrate w.r.t. x to get R R x u sin ydxdy x + y y 4c 7 R u y sin ydxdy R + R u sin ydxdy 1 + c 7 (3.6 B p R x + y y By continuity this inequality can be extended for functions in H 1 \p with Dirichlet boundary condition at the point p. In particular it holds if u =w, where w C with bounded support such that w = in a neighborhood of the point p. By Lemma 3.4 and (3.4 we have R R x w sin ydxdy x + y y c 8 R i + Aw sin y dx dy (3.61 where c 8 = 4R c 7 + R + 4R c 7 (3.6 cos y +R Put m = R and let be given by (3.8. We write w = w + w1 and use (3.7 to get w sin ydxdy 16 w x + y y x sin ydxdy+ 16 w sin ydxdy m w sin ydxdy + x + y y m 16 w x sin ydxdy+ c 9 m w sin ydxdy x + y y (3.63

Stability of the Magnetic Schrödinger Operator in a Waveguide 553 where c 9 = 18 + 3. Inequality (3.63 can be extended by continuity to functions R from H 1 \p with Dirichlet boundary condition at the point p. Finally, if w = v, where v C with bounded support such that v = in a neighbourhood of the point p, we get by (3.4 that Using inequality (3.61 we have v sin ydxdy 16 i + Av sin ydx dy x + y y v sin ydxdy +c 9 (3.64 x + y y m v sin ydxdy c x + y y 1 i + Av sin y dx dy (3.65 where the constant c 1 = 16 + c 8c 9. This proves inequality (3.39 with R R cos ( y + R c AB = 8 ( (3.66 R + c 7 + 1 + c 7 9R + 16 where and c 7 are given by (3.41 and (3.51, respectively. 4. Stability of the Essential Spectrum We will prove that under some assumptions on the magnetic vector potential a magnetic field will not change the essential spectrum of the Laplacian. Let be a subset of with piecewise continuously differentiable boundary and let us assume that there is a bounded set such that \ consists up to translations and rotations of two half strips 1 and. By a half strip we denote the set. Let M be the self-adjoint operator associated with the quadratic form i + Au dx dy (4.1 on the domain HA 1, where A will be specified below. Theorem 4.1. If the magnetic vector potential A = a 1 a is such that a j L loc for j = 1 and the functions A and div A exist in and are in L, then ess M = 1 (4. Proof. We can without loss of generality assume that =. To prove that 1 ess M we construct Weyl sequences. Assume that is a nonnegative real number. Let h n n=1 be a singular sequence of real-valued test functions for the operator d in L at such that supp h dx n n and such that

554 Ekholm and Kovařík h n and h n are uniformly bounded in n. For instance, let C be a non-negative function such that L = 1 and supp 1 1. Let if x<nor x n x n x = nn 1 nn + 1 if n x< n 1 (4.3 x nn 1 + n nn + 1 if x<n n 1 then h n can be chosen as a subsequence of n x cos x such that the functions from the subsequence have disjoint support. Construct the functions g n x y = h n x sin y (4.4 We will prove that g n is a singular sequence for M at 1 +. Clearly g n M and g n L = h n x sin ydxdy= h n L > (4.5 for every n. Let u be any function in L ; then u g n L = sin y n h n xux ydx dy (4.6 The latter follows since u yis in L for a.e. y. Finally we must show that M + 1g n, as n. There is a constant c depending on h n and h n such that ( M 1 + g n = c L h n h n dx (4.7 n + A +div A dxdy (4.8 as n. We have proved that 1 + ess M for all nonnegative, i.e., 1 ess M. To prove the reverse inclusion, ess M 1, it will be enough to prove that inf ess M 1. We study the operator M N being M with additional magnetic Neumann boundary conditions at the intersections 1 and. Then M N can be written as a direct sum of three operators, M 1 M M in L 1 L L. Since the magnetic vector potential is in L loc, it follows from Hundertmark and Simon (4 and Theorem. in Avron et al. (1978 that the spectrum of M is discrete. By the maximin principle we have n inf ess M inf ess M N min { inf M 1 inf M } (4.9 By the diamagnetic inequality we get inf M j inf DN = 1 (4.1

Stability of the Magnetic Schrödinger Operator in a Waveguide 555 for j = 1, where DN denotes the Laplace operator in L j with Dirichlet boundary conditions at j and Neumann boundary conditions at the intersection j. Hence the proof is complete. Corollary 4.. Assume that the operator M is associated with a magnetic field B L, then essm = 1. Proof. We can choose such that Bx y = for all x y. In this case we can gauge away the magnetic vector potential in. It is clear that the assumptions of Theorem 4.1 hold. 5. Local Enlargement of the Waveguide As already mentioned in Section., we use the Hardy inequalities in order to prove the stability of the spectrum of the magnetic Schrödinger operator under small enlargements of the waveguide. We will consider the case of a bounded compactly supported magnetic field and the Aharonov Bohm field separately. Let f,, and M d be given as in Section..1. In Bulla et al. (1997 it was proven that the Friedrich extension of 1 defined on C had negative eigenvalues for all >. For small enough values of > there is a unique simple negative eigenvalue E, the function E is analytic at =, and E = ( fs ds + 3 (5.1 We will show that with a compactly supported magnetic field or an Aharonov Bohm field added to the model a certain strength of the perturbation is needed for those eigenvalues to appear. 5.1. Compactly Supported Fields We remark that if the magnetic field B has compact support then there exists a magnetic vector potential A = a 1 a such that a 1 a L ; choose, for instance, a 1 and a from (3.3 and (3.4. Proof of Theorem.3. From Corollary 4. we know that the essential spectrum of M d coincides with the half-line 1. It will be sufficient to prove that M d 1is nonnegative. We denote by d the quadratic form associated with M d, i.e., ( d = is + a 1 + i t + a ds dt (5. with d = H 1. Define to be the unitary operator given by U L L (5.3 U x y = 1 + fxx 1 + fxy (5.4

556 Ekholm and Kovařík The operator M d is unitary equivalent to the operator M = U M d U 1 (5.5 defined on the set U M d in L. The form associated with M is then given by = d U 1 (5.6 defined on the space = U d. For convenience let gs = 1 + fs, then = d U 1 ( = i s + a 1 s tgs 1 s gs 1 t = + i t + a s tgs 1 s gs 1 t ds dt (5.7 ( ig x gx x y i xx y + iyg x gx yx y +ã 1 x yx y + i gx yx y +ã x yx y dx dy where Ãx y = ã 1 x y ã x y = Ax gxy. Straightforward calculation gives ( = i x +ã 1 + i y +ã y g g x + x 1 ( g yg 4 g g x y + y x (5.8 + y g + 1 g y + i yg ã 1 + fã g y y dx dy Let be the quadratic form associated with the Schrödinger operator with the magnetic vector potential à in the space L. We have = L L ( y f f f + g y 1 ( f 4 g yf g x y + y x f + i yf ã 1 + fã g y y g x + x dx dy (5.9 Without loss of generality we can assume that 1. Let be the characteristic function of the support of f. The following lower bound holds true: L L (c ( 11 x + y + c 1 dx dy (5.1

Stability of the Magnetic Schrödinger Operator in a Waveguide 557 where the constants are given by ( 1 c 11 =f + +a f + + + a 1 f (5.11 c 1 = 1 4 f + 1 f + a 1 f +a f (5.1 By the pointwise inequality x + y ( i + Ã +Ã (5.13 and Theorem.1 we get ( L ( 1 c 11 L ( ch + c 131 + d dx dy (5.14 1 + x where d = max supp f and c 13 = 1 +a 1 +a c 11 + c 1 (5.15 and c H is the constant from (.3. Put = c H (5.16 c 13 1 + d then the right-hand side of (5.14 is positive for all. The following corollary is an immediate consequence of the previous theorem and Corollary 3.3. It shows the asymptotical behavior of for weak magnetic fields and the dependence on the distance between the magnetic field and the geometrical perturbation. Corollary 5.1. If we replace the magnetic field B by B, where, then k 4 k 13 c 5 1 + d + 4 (5.17 as, where k 13 = lim c 13 =f + f + 4 1 f + 1 + f (5.18 and the other constants are given in (3.33, (3.37, and (5.15. Corollary 5.1 gives us a lower bound on. In the next theorem we will present an upper bound on, i.e., we will give a relation between and for which bound states exist. Without loss of generality we assume that includes a small triangle spanned by the points s 1 s 1, and 1 + with s >.

558 Ekholm and Kovařík Theorem 5.. that Let the magnetic field B be replaced by B, where, and assume s + (5.19 4A L as, where A is any magnetic vector potential associated with B. Then the operator M d has at least one eigenvalue below the essential spectrum. Proof. Define the trial function introduced in Bulla et al. (1997, as follows: sin ye ( sx s x s <y< ( y x y = sin 1 + ( x <s <y< 1 + ( 1 x 1 x s s otherwise. (5. Let = L. A simple calculation gives = 1 s + 3 (5.1 for. In order to prove that the discrete spectrum of M d enough to show that the inequality is nonempty, it is i + A < 1 (5. is satisfied for certain values of and. By (3.3 and (3.4 it follows that A L. Since = 1, we have i + A + A = 1 s + A + 3 (5.3 Taking into account the fact that ( 1 = s + s + s (5.4 we get s 4A + (5.5 and the proof is complete. Corollary 5.1 together with Theorem 5. shows that the order in the asymptotical behavior of the constant c H given in Corollary 3.3 is sharp.

Stability of the Magnetic Schrödinger Operator in a Waveguide 559 5.. The Aharonov Bohm Field For simplicity we assume that supp f. Proof of Theorem.4. Since div A = and A L 1, it follows from Theorem 4.1 that the essential spectrum of M d equals 1. We will prove that M d 1 is nonnegative. Let U be the unitary mapping given by (5.3 and (5.4. The operator M d is unitary equivalent to M = U M d U 1 (5.6 defined on the set U M d in L. The quadratic form associated with M d is d = i s + a 1 + i t + a ds dt (5.7 defined on d = H 1 A. Hence the form associated with M is given by = d U 1 (5.8 defined on the space = U d. Put gs = 1 + fs and let be the quadratic form associated with the Schrödinger operator with the magnetic vector potential à in the space L, where Ãx y = ã 1 x y ã x y = Ax gxy. Without loss of generality we assume that 1. Similarly as in arriving to (5.9 we get = L L ( y f f f + g y 1 ( f 4 g L yf g x y + y x f + i yf ã 1 + fã g y y g x + x dx dy (c 14 ( x + y + ( c 15 + c 16 ã 1 +ã dx dy (5.9 where c 14 = f + 3f +f, c 15 = 1 4 f + 1 f, c 16 = f + f, and is the characteristic function of the support of f. We have L L ( c 14 i + à ( c14 + c + 15 d + x + y y + c 14 + c 16 ã 1 +ã dx dy (5.3

56 Ekholm and Kovařík where d = max supp f. We use the pointwise inequality x (ã 1 x y +ã x y 4 d + x + y y (5.31 to arrive at L c L 14 L c 17 d + c 18 x + y y dx dy (5.3 where c 17 = c 14 + c 15 + 4 c 14 + c 16 and c 18 = c 14 + c 15 + 4 c 14 + c 16. From Theorem. we conclude ( 1 L c 14 L ( cab + c 17d + c 18 dx dy x + y y for, where c AB is the constant from (3.66 and = c AB (5.33 c 17 d + c 18 6. Curved Waveguides Let, and M c be defined as in Section... Duclos and Exner (1995 gave a proof based on ideas of Goldstone and Jaffe (199 of existence of bound states below the essential spectrum for the Schrödinger operator in with Dirichlet boundary conditions, assuming that, see also Bulla and Renger (1995. The results are given in terms of the curvature and not of the functions a and b. These functions a and b can be constructed from uniquely up to rotations and translations from the identities x ( x1 ax = a + cos x dx dx 1 (6.1 x ( x1 bx = b + sin x dx dx 1 (6. Our aim is to prove that if we introduce an appropriate magnetic field into the system it will make the threshold of the bottom of the essential spectrum stable if the parameter is small enough. 6.1. Compactly Supported Field Proof of Theorem.5. From Corollary 4. we know that the essential spectrum of M c is 1. Again we will show that M c 1 is nonnegative.

Stability of the Magnetic Schrödinger Operator in a Waveguide 561 The quadratic form c associated with M c is given by ( c = is + a 1 + i t + a ds dt (6.3 on c = H 1. Define the unitary operator U L L (6.4 as U x y = 1 + yx ax yb x bx + ya x (6.5 The operator M c is unitary equivalent to the operator M = U M c U 1 (6.6 acting on the dense subspace M = U M c of the Hilbert space L.We calculate the quadratic form associated with M. Our change of variables gives us the Jacobian in the form s t x y = ( a x yb x b x + ya x b x a x (6.7 Hence we have s = 1 + y 1 a x x b x + ya x y t = 1 + y 1 b x x a x yb x y (6.8 and = c U 1 ( [ ix b x + ya x y = 1 + yx [ ib + x x + a x yb x y 1 + yx ]( x y +ã 1 x y 1 + yx ]( x y +ã x y 1 + yx 1 + yxdx dy (6.9 where Ãx y = ã 1 x y ã x y = Aax yb x bx + ya x. We continue without writing arguments of the functions and use the identities a a + b b =

56 Ekholm and Kovařík and a + b =, ( x = 1 + y ia ã 1 + b ã 1 + y x x + y i b + ya ã 1 + a yb ã 1 + y y y y 1 + y 3 x + x 1 + y y + y ( y + 41 + y + 4 41 + y +ã 1 +ã dx dy (6.1 We write the form q as a perturbation of = i x + a ã 1 + b ã + i y + b ã 1 + a ã dx dy (6.11 i.e., = L L ( y + y 1 + y x iya ã 1 + b ã x x ( iy b ã 1 + a ã + a ã 1 b ã 1 + y y y We can easily arrive at the estimate y + 1 + y 3 x + x + 1 + y y + y ( y 41 + y + dx dy (6.1 4 41 + y L L c 19 x + y + c dx dy (6.13 where is the characteristic function of the support of and c 19 = + 1 +a 1 +a + (6.14 ( 1 c = + 3a 1 + 3a + (6.15 By using Theorem.1 and (5.13 we get L ( 1 c 19 ( L ( ch + c 11 + d dx dy (6.16 1 + x

Stability of the Magnetic Schrödinger Operator in a Waveguide 563 where d = max supp and c 1 = 1 +a 1 +a c 19 + c (6.17 The right-hand side is positive if, with = c H (6.18 c 1 1 + d Hence the discrete spectrum of the operator M c is empty. The following corollary gives a lower bound on as a function of. Corollary 6.1. If we replace the magnetic field B by B, where, then k 4 c 5 k 1 1 + d + 4 (6.19 as, where k 1 = lim c 1 = + 4 + 1 + 3 (6. The constants are given in (3.33, (3.37, and (6.17. 6.. The Aharonov Bohm Field For simplicity we assume that supp. Proof of Theorem.6. Since div A = and A L 1, it follows from Theorem 4.1 that the essential spectrum of M c equals 1. It will be enough to prove that M c 1 is nonnegative. Denote by M the operator U M c U 1, where U is defined in (6.4 and (6.5. Let be the form associated with M defined on the domain = U c. Following the calculations in (6.9 (6.1, we get = L L (6.1 ( y + y 1 + y x iya ã 1 + b ã x x ( iy b ã 1 + a ã + a ã 1 b ã 1 + y y y y + 1 + y 3 x + x + 1 + y y + y ( y 41 + y + dx dy 4 41 + y Without loss of generality we can assume that 1. Hence L L (6. c x + y + c 3 + c 4 ã 1 +ã dx dy

564 Ekholm and Kovařík where c = 3 + + (6.3 c 3 = 1 + (6.4 c 4 = 1 + (6.5 By the inequality (5.13, Theorem., and the fact that xã 1 x y +ã x y d + (6.6 disty x + y y where d = max supp, we obtain with If we choose L ( 1 c L ( cab c 5 dx dy (6.7 x + y y c 5 = d + c + c 3 + disty c + c 4 (6.8 = c AB c 5 (6.9 it follows that the right hand side of (6.7 is positive. Acknowledgments We would like to thank Timo Weidl for his permanent support and numerous stimulating discussions throughout the project. Many useful comments and remarks of Denis I. Borisov and Pavel Exner are also gratefully acknowledged. T.E. has been partially supported by ESF programme SPECT. References Avron, J., Herbst, J., Simon, B. (1978. Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45:847 883. Balinsky, A. A. (3. Hardy type inequalities for Aharonov Bohm magnetic potentials with multiple singularities. Math. Res. Lett. 1:169 176. Balinsky, A. A., Evans, W. D., Lewis, R. T. (in press. On the number of negative eigenvalues of Schrödinger operators with an Aharonov Bohm magnetic field. mp_arc -45. Balinsky, A. A., Laptev, A., Sobolev, A. V. (4. Generalized Hardy inequality for the magnetic Dirichlet forms. J. Stat. Phys. 116(1/4:57 51. Birman, M. S., Solomyak, M. Z. (1987. Spectral Theory of Self-Adjoint Operators in Hilbert space. Dordrecht: D. Reidel.

Stability of the Magnetic Schrödinger Operator in a Waveguide 565 Birman, M. S., Solomyak, M. Z. (199. Schrödinger operator. Estimates for number of bound states as function-theoretical problem. Amer. Math. Soc. Transl. ( 15:1 54. Blankenbelcer, R., Goldberger, M. L., Simon, B. (1977. The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians. Ann. Phys. 18:69 78. Borisov, D., Exner, P., Gadyl shin, R. R., Krejčiřík, D. (1. Bound states in weakly deformed strips and layers. Ann. Henri Poincaré :553 57. Bulla, W., Renger, W. (1995. Existence of bound states in quantum waveguides under weak conditions. Lett. Math. Phys. 35(1:1 1. Bulla, W., Gesztesy, F., Renger, W., Simon, B. (1997. Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 15(5:1487 1495. Duclos, P., Exner, P. (1995. Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7:73 1. Evans, W. D., Lewis, R. T. (4. On the Rellich inequality with magnetic potentials. mp_arc 4-93. Exner, P., Šeba, P. (1989. Bound states in curved quantum waveguides. J. Math. Phys. 3:574 58. Goldstone, J., Jaffe, R. L. (199. Bound states in twisting tubes. Phys. Rev. B45:141 1417. Hardy, G. H. (19. Note on a theorem of Hilbert. Math. Zeit. 6:314 317. Hundertmark, D., Simon, B. (4. A diamagnetic inequality for semigroup differences. J. Reine Angew. Math. 571:17 13. Kato, T. (1973. Schrödinger operators with singular potentials. Israel J. Math. 13:135 148. Laptev, A., Weidl, T. (1999. Hardy inequalities for magnetic Dirichlet forms. Oper. Theory Adv. Appl. 18:99 35. Mantoiu, M., Richard, S. (. Absence of singular spectrum for Schrödinger operators with anisotropic potentials and magnetic fields. J. Math. Phys. 41:73 74. Melgaard, M., Ouhabaz, E.-M., Rozenblum, G. (4. Spectral properties of perturbed multivortex Aharonov-Bohm Hamiltonians. Ann. Henri Poincaré 5:979 11. Simon, B. (1979. Maximal and minimal Schrödinger forms. J. Operator Theory 1:37 47. Weidl, T. (1997. Remarks on virtual bound states for semi-bounded operators. Comm. PDE 4(1:5 6.