WEAKLY COUPLED BOUND STATESIN QUANTUM WAVEGUIDESW. Bulla1, F. Gesztesy2, W. Renger2, and B. Simon3Abstract. We study the eigenvalue spectrum of Dirichlet Laplacians which modelquantum waveguides associated with tubular regions outside of a bounded domain.Intuitively, our principal new result in two dimensions asserts that any domain obtained by adding an arbitrarily small \bump" to the tube 0 = R � (0; 1) (i.e., % 0, � R2 open and connected, = 0 outside a bounded region) producesat least one positive eigenvalue below the essential spectrum [�2;1) of the DirichletLaplacian ��D . For jn0j su�ciently small (j : j abbreviating Lebesgue measure),we prove uniqueness of the ground state E of ��D and derive the \weak coupling"result E = �2 � �4jn0j2 +O(jn0j3). As a corollary of these results we obtainthe following surprising fact: Starting from the tube 0 with Dirichlet boundaryconditions at @0, replace the Dirichlet condition by a Neumann boundary condi-tion on an arbitrarily small segment (a; b) � f1g, a < b of @0. If H(a; b) denotesthe resulting Laplace operator in L2(0), then H(a; b) has a discrete eigenvalue in[�2=4; �2) no matter how small jb� aj > 0 is.x1. IntroductionOur goal in this paper is to study the bound state spectra of the Dirichlet Lapla-cian ��D for open regions � Rn which are tubes outside of a bounded region(quantum waveguides). (Following the traditional notation in quantum physics, wedenote the Laplacian by �� as opposed to � in the following.) In particular, let0 � R2 be de�ned by 0 = R � (0; 1):Consider open connected sets such that:(i) For some R > 0, \ fx 2 R2 j jxj > Rg = 0 \ fx 2 R2 j jxj > Rg.(ii) 0 � , 0 6= .Because of condition (i),�ess(��D ) = �ess(��D0) = [�2;1): (1)Then one of our main goals will be to prove1 Institute for Theoretical Physics, Technical University of Graz, A-8010 Graz, Austria. E-mailfor W.B.: bulla@itp.tu-graz.ac.at2 Department of Mathematics, University of Missouri, Columbia, MO 65211. E-mail for F.G.:mathfg@ mizzou1.missouri.edu; e-mail for W.R.: walter@mumathnx3.cs.missouri.edu3 Division of Physics, Mathematics, and Astronomy, California Institute of Technology,Pasadena, CA 91125. This material is based upon work supported by the National Science Foun-dation under Grant No. DMS-9401491. The Government has certain rights in this material.To be submitted to Commun. Math. Phys. Typeset by AMS-TEX1

2 W. BULLA, F. GESZTESY, W. RENGER, AND B. SIMONTheorem 1.1. If obeys (i), (ii), then ��D has at least one eigenvalue in (0; �2).Actually, the eigenvalue lies in [ �24R2 ; �2) since � R � (�R;R) impliesinf spec(��D ) � inf spec(��DR�(�R;R)) = �24R2 .We will focus especially on the particular case = �;where � = f(x; y) 2 R2 j 0 < y < 1 + �f(x)g (2)and where f is a C1(R) function of compact support with f � 0. Since2 inf spec(��D ) decreases as increases and every obeying (i), (ii) has 0 �� � for some f , it su�ces to prove Theorem 1.1 for � of the form (2). Indeed,it su�ces to prove the result for � su�ciently small.We will prove a much more detailed result in these � regions for � small enough.Actually, we can replace f � 0 by the weaker requirement that RR f(x) dx > 0.Theorem 1.2. Let � be given by (2) where f is a C10 (R) function withRR f(x) dx > 0. Then for all small positive �, ��D� has a unique eigenvalueE(�) in (0; �2), it is simple, E(�) is analytic at � = 0, andE(�) = �2 � �4�2�ZR f(x) dx�2 + O(�3): (3)This is the main result of this paper, which we'll prove in Section 2 using acalculation in the appendix. The technique used in our proof is closely patternedafter the theory of bound states of � d2dx2 + �V (x) for � small as developed in[2],[9],[10],[13]. The key idea there is that (� d2dx2 + k2)�1 has a well-behaved limitas k # 0 except for a divergent rank one piece. In exactly the same way, (��D0 ��2 + k2)�1 has a nice limit as k # 0 except for a rank one piece.Theorem 1.1 (or 1.2) leads to the following remarkable result which, roughlyspeaking, asserts that if on an arbitrarily small segment in the boundary @0 of0 the original Dirichlet boundary condition is replaced by a Neumann boundarycondition, at least one additional eigenvalue is instantly created in the interval(0; �2).Corollary 1.3. Let 0 = R� (0; 1) and denote by H(a; b) in L2(0) the Laplacianon 0 with a Neumann boundary condition on the segment (a; b)�f1g; �1 < a <b <1, and Dirichlet boundary conditions on @0nf(a; b)�f1gg. Then H(a; b) hasa discrete eigenvalue in [�24 ; �2) no matter how small jb� aj > 0 is.Proof. Clearly H(a; b) � 0 and �ess(H(a; b)) = [�2;1). Enlarge 0 to � of thetype (2) with � > 0 su�ciently small and some 0 � f 2 C1(R) with supp(f) =[a; b], f > 0 on (a; b). By Theorem 1.1, ��D� has at least one eigenvalue E� 2(0; �2). Next, decouple 0 and �n0 by a Neumann boundary condition alongthe segment (a; b) � f1g. Denoting the resulting Laplace operator by H� , we

WEAKLY COUPLED BOUND STATES IN QUANTUM WAVEGUIDES 3obtain the direct sum decomposition H� = H(a; b) � �e�(a; b) with respect toL2(�) = L2(0) � L2(�n0), where �e�(a; b) has Dirichlet (resp. Neumann)boundary conditions on @�n@0 (resp. (a; b) � f1g). By Neumann decoupling(see, e.g., [11], p.270)0 � inf spec(H�) � inf spec(��D�) � E� < �2:Choosing f appropriately such that inf spec(�e�(a; b)) > �2 (e.g., choose f suchthat �n0 is a smoothed out rectangle of the type (a; b)� (1; 1+ c) with 0 < c�jb� aj), one obtains0 � inf spec(H(a; b)) � inf spec(��D�) � E� < �2:That actually inf spec(H(a; b)) � �24 follows as in the proof of Corollary 1.4. �We have a number of remarks concerning Theorem 1.2:(1) � RR f(x) dx is exactly the area of �n0.(2) In thinking about the higher-dimensional analogs, one needs to realize thereare two independent dimensions in the above examples: the dimension of the crosssection and the number of unbounded dimensions. In general, one can considerK � Rn , a bounded connected open set and 0 = R` �K. With minor changes,our analysis extends to general (n;K) so long as ` = 1, that is, for 0 a long tube.(3) In the notation of point (2), the results are ` dependent. For ` = 2, thatis, 0 a long slab, there are still weakly coupled states, but as in [13], the bindingis only O(e�c=�). For ` � 3, there will be no bound state if too small a bump isadded.(4) If one uses the one-dimensional Schr�odinger operator [13] as a guide, onemight guess that if RR f(x) dx = 0, then ��D� has a bound state for all su�cientlysmall � but since ��D� has second-order terms not found in the one-dimensionalcase, that is not totally clear.(5) However, if RR f(x) dx < 0, then by our analysis, ��D� has no spectrum in[0; �2) if � is too small.(6) Since � isn't monotone if f isn't positive, we cannot be sure that if f issomewhere negative then �=1 has bound states even if RR f(x) dx > 0. Indeed, iff is very close to �1 on a long region, we expect that ��D�=1 has no bound states.(7) We owe to Mark Ashbaugh the following observation:Corollary 1.4. Let ~ = fR� (0; 2)gnfR�f1gg[f(a; b)�f1gg, �1 < a < b <1(i.e., ~ consists of two copies of 0 with the boundary between them removed in(a; b)�f1g) and denote by ��D~ the associated Dirichlet Laplacian in L2(~). Then��D~ has a discrete eigenvalue in [�24 ; �2) independently of the size jb� aj > 0 ofthe slit (a; b)� f1g.Proof. ~ has re ection symmetry under y ! 2 � y. Thus, ��D~ is a direct sumof operators even and odd under this symmetry and so ��D~ �= H(a; b) � ��D0 ,where H(a; b) is the operator in Corollary 1.3 (since even is equivalent to Neumannand odd to Dirichlet boundary conditions) and �= abbreviates unitary equivalence.

4 W. BULLA, F. GESZTESY, W. RENGER, AND B. SIMONSince �ess(H(a; b)) = �ess(��D0) = [�2;1) = �ess(��D~ ), it su�ces to prove that��D~ has spectrum in [�24 ; �2).Let b = 0 [ f(x; y) 2 R2 j a < x < b; 0 < y < 2g. Then b � ~ soinf spec(��D~ ) � inf spec(��Db ) < �2 by Theorem 1.1. The lower bound thenfollows from ~ � R � (0; 2). �As is obvious from ��D~ �= H(a; b)���D0 , either one of Corollaries 1.3 and 1.4can be used to prove the other given the result of Theorem 1.1 (or 1.2). It seemsdi�cult, however, to prove Corollary 1.3 (or 1.4) directly, i.e., without the trick ofenlarging (or doubling) the domain and appealing to Theorem 1.1 (or 1.2).Remark 1.5. An alternative proof of Theorem 1.1 can be based on the followingtrial function argument. Without loss of generality assume that contains a smallneighbourhood of the point (0; 1). Thus there are a; b > 0 such that the trianglespanned by the points (�a; 1), (a; 1) and (0; 1 + b) is in . De�ne on �;�(x; y) = 8><>: sin(�y)e��(jxj�a); jxj > a; 0 < y < 1sin( �y1+�(1� jxja ) ); jxj � a; 0 < y < 1 + �(1� jxja );0; otherwise (4)where 0 < � < b and � > 0. This trial function certainly vanishes on @ and at 1and it is in the form domain Q(��D ). By a straightforward calculation we obtainE( �;�) = (r �;�;r �;�)( �;�; �;�) = �2(1� 2a��) +O(�2�) + O(�2):If we �rst choose � and then � small enough we getE( �;�) < �2 = inf �ess(��D ):Since inf spec(��D ) < E( �;�) and ��D > 0 this proves Theorem 1.1. Note thatsin(�y) in (4) represents the function u(x; y) in (7) used prominently in Lemma 2.2and in the proof of Theorem 1.2.Spectral properties of quantum waveguides received considerable attention re-cently. While a complete bibliography is beyond the scope of this paper, the in-terested reader is referred to [1],[3]{[7],[12] and the literature cited therein. In par-ticular, a weak coupling mechanism di�erent from the one discussed in the presentpaper, based on arbitrarily small bending of tubes, has been studied in detail in [4]and[12].Without entering into further details we remark that Theorem 1.1 admits a va-riety of extensions. For instance, and 0 need not coincide outside a sphere ofradius R as assumed in our condition (i), only needs to approach 0 asymptot-ically (still assuming condition (ii)) since equality of the essential spectra of ��Dand ��D0 as recorded in (1) is the crucial property in question. In addition, could have various further branches running o� to in�nity as long as the asymptotic

WEAKLY COUPLED BOUND STATES IN QUANTUM WAVEGUIDES 5width of these branches is less than or equal to one in order to guarantee the valid-ity of (1). Moreover, combining our results with the ones in [4] and [12] producesthe same ground state e�ect for a bent tube of constant width one (and again ad-ditional bent branches running o� to in�nity of asymptotic widths not larger thanone can be accommodated).x2. Weak Coupling AnalysisWe'll study ��D� by a perturbation method. Since L2(�) is � dependent, itis di�cult to use perturbation theory directly, so we'll map all the operators ontothe same space. Let U� : L2(�)! L2(0) by(U� )(x; y) =p1 + �f(x) (x; (1 + �f(x))y):Then U� is unitary and H� = U�(��D� )U�1� � �2acts in L2(0). We subtract �2 so that �ess(H�) = [0;1).A straightforward calculation found in the appendix (cf. (A.6)) proves thatH� = H0 + � 3Xi=1 A�iBi + �2 8Xi=4 A�iBi ; (5)where each Ai and Bi is a �rst-order di�erential operator with coe�cients whichhave compact support and (g is a C1 function chosen such that g � 1 on supp f)(i) A�1 = 2f(x) @@y ; B1 = g(x) @@y ;(ii) A�2 = f 00(x); B2 = g(x)�y @@y + 12�;(iii) A�3 = �2y @@y + 1�g(x); B3 = f 0(x) @@x(we'll see below that to leading order only A�1B1 matters).Rewrite (5) as follows. De�ne C(�), D : L2(0)! L2(0) C 8 by(C')i = � Ai' i = 1; 2; 3�Ai'; i = 4; : : : ; 8(D')i = Bi'; i = 1; : : : ; 8:Then (5) becomes H� = H0 + �C�(��)D:

6 W. BULLA, F. GESZTESY, W. RENGER, AND B. SIMONLemma 2.1. Let k 2 C , Re k > 0 and � 2 R. Then �k2 is an eigenvalue of H� ifand only if �D(H0 + k2)�1C� � Q�has �1 as an eigenvalue.Proof. If Q� � � , then ��(H0+ k2)�1C� � ' is seen to satisfy H�' = �k2'.Conversely, if H�' = �k2', then ' 2 Q(H�) � D(D) so = D' is in L2(0) andQ� = � . �Lemma 2.2. Let h be a C1 function of compact support in R. Thenh(H0 + k2)�1h = (hu; � )hu2k + A(k); (6)where u is the function u(x; y) = 2 12 sin(�y) (7)and A(k) is a bounded operator{valued function of k, which can be analyticallycontinued from fk 2 C j Re k > 0g to a region that includes k = 0. Indeed, even(H0 + 1)1=2A(k)(H0 + 1)1=2 has an analytic continuation into such a region.Moreover, (H0+1)1=2A(k)(H0+1)1=2 is bounded uniformly in fk 2 C j jArg kj <�=3g [ fa small disk about k = 0g (�3 can be replaced by any number strictly lessthan �2 ).Proof. Let H0 � L2(0) be the space of L2(0) functions of the form '(x) sin(�y),sin(�y) being chosen as the lowest eigenfunction of (� d2dy2 )D. Let P0 be the pro-jection onto H0. Then (H0 + k2)�1(1 � P0) has an analytic continuation into theregion fk 2 C j �k2 2 C n[3�2 ;1)g since the lowest point in the spectrum ofH0(1� P0) � (1� P0)L2(0) is 3�2.On the other hand, h(H0 + k2)�1P0h has the explicit integral kernel:(2k)�1h(x)h(x0)u(y)u(y0)e�kjx�x0j = a1(k) + a2(k);where a1(k) is obtained by replacing e�kjx�x0j by 1 and a2(k) by using e�kjx�x0j�1in its place. The �rst term is the explicit rank one piece in (6) and the second termis analytic as a Hilbert-Schmidt kernel at k = 0.It is easy to modify this argument to accommodate the extra factors of (H0+1)1=2and prove the boundedness. �Proof of Theorem 1.2. Consider �rst the operator onL2(0) C 8 : L0 = (C(� = 0)u; � )Du;where u is given by (7). Then L0 is a rank one operator, so it has a single eigenvalueat e0 = Tr(L0): (8)

WEAKLY COUPLED BOUND STATES IN QUANTUM WAVEGUIDES 7But Ci(0) = 0 for i = 4; : : : ; 8, B3u = 0, and (A2u;B2u) = 0 since RR f 00(x) dx = 0.It follows thate0 = (A1u;B1u) = �2�ZR f(x) dx�Z 10 2�2 cos2(�y) dy= �2�2 ZR f(x) dx:Let k = �`. Then by Lemma 2.2,�D(H0 + k2)�1C� = Q(�; `)has the form: Q(�; `) = 12` L� + �M(�; `);where(i) L� is rank one and L� = L0 + �~L.(ii) M(�; `) = DA(�; `)C�, where A is given by Lemma 2.2 and h is chosen suchthat h � 1 in a neighborhood of supp f .By Lemma 2.2, we are interested in when Q(�; `) has �1 as an eigenvalue for� > 0 and ` > 0. Since M is uniformly bounded in � on a sector about (0;1),this can happen where � is small if e02` is near �1, that is, ` is near �e02 > 0 sinceRR f(x) dx > 0 by hypothesis.For such ` and � small, Q(�; `) has exactly one eigenvalue near �1, call it E(�; `),which by eigenvalue perturbation theory ([8], Ch.2, [11], Ch.XII) is jointly analyticin �; `. Let F (�; `) = 2`(E(�; `) + 1):Since 2`Q(�; `)j�=0 is independent of `, @(2`E(�;`))@` ����=0 = 0 and so@F (�;`)@` ����=0;`=�e0=2 = 2 6= 0. It follows by the implicit function theorem that for �su�ciently small, there is an analytic function `(�) = � e02 +O(�) so that for � > 0and ` in the sector jArg `j < �3 , �1 is an eigenvalue of Q(�; `) if and only if ` = `(�).Since H� for � real has only real eigenvalues, `(�) must be real for � > 0. Thus H�has a unique eigenvalue, e(�), in (�1; 0) given by e(�) = ��2(� e02 )2 + O(�3) asclaimed. �Acknowledgments. It is a pleasure to thank Mark Ashbaugh and ThomasHo�mann-Ostenhof for numerous discussions on this subject. W. B. is indebtedto the Department of Mathematics of the University of Missouri, Columbia forthe hospitality extended to him during a stay in the spring of 1994. Furthermore,he gratefully acknowledges the �nancial support for this stay by the TechnischeUniversit�at Graz, Austria.

8 W. BULLA, F. GESZTESY, W. RENGER, AND B. SIMONAppendix: Calculating eH� = U�(��D�)U�1� .We use coordinates (x; y) on 0 and (s; u) on �. Thus U� becomesU� : L2(�) ! L2(0)~ (s; u) 7! (x; y) =p1 + �f(x) ~ (x; (1 + �f(x))y): (A.1)The coordinate transformation isx = s; y = (1 + �f(s))�1u: (A.2)The form associated with ��D� is given byqD� : Q(��D�)�Q(��D�) ! C('; ) 7! (r';r ); (A.3)where Q(��D�) = H1;20 (�) is the local Sobolev space (i.e., the completion of C10under the norm k �kr = (kr�k2+k �k2)1=2). By unitary equivalence, the quadraticform associated with eH� isq eH� : U�Q(��D�)� U�Q(��D�) ! C('; ) 7! qD�(U�1� ';U�1� ): (A.4)The form domain of eH� is U�H1;20 (�) = U�C10 (�), where the bar denotes com-pletion under the norm k � kq = (q eH�( � ; � ) + k � k2)1=2.Next we calculate the quadratic form q eH�('; ) for '; 2 C10 (0). We use theshorthand c(x) = 1 + �f(x) and use subscripts to denote partial derivatives.q eH�('; ) = qD�(U�1� ';U�1� )= Z� (@sc(s)�1=2 '(s; uc(s)) )(@sc(s)�1=2 (s; uc(s))) dsdu+ Z� (@uc(s)�1=2 '(s; uc(s)) )(@uc(s)�1=2 (s; uc(s))) dsdu= Z0 ���12 c0(x)c(x) '(x; y) + 'x(x; y)� yc0(x)c(x) 'y(x; y)���12 c0(x)c(x) (x; y) + x(x; y)� yc0(x)c(x) y(x; y)�+ 1c(x)2 'y(x; y) y(x; y)�dxdy

WEAKLY COUPLED BOUND STATES IN QUANTUM WAVEGUIDES 9= Z0 �'x(x; y) x(x; y) + 1 + y2c0(x)2c(x)2 'y(x; y) y(x; y)� yc0(x)c(x) ('x(x; y) y(x; y) + 'y(x; y) x(x; y))� c0(x)2c(x) ('x(x; y) (x; y) + '(x; y) x(x; y))+ yc0(x)22c(x)2 ('y(x; y) (x; y) + '(x; y) y(x; y))+ c0(x)24c(x)2 '(x; y) (x; y)�dxdy: (A.5)By partial integration we get the operatoreH� = � @2@x2 � 1 + y2�2f 0(x)2c(x)2 @2@y2 + �y�f 00(x)c(x) � 3y�2f 0(x)2c(x)2 � @@y+ 2y�f 0(x)c(x) @@x @@y + �f 0(x)c(x) @@x + �f 00(x)2c(x) � 3�2f 0(x)24c(x)2= ��D0 + ��2f(x) @2@y2 + yf 00(x) @@y + 2yf 0(x) @@x @@y + f 0(x) @@x + f 00(x)2 �� �2�3f(x)2 + 2�f(x)3 + y2f 0(x)2(1 + �f(x))2 @2@y2 + � yf(x)f 00(x)(1 + �f(x)) + 3yf 0(x)2(1 + �f(x))2� @@y+ 2yf(x)f 0(x)(1 + �f(x)) @@x @@y + f(x)f 0(x)(1 + �f(x)) @@x + f(x)f 00(x)2(1 + �f(x)) + 3f 0(x)24(1 + �f(x))2�:(A.6)Since we assumed f 2 C10 (R), clearly C10 (0) � D( eH�) = U�D(��D�).Actually, U�C10 (�) = C10 (0). From (A.5) we infer that the norm k � kq isequivalent to the norm k � kr, i.e.,Q( eH�) = U�C10 (�) = H1;20 (0): (A.7)References[1] M.S. Ashbaugh and P. Exner, Lower bounds to bound state energies in bent tubes,Phys. Lett. A 150 (1990), 183{186.[2] R. Blankenbecler, M.L. Goldberger, and B. Simon, The bound states of weakly coupled long{range one{dimensional quantum Hamiltonians, Ann. Phys. 108 (1977), 69{78.[3] P. Duclos and P. Exner, Curvature vs. thickness in quantum waveguides, Czech. J. Phys. 41(1991), 1009{1018;; erratum 42 (1992), 344.[4] P. Duclos and P. Exner, Curvature{induced bound states in quantum waveguides in two andthree dimensions, Rev. Math. Phys. 7 (1995), 73{102.[5] P. Exner, Bound states in quantum waveguides of a slowly decaying curvature, J. Math. Phys.34 (1993), 23{28.[6] P. Exner and P. �Seba, Bound states in curved quantum waveguides, J. Math. Phys. 30 (1989),2574{2580.

10 W. BULLA, F. GESZTESY, W. RENGER, AND B. SIMON[7] P. Exner, P.�Seba, and P. �S�tov���cek, On existence of a bound state in an L{shaped waveguide,Czech. J. Phys. B 39 (1989), 1181{1191.[8] T. Kato, Perturbation Theory for Linear Operators, corr. printing of the 2nd ed., SpringerVerlag, Berlin, 1976.[9] M. Klaus, On the bound state of Schr�odinger operators in one dimension, Ann. Phys. 108(1977), 288{300.[10] M. Klaus and B. Simon, Coupling constant thresholds in nonrelativistic quantum mechanics.I: Short{range two{body case, Ann. Phys. 130 (1980), 251{281.[11] M. Reed and B. Simon,Methods of Modern Mathematical Physics. IV. Analysis of Operators,Academic Press, New York, 1978.[12] W. Renger and W. Bulla, Existence of bound states in quantum waveguides under weakconditions, Lett. Math. Phys. 35 (1995), 1{12.[13] B. Simon, The bound state of weakly coupled Schr�odinger operators in one and two dimen-sions, Ann. Phys. 97 (1976), 279{288.