AP07_2-3.vp In this paper we consider an example of a quantum waveguide with a small ��-symmetric perturbation. The perturbed system is weakly non-self-adjoint and we employ general results of [1, 2] to study the problem. The main aim is to show that the technique suggested in the cited works can be used effectively in the perturbation theory for ��-symmetric operators. Let x x x� ( , )1 2 be Cartesian coordinates in � 2, � �� � � � �x x: � �2 22 be an infinite straight strip. By V V x� ( ) we denote a real-val- ued function defined on � having bounded support and belonging to L�( )� . We assume that it satisfies the following assumption, V x x V x x( , ) ( , )� � �1 2 1 2 , x � �. (1) The main object of our study is the operator �� �: � � � i V (2) on � subject to the Dirichlet boundary condition. We define it rigorously as an unbounded operator in L2( )� with the do- main W2 0 2 , ( )� , where the latter is a subspace of W2 2( )� of the functions vanishing on ��. The symbol � indicates a small positive parameter. The Dirichlet Laplacian on � is a closed operator in L2( )� and the multiplication operator by V is relatively bounded. Because of this the operator �� is closed. It can also be shown that it is m-sectorial. The main property of the operator �� is ��-symmetricity expressed by the identity � �� � * � �� � 1, where ( )( ) ( , )�u x u x x� � 1 2 . Our aim is to study the spectrum of the operator ��. We will focus our attention on the continuous, residual and point spectrum of this operator. We define these subsets of the spectrum in accordance with [3]. Namely, the continuous spectrum is introduced in terms of singular sequence, the point spectrum is the set of all eigenvalues, and the residual spectrum is the complement of the continuous and point spectrum with reference to the whole spectrum. Our first result describes the continuous and residual spectrum of H�. Theorem 1 The residual spectrum of �� is empty and the continuous one co- incides with �1, � . The proof is the same as the proof of similar results in [4]; we therefore do not give the proof here. It is well-known that the spectrum of operator �0 is purely continuous and coincides with �1, � . The small pertur- bation �V can generate an eigenvalue converging to the threshold of the continuous spectrum. Our second theorem deals with the existence and asymptotic behaviour of such an eigenvalue. Before formulating it, we introduce auxiliary notations. We denote � � � � � j j j x jx v x V x x x x u ( ): sin , ( ): ( ) ( ) ( ) , 2 2 1 1 2 2 2 0 1 2 � � � d ( ): ( ) , ( ): x x t v t t u x j ej j x t 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 � � � � � � � � � d � v t tj ( ) .1 1d � � Employing these functions, we define ~ ( ): ( ) ( ) ( ( ) ( )) ( ), ~( ) V x v x x V x v x x u x j j j � � � � � 1 2 2 1 1 1 2� � : ( ) ( ).� � � u x xj j j 1 2 2 � The function ~u is well-defined and belongs to W2 2( )� . It can be shown that it is an exponentially decaying solution to the problem � � � � � � � � ~ ~ ~, ~ , . u u V x u x0 � (3) Finally, we introduce a number K by the formula © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 57 Acta Polytechnica Vol. 47 No. 2–3/2007 On a Quantum Waveguide with a Small �� -symmetric Perturbation D. Borisov We consider a quantum waveguide with a small �� -symmetric perturbation described by a potential. We study the spectrum of such a system and show that the perturbation can produce eigenvalues near the threshold of the continuous spectrum. Keywords: waveguide, �� -symmetric potential, spectrum K u V u L L : ( ~ ,~) ( ) ( ) � � �1 2 2 2� � . We will show below that the first norm in this formula is well-defined. Theorem 2 If K>0, there exists the unique eigenvalue of the operator ��, converging to the threshold of the continuous spectrum. This eigen- value is simple, real and satisfies the asymptotic formula � � �� � � � 1 4 0 4 2 5K �( ), . (4) If K<0 , the operator �� has no eigenvalues converging to the threshold of the continuous spectrum as � � 0 . In particular, if V x v x( ) ( )� 1 1 (5) the number K is positive, and if v1 0� (6) the number K is negative. Proof. We introduce the numbers k V x x v x x k u v 1 2 1 2 2 2 : , : � � � � � � i ( ) (x ) d ( ) d 1 ( 1 2 1 1 1 1 1 2 � � � � ~~ ~~ .V u x u v x V u x) d d d1 1 1 � � � � �� � � � � �� � � � �� 1 2 � It follows from [2, Th. 1] that if Re k1 0� , or Re k1 0� , Re k2 0� , (7) there exists the unique eigenvalue of �� converging to the threshold of the continuous spectrum, and the asymptotics of this eigenvalue reads as follows � � � �� � �� � � � 1 0 2 1 2 2 3k k k k, ( ),� . (8) It also follows from [2, Th. 1] that if Re k1 0� , or Re k1 0� , Re k2 0� (9) the operator �� has no eigenvalues converging to the thresh- old as � � 0. Thus, it is sufficient to calculate the numbers k1, k2. The identity (1) implies that v1 is an odd function, and hence k1 0� . (10) Therefore, it is sufficient to calculate k2 and check its sign. The mean value of v1 being zero, the function u1 is constant as x1 is large enough. This allows us to write v u x u u x u x1 1 1 1 1 1 1 2 1d d d � � � � � �� � �� � �( ) . (11) Hence, k K 2 2 � . We substitute this formula and (10) into (8), and by (7), (9) we conclude that if K � 0 , there exists the unique eigenvalue of �� satisfying (4). If K � 0, the operator has no eigenvalues converging to the threshold of the continuous spectrum. Using (1), one can check easily that � is an eigenvalue of �� as well. It converges to the threshold and by the uniqueness of such an eigenvalue we conclude that this eigenvalue is real. Let us prove that the conditions (5), (6) are sufficient for the eigenvalue to be present or absent. Assume first (5). In this case ~ V � 0, ~u � 0 and K u L � � � 1 2 01 2 2 ( )� (12) If the relation (6) holds true, the function u1 is identically zero, and K V u L: ( ~ ,~) ( )� � 2 � Employing (3), by analogy with (11) in the same way we check ~~ ~ ~ ( ) ( ) V u x u u L L d � � �� � � �2 2 2 2 . (13) It follows from the definition of the function ~( , )u x1 � that � � � � ~ ( , ) ~ ( , ) ( , ) u x x u L L 2 1 0 2 0 2 2 2 4� � , and hence � �~ ~ ( ) ( ) u u L L2 2 2 2 4 � � . In view of (12), (13) and this estimate we conclude that K u L � � �3 0 2 2~ ( )� . In conclusion, we observe that the results of [1, 2] allow one also to study also the existence of the eigenvalues emerg- ing from the higher thresholds in the continuous spectrum that are j2. It was shown in [2], that if it exists, this eigenvalue is unique. As in Theorem 2, this fact implies that the eigen- value is real and therefore in this case we are dealing with embedded eigenvalues. Acknowledgments This work has been supported in parts by RFBR (05-01-97912) and by the Czech Academy of Sciences and Ministry of Education, Youth and Sports (LC06002). The au- thor has also been supported by a Marie Curie International Fellowship within the 6th European Community Framework (MIF1-CT-2005-006254) and gratefully acknowledges sup- port from the Deligne 2004 Balzan Prize in Mathematics and from the the grant of Republic Bashkortostan for young scien- tists and young scientific collectives. References [1] Gadyl’shin, R.: On Regular and Singular Perturbation of Acoustic and Quantum Waveguides, Comptes Rendus Méchanique, Vol. 332 (2004), No. 8, p. 647–652. [2] Gadyl’shin, R.: Local Perturbations of Quantum Wave- guides, Theor. Math. Phys., Vol. 145 (2005), No. 3, p. 1678–1690. 58 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 [3] Glazman, I. M.: Direct Methods of Qualitative Spectral Anal- ysis of Singular Differential Operators. London, Oldbourne Press, 1965. [4] Borisov, D., Krejčiřík, D.: ��-symmetric Waveguide, Submitted. Preprint: arXiv:0707.3039. Dr. Denis I. Borisov phone: +49 371 431 3215 e-mail: borisovdi@yandex.ru, BorisovDI@ic.bashedu.ru Faculty of Mathematics Chemnitz University of Technology Reichenhainer str. 39 D-09107 Chemnitz, Germany © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 59 Acta Polytechnica Vol. 47 No. 2–3/2007