Acta Polytechnica doi:10.14311/AP.2014.54.0093 Acta Polytechnica 54(2):93–100, 2014 © Czech Technical University in Prague, 2014 available online at http://ojs.cvut.cz/ojs/index.php/ap EIGENVALUE COLLISION FOR PT-SYMMETRIC WAVEGUIDE Denis Borisova, b a Institute of Mathematics of Ufa Scientific Center of RAS, Chernyshevskogo, str. 112, 450008, Ufa, Russian Federation b Bashkir State Pedagogical University, October St. 3a, 450000, Ufa, Russian Federation correspondence: borisovdi@yandex.ru Abstract. We consider a model of a planar PT -symmetric waveguide and study the phenomenon of the eigenvalue collision under perturbation of the boundary conditions. This phenomenon was discovered numerically in previous works. The main result of this work is an analytic explanation of this phenomenon. Keywords: PT-symmetric operator, eigenvalues, perturbation, asymptotics. 1. Introduction and main results In this paper we study a problem in the theory of PT -symmetric operators which has been studied rather intensively after the pioneering works [12–21]. Our model is introduced as follows. Let x = (x1,x2) be Cartesian coordinates in R2, let Ω be the strip {x : −d < x2 < d}, d > 0, and let α = α(x1) be a function in W 1∞(R). We consider the operator Hα in L2(Ω) acting as Hαu = −∆u on the functions u ∈ W 22 (Ω) satisfying the non-Hermitian boundary conditions( ∂ ∂x2 + iα ) u = 0 on ∂Ω. (1.1) It was shown in [1] that this operator is m-sectorial, densely defined, and PT -symmetric, namely, PTHα = HαPT , (1.2) where (Pu)(x) = u(x1,−x2), and T is the operator of complex conjugation, T u = u. It was also proven in [1] that H∗α = H−α, H ∗ α = THαT = PHαP. (1.3) A non-trivial question related to Hα is the behavior of its eigenvalues. As α(x1) is a small regular localized perturbation of a constant function, sufficient conditions were obtained in [1] for the existence and absence of isolated eigenvalues near the threshold of the essential spectrum. Similar results for both regularly and singularly perturbed models were obtained in [2–6]. Numerical experiments performed in [6, 7] provided a very non-trivial picture of the distribution of the eigenvalues. An interesting phenomenon discovered numerically in [6, 7] was the eigenvalue collision. Namely, let t ∈ R be a parameter, then as t increases, operator Htα can have two simple real isolated eigenvalues meeting at some point. Then two cases are possible. In the first of them, these eigenvalues stay real as t increases and they just pass along the real line. In the second case, the eigenvalues become complex as t increases and they are located symmetrically w.r.t. the real axis. The present paper is devoted to an analytic study of this phenomenon. Suppose λ0 ∈ R is an isolated eigenvalue of Hα, ε is a small real parameter, β ∈ W 2∞(R) is some function. Denote Γ± := {x : x2 = ±d}. Our first main result describes the case when λ0 is an eigenvalue of geometric multiplicity two. Theorem 1.1. Assume λ0 ∈ R is a double eigenvalue of Hα, ψ±0 are the associated eigenfunctions satisfying (ψ±0 ,T ψ ± 0 )L2(Ω) = 1, (ψ + 0 ,T ψ − 0 )L2(Ω) = 0. (1.4) Suppose also (b11 − b22)2 + 4b212 6= 0, (1.5) b11 = i ∫ Γ+ β(ψ+0 ) 2 dx1 − i ∫ Γ− β(ψ+0 ) 2 dx1, b22 = i ∫ Γ+ β(ψ−0 ) 2 dx1 − i ∫ Γ− β(ψ−0 ) 2 dx1, b12 = i ∫ Γ+ βψ+0 ψ − 0 dx1 − i ∫ Γ− βψ+0 ψ − 0 dx1. (1.6) 93 http://dx.doi.org/10.14311/AP.2014.54.0093 http://ojs.cvut.cz/ojs/index.php/ap Denis Borisov Acta Polytechnica Then for all sufficiently small ε the operator Hα+εβ has two simple isolated eigenvalues λ±ε converging to λ0 as ε → 0. These eigenvalues are holomorphic w.r.t. ε and the first terms of their Taylor series are λ±ε = λ0 + ελ ± 1 + O(ε 2), λ±1 = 1 2 (b11 + b22) ± 1 2 ( (b11 − b22)2 + 4b212 )1/2 . (1.7) The second main result is devoted to the case when the geometric multiplicity of λ0 is one but the algebraic multiplicity is two. Theorem 1.2. Let λ0 ∈ R be a simple eigenvalue of Hα and let ψ0 be the associated eigenfunction. Assume that the equation (Hα −λ0)φ0 = ψ0 (1.8) is solvable and there exists a solution satisfying (φ0,T ψ0)L2(Ω) 6= 0, (φ0,ψ0)L2(Ω) = 0. (1.9) Then eigenfunction ψ0 can be chosen so that (φ0,T ψ0)L2(Ω) = 1, (φ0,ψ0)L2(Ω) = 0, (1.10) ψ0 = PT ψ0, φ0 = PT φ0. (1.11) Suppose then that this eigenfunction obeys the inequality∫ Γ+ β Re ψ0 Im ψ0 dx1 6= 0. (1.12) Then for all sufficiently small ε the operator Hα+εβ has two simple isolated eigenvalues λ±ε converging to λ0 as ε → 0. These eigenvalues are real as ε ∫ Γ+ β Re ψ0 Im ψ0 dx1 < 0 (1.13) and are complex as ε ∫ Γ+ β Re ψ0 Im ψ0 dx1 > 0. (1.14) Eigenvalues λ±ε are holomorphic w.r.t. ε1/2 and the first terms of their Taylor series read as λ±ε = λ0 + ε 1/2λ±1/2 + O(ε), λ ± 1/2 = ±2 ( − ∫ Γ+ β Re ψ0 Im ψ0 dx1 )1/2 . (1.15) Let us discuss the results of these theorems. The typical situation of the eigenvalue collision is that two simple eigenvalues of Hα+εβ converge to the same limiting eigenvalue λ0 of Hα as ε → 0. Then it is a general fact from the regular perturbation theory that the algebraic multiplicity of λ0 should be two. The above theorems address two possible situations. In the first of them the geometric multiplicity of λ0 is two, i.e., there exist two associated linearly independent eigenfunctions. As we see from Theorem 1.1, in this situation the perturbed eigenvalues are holomorphic w.r.t. ε and their first terms in the Taylor series are given by (1.7 right). The numbers λ±1 are some fixed constants and they can be either complex or real. But an important issue is that here when changing the sign of ε, the eigenvalues can not bifurcate from real line to the complex plane or vice versa. This fact is implied by (1.7 right), namely, if λ±1 are complex numbers, then λ ± ε are also complex for both ε < 0 and ε > 0. Thus, in this case we do not face the above-mentioned phenomenon of the eigenvalue collision discovered numerically in [6], [7]. If λ±1 are real, then we need to calculate the next terms of their Taylor series to see whether they are complex or real. Once all the terms in the Taylor series are real, we deal with two real eigenvalues which just pass one through the other staying on the real line. Nevertheless, in view of formulae (1.6) we believe that choosing appropriate β we can get almost any value for the quantity in (1.5). In a particular interesting case β = α the author does not know a way of identifying the sign of (b11 − b22)2 + 4b212 or proving the reality of the eigenvalues λ±ε . Theorem 1.2 treats the case when the geometric multiplicity of λ0 is one. Then the Taylor series for the perturbed eigenvalues are completely different from Theorem 1.1 and here the expansions are made w.r.t. ε1/2. And the presence of this power perfectly explains the studied phenomenon. Namely, once ε is positive, the same is true for ε1/2, while for negative ε the square root ε1/2 is pure imaginary. This is exactly what is needed, once ε changes the sign, real eigenvalues become complex and vice versa. Unfortunately, we cannot even analytically prove for our model the existence of such eigenvalues. We can just state that once λ0 has geometric multiplicity one and the associated eigenfunction ψ0 satisfies the identity (ψ0,T ψ0)L2(Ω) = 0, then equation (1.8) is solvable (see Lemma 2.1). And numerical results in [6], [7] show that this is quite a typical situation. Our next main result provides another criterion identifying the solvability of equation (1.8) 94 vol. 54 no. 2/2014 Eigenvalue Collision for PT-Symmetric Waveguide Theorem 1.3. Suppose λ0 is a simple eigenvalue of Hα, the associated eigenfunction satisfies the estimate∑ γ∈Z2+ |γ|62 ∣∣∣∂γψ0 ∂xγ (x) ∣∣∣ 6 C 1 + |x1|3 , x ∈ Ω. (1.16) Then equation (1.8) is solvable if and only if∫ R2 K(x1,y1) ( α(x1) −α(y1) ) Re ψ0(x1,d) Im ψ0(y1,d) dx1 dy1 = 0, (1.17) where K(x1,y1) := { x1 if y1 < x1, −y1 if y1 > x1. Here ψ0 is chosen so that it satisfies the first identity in (1.11). Assumption (1.16) is not very restrictive since usually eigenfunctions associated with isolated eigenvalues of elliptic operators decay exponentially at infinity. The main condition here is (1.17). As we shall show later in Lemma 2.1, equation (1.8) is solvable if and only if (ψ0,T ψ0)L2(Ω) = 0. And we rewrite this identity to (1.17) by calculating (ψ0,T ψ0)L2(Ω). The left hand side in (1.17) is simpler in the sense that it involves only boundary integrals while (ψ0,T ψ0)L2(Ω) is in fact the integral over the whole strip Ω. 2. Proofs of main results In L2(Ω) we introduce the unitary operator (Uεβf)(x) := e−iεβ(x1)x2f(x). Then it is easy to see that the spectra of Hα+εβ and U−1εβ Hα+εβUεβ coincide and U−1εβ Hα+εβUεβ = Hα −εLε, (2.1) Lε := −2iβ′x2 ∂ ∂x1 − 2iβ ∂ ∂x2 −εβ2 −ε(β′)2x2 − iβ′′x2. (2.2) In the proofs of the main results we shall make use of several auxiliary lemmata. Lemma 2.1. Under the hypothesis of Theorem 1.2 the equation (Hα −λ0)u = f (2.3) is solvable if and only if (f,T ψ0)L2(Ω) = 0. (2.4) Under the hypothesis of Theorem 1.1 equation (2.3) is solvable if and only if (f,T ψ±0 )L2(Ω) = 0. (2.5) Proof. By (1.3) we see that under the hypotheses of both Theorems 1.1 and 1.2, λ0 is an eigenvalue of H∗α with the associated eigenfunction(s) T ψ0 or T ψ±0 . Then the lemma follows from [8, Ch. III, Sec. 6.6, Rem. 6.23]. Lemma 2.2. Suppose the hypothesis of Theorem 1.2. Then eigenfunction ψ0 can be chosen so that relations (1.10), (1.11), and (ψ0,T ψ0)L2(Ω) = 0 (2.6) hold true. The functions Re ψ0 and Re φ0 are even w.r.t. x2 and Im ψ0 and Im φ0 are odd w.r.t. x2. Proof. Identity (2.6) follows directly from (2.4) applied to equation (1.8). Since λ0 is a real simple eigenvalue and equation (1.8) has a unique solution satisfying the second identity in (1.10), by (1.2) we have (1.11) and thus Re ψ0 and Re φ0 are even, while Im ψ0 and Im φ0 are odd w.r.t. x2. Employing this fact and (1.8), we obtain (φ0,T ψ0)L2(Ω) = − ∫ Ω φ0(∆ + λ0)φ0 dx = i ∫ Γ+ αφ20 dx1 − i ∫ Γ− αφ20 dx1 + ∫ Ω ((∂φ0 ∂x1 )2 + (∂φ0 ∂x2 )2 −λ0φ20 ) dx = −4 ∫ Γ+ α Re φ0 Im φ0 dx1 + ∫ Ω ( |∇Re φ0|2 −|∇Im φ0|2 ) dx−λ0 ∫ Ω ( |Re φ0|2 −|Im φ0|2 ) dx ∈ R. (2.7) Hence, multiplying function ψ0 and φ0 by an appropriate constant, we can easily get the first identity in (1.10) not spoiling other established properties of φ0 and ψ0. 95 Denis Borisov Acta Polytechnica Lemma 2.3. Suppose the hypothesis of Theorem 1.2. Then for λ close to λ0 the resolvent (Hα −λ)−1 can be represented as (Hα −λ)−1 = P−2 (λ−λ0)2 + P−1 λ−λ0 + Rα(λ), (2.8) P−2 = ψ0`2, P−1 = φ0`2 + ψ0`1, `2f := −(f,T ψ0)L2(Ω), `1f := − ( f,T (φ0 −ψ0) ) L2(Ω) , (2.9) where Rα(λ) is the reduced resolvent which is a bounded and holomorphic in the λ operator. Proof. We know by [8, Ch. III, Sec. 6.5] (see also the remark on space M′(0) in the proof of Theorem 1.7 in [8, Ch. VII, Sec. 1.3]) that (Hα −λ)−1 can be expanded into the Laurent series (Hα −λ)−1 = N∑ n=1 P−n (λ−λ0)n + Rα(λ), where N is a fixed number independent of λ, Rα is the reduced resolvent which is a bounded and holomorphic in λ operator. Given any f ∈ L2(Ω), we then have u = (Hα −λ)−1f = N∑ n=1 u−n (λ−λ0)n + ∞∑ n=0 (λ−λ0)nun. We substitute this formula into the equation (Hα −λ)u = f and equate the coefficients at the like powers of (λ−λ0): (Hα −λ0)u−N = 0, (Hα −λ0)u−k = u−k−1, k = 1, . . . ,N − 1, (Hα −λ0)u0 = f + u−1, (Hα −λ0)u1 = u0. (2.10) This implies that u−N = ψ0`2f, u−N+1 = φ0`2f + ψ0`1f, where `i are some functionals on L2(Ω). If N > 2, then by (1.9) and Lemma 2.1 the equation for u−N+2 is unsolvable. Hence, we can assume N = 2. Writing then the solvability condition (2.4) for equations (2.10) and taking into consideration the identity in (1.10), we arrive easily to the formula for `2 in (2.9) and `1f := −(U0,T ψ0)L2(Ω), (2.11) where U0 is the solution to the equation (Hα −λ0)U0 = f + ψ0`2f (2.12) satisfying (U0,ψ0)L2(Ω) = 0. (2.13) It follows from (1.3) and (1.8) that (U0,T ψ0)L2(Ω) = ( U0,T (Hα −λ0)φ0 ) L2(Ω) = ( U0, (Hα −λ0)∗T φ0 ) L2(Ω) = ( (Hα −λ0)U0,T φ0 ) L2(Ω) = (f + ψ0`2f,T φ0)L2(Ω). These identities, the above obtained formula for `2, and (2.6), (2.11) imply formula (2.12) for `1. Lemma 2.4. Suppose the hypothesis of Theorem 1.1. Then for λ close to λ0 the resolvent (Hα −λ)−1 can be represented as (Hα −λ)−1 = P−1 λ−λ0 + Rα(λ), (2.14) P−1 = ψ+0 `+ + ψ − 0 `−, `±f := −(f,T ψ ± 0 )L2(Ω), (2.15) where Rα(λ) is the reduced resolvent which is a bounded and holomorphic in λ operator. The proof of this lemma is similar to that of Lemma 2.3, we just should bear in mind that due to (1.4) and Lemma 2.1 the equations (Hα −λ0)u = ψ±0 are unsolvable. We proceed to the proofs of Theorems 1.1, 1.2, 1.3. 96 vol. 54 no. 2/2014 Eigenvalue Collision for PT-Symmetric Waveguide Proof of Theorem 1.2. The proof is based on the modified version of the Birman-Schwinger principle suggested in [9] in the form developed in [10]. In view of (2.1), the eigenvalue equation for Hα+εβ is equivalent to the same equation for Hα −εLε. The latter equation can be written as (Hα −λε)ψε = εLεψε. (2.16) We then invert the operator (Hα −λε) by Lemma 2.3 and obtain ψε = ε P−2Lεψε (λε −λ0)2 + ε P−1Lεψε λε −λ0 + εRα(λε)ψε. By Lemma 2.3 the operator Rα(λ) is bounded uniformly in λ close to λ0 and hence the inverse A(z,ε) :=( I −εRα(λ0 + z) )−1 is well-defined and is uniformly bounded for all λ close to λ0 and for all sufficiently small ε. We apply this operator to the latter equation and get ψε = ε z2ε A(λ0 + zε,ε)P−2Lεψε + ε zε A(λ0 + zε,ε)P−1Lεψε, (2.17) where we denote zε := λε −λ0. Then we apply functionals `2Lε, `1Lε to the obtained equation and it results in( ε zε A11(zε,ε) − 1 ) X1 + ε z2ε ( A11(zε,ε) + zεA12(zε,ε) ) X2 = 0, ε zε A21(zε,ε)X1 + ( ε z2ε ( A21(zε,ε) + zεA22(zε,ε) ) − 1 ) X2 = 0, (2.18) where Xi = `iLεψε, and Ai1(z,ε) := `iLεA(λ0 + z,ε)ψ0, Ai2(z,ε) := `iLεA(λ0 + z,ε)φ0, i = 1, 2. The obtained system of equations is linear w.r.t. (X1,X2). We need a non-zero solution to this system since otherwise by (2.17) we would get ψε = 0 and ψε then cannot be an eigenfunction. System (2.18) has a nonzero solution if its determinant vanishes. It implies the equation z2ε −ε ( A11(zε,ε) + A22(zε,ε) ) zε −εA21(zε,ε) + ε2 ( A11(zε,ε)A22(zε,ε) −A12(zε,ε)A21(zε,ε) ) = 0, which is equivalent to the following two zε = G±(zε,ε1/2), (2.19) where G±(z,κ) := κ2(A11(z,κ2) + A22(z,κ2)) 2 ±κ ( A21(z,κ2) + κ2 4 ( A11(z,κ2) −A22(z,κ2) )2 + κ2A12(z,κ2)A21(z,κ2))1/2. (2.20) Here the branch of the square root is fixed by the restriction 11/2 = 1. It is clear that the functions Aij are jointly holomorphic w.r.t. sufficiently small z and ε. Moreover, by (2.2) A21(0,ε) = `2LεA(0,ε)ψ0 = i`2 ( −2β′x2 ∂ ∂x1 − 2β ∂ ∂x2 −β′′x2 ) ψ0 + O(ε). (2.21) To calculate the first term on the right hand side of this identity, we first observe that by the equation for ψ0 we have − ( 2β′x2 ∂ ∂x1 + 2β ∂ ∂x2 + β′′x2 ) ψ0 = −(∆ + λ0)βx2ψ0 =: g. Now we find i`2g by integration by parts i`2g = ∫ Ω ψ0(∆ + λ0)βx2ψ0 dx = i ∫ Γ+ ( ψ0 ∂ ∂x2 βx2ψ0 −βx2ψ0 ∂ψ0 ∂x2 ) dx1 − i ∫ Γ− ( ψ0 ∂ ∂x2 βx2ψ0 −βx2ψ0 ∂ψ0 ∂x2 ) dx1 = i ∫ Γ+ βψ20 dx1 − i ∫ Γ− βψ20 dx1. (2.22) Together with Lemma 2.2 this implies i`2g = −4 ∫ Γ+ β Re ψ0 Im ψ0 dx1. (2.23) 97 Denis Borisov Acta Polytechnica Hence, by (2.20), (2.22), (1.12), and the properties of functions Aij we conclude that functions G± are jointly holomorphic w.r.t. sufficiently small z and κ. Applying the Rouché theorem as in [10, Sec. 4], we conclude that for all sufficiently small κ each of the functions z 7→ z − G±(z,κ) has a simple zero z±(κ) in a small neighborhood of the origin. By the implicit function theorem these zeroes are holomorphic w.r.t. κ. Thus, the desired solutions to equations (2.19) are z±(ε1/2), and these functions are holomorphic w.r.t. ε1/2. Moreover, it follows from (2.19), (2.20), (2.21), (2.22), (2.23) that z±(ε1/2) = G±(0,ε1/2) + O(ε) = ±ε1/2A 1/2 21 (0,ε) + O(ε) and then the sought eigenvalues are λ±ε = λ0 + z±(ε1/2). These eigenvalues are holomorphic w.r.t. ε1/2 and obey (1.15). Let us prove that these eigenvalues are real as (1.13) holds true and are complex once (1.14) is satisfied. The latter statement follows easily from formulae (1.15) since in this case ε1/2λ±1/2 are two imaginary numbers. To prove the reality, as one can easily make sure, it is sufficient to prove that functions G±(z,κ) are real for real z and κ. Then the existence of a real root is implied easily by the implicit function theorem for real functions. In view of definition (2.20) of G±, the desired fact is yielded by the similar reality of Aij. Let us prove the latter. It follows from Lemma 2.3 that for each f ∈ L2(Ω) the function Rα(λ)f = (Hα −λ)−1f − P−2f (λ−λ0)2 − P−1f λ−λ0 solves the equation (Hα −λ)Rα(λ)f = f + ψ0`1f + φ0`2f. (2.24) Employing definition (2.2) of Lε, we check easily that PTLε = LεPT . This identity and (1.11), (2.24) yield that for z ∈ R, κ ∈ R PTLεA(λ0 + z,κ)ψ0 = LεA(λ0 + z,κ)ψ0, PTLεA(λ0 + z,κ)φ0 = LεA(λ0 + z,κ)φ0. Using (1.11) once again, for z ∈ R, κ ∈ R we get A11(z,κ) = ( PTLεA(λ0 + z,κ)ψ0,Pψ0 ) L2(Ω) = ( TLεA(λ0 + z,κ)ψ0,T ψ0 ) L2(Ω) = A11(z,κ). The reality of other functions Aij can be proven in the same way. The proof is complete. Proof of Theorem 1.1. The main ideas here are the same as in the proof of Theorem 1.2, so, we focus only on the main milestones. We again begin with (2.1) and invert (Hε −λε) by Lemma 2.2. It leads us to an analogue of equation (2.17), ψε = ε zε A(λ0 + zε,ε)P−1Lεψε, (2.25) where operator A is introduced in the same way as above. We then apply functionals `±Lε to this equation( ε zε B11(zε,ε) − 1 ) X1 + ε zε B12(zε,ε)X2 = 0, ε zε B21(zε,ε)X1 + ( ε zε B22(zε,ε) − 1 ) X2 = 0, (2.26) B11(z,ε) := `+LεA(λ0 + z,ε)ψ+0 , B12(z,ε) := `+LεA(λ0 + z,ε)ψ − 0 , B21(z,ε) := `−LεA(λ0 + z,ε)ψ+0 , B22(z,ε) := `−LεA(λ0 + z,ε)ψ − 0 . The determinant of system (2.26) should again vanish and it implies the equation z2ε −ε ( B11(zε,ε) + B22(zε,ε) ) + ε2 ( B11(zε,ε)B22(zε,ε) −B12(zε,ε)B21(zε,ε) ) = 0, which splits into other two zε = Q±(zε,ε), (2.27) Q±(z,ε) := ε 2 ( B11(zε,ε) + B22(zε,ε) ) ± ε 2 ( (B11(z,ε) −B22(z,ε))2 + 4B12(z,ε)B21(z,ε) )1/2 . Here the branch of the square root is fixed by the restriction 11/2 = 1. Let us prove that this square root is jointly holomorphic w.r.t. z and ε. Integrating by parts as in (2.22) and employing (1.1), one can make easily sure that Bii = bii + O(ε), i = 1, 2, B12(0,ε) = b12 + O(ε), B21(0,ε) = b21 + O(ε). (2.28) Hence, by assumption (1.5), functions Q± are jointly holomorphic w.r.t. z and ε. Proceeding now as in the proof of Theorem 1.2, we arrive at the statement of Theorem 1.1. 98 vol. 54 no. 2/2014 Eigenvalue Collision for PT-Symmetric Waveguide Proof of Theorem 1.3. Denote ψ(x) := 1 2 ∫ x1 −∞ tψ0(t,x2) dt. In view of (1.16) this function is well-defined. Throughout the proof we shall deal with several integrals of such kind and all of them will be well-defined due to (1.16). In what follows we shall not stress this fact anymore. Employing the equation for ψ0, integrating by parts, and bearing in mind estimates (1.16), we get (∆ + λ0)ψ = ψ0 + 1 2 x1 ∂ψ0 ∂x1 + 1 2 ∫ x1 −∞ t ( ∂2 ∂x22 + λ0 ) ψ0(t,x2) dt = ψ0 + 1 2 x1 ∂ψ0 ∂x1 − 1 2 x1 ∫ x1 −∞ ∂2ψ0 ∂x21 (t,x2) dt = ψ0. The proven equation for ψ allows us to integrate once again,∫ Ω ψ20 dx = ∫ Ω ψ0(∆ + λ0)ψ dx = ∫ Γ+ ( ψ0 ∂ψ ∂x2 −ψ ∂ψ0 ∂x2 ) dx1 − ∫ Γ− ( ψ0 ∂ψ ∂x2 −ψ ∂ψ0 ∂x2 ) dx1 = ∫ Γ+ ψ0 ( ∂ψ ∂x2 + iαψ ) dx1 − ∫ Γ− ψ0 ( ∂ψ ∂x2 + iαψ ) dx1. Now we employ identity (1.11) and boundary condition (1.1) for ψ0 to simplify the sum of these integrals,∫ Ω ψ20 dx = − ∫ Γ+ dx1 Re ψ0(x1,d)x1 ∫ x1 −∞ ( α(x1) −α(y1) ) Im ψ0(y1,d) dy1 − ∫ Γ+ dx1 Im ψ0(x1,d)x1 ∫ x1 −∞ ( α(x1) −α(y1) ) Re ψ0(y1,d) dy1 = − ∫ Γ+ dx1 Re ψ0(x1,d)x1 ∫ x1 −∞ ( α(x1) −α(y1) ) Im ψ0(y1,d) dy1 + ∫ Γ+ dx1 Re ψ0(x1,d)x1 ∫ +∞ x1 ( α(x1) −α(y1) ) Im ψ0(y1,d) dy1 = − ∫ R2 K(x1,y1) ( α(x1) −α(y1) ) Re ψ0(y1,d) Im ψ0(y1,d) dx1 dy1. By (2.4) we then conclude that equation (1.8) is solvable if and only if identity (1.17) holds true. Remark 2.5. The idea of the latter proof was borrowed from the proof of Lemma 2.2 in [11], see also proof of Lemma 3.6 in [10]. Acknowledgements The author thanks M. Znojil for valuable discussions that stimulated him to write this paper. The work is partially supported by RFBR, by a grant of the President of Russia for young scientists — doctors of science (MD-183.2014.1) and by the Dynasty foundation fellowship for young mathematicians. References [1] D. Borisov, D. Krejčiřík. PT -symmetric waveguide.Integral Equations and Operator Theory. 2008. V. 62. No. 4. P. 489-515. doi: 10.1007/s00020-008-1634-1 [2] D. Borisov. On a PT -symmetric waveguide with a pair of small holes.Proceedings of Steklov Institute of Mathematics. 2013. V. 281. No. 1 supplement. P. 5-21; translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2012. V. 18, No. 2. P. 22-37. doi: 10.1134/S0081543813050027 [3] D. Borisov. Discrete spectrum of thin PT -symmetric waveguide.Ufa Mathematical Journal. 2014. vol. 6, no. 1, pp. 29–55. doi: 10.13108/2014-6-1-29 [4] D. Borisov. On a quantum waveguide with a small PT -symmetric perturbation.Acta Polytechnica. 2007. No. 2-3. P. 59-61. [5] D. Borisov, D. Krejčiřík. The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions.Asymptotic Analysis. 2012. V. 76. No. 1. P. 49-59. doi: 10.3233/ASY-2011-1061 [6] D. Krejčiřík and P. Siegl. PT -symmetric models in curved manifolds.Journal of Physics A: Mathematical and Theoretical. 2010. V. 43, No. 48. id 485204. doi: 10.1088/1751-8113/43/48/485204 [7] D. Krejčiřík and M. Tater. Non-Hermitian spectral effects in a PT -symmetric waveguide.Journal of Physics A: Mathematical and Theoretical. 2008. V. 41, No. 24. id. 244013. doi: 10.1088/1751-8113/41/24/244013 [8] T. Kato. Perturbation theory for linear operators. Classics in Mathematics, Springer-Verlag, Berlin. 1995. 99 http://dx.doi.org/10.1007/s00020-008-1634-1 http://dx.doi.org/10.1134/S0081543813050027 http://dx.doi.org/10.13108/2014-6-1-29 http://dx.doi.org/10.3233/ASY-2011-1061 http://dx.doi.org/10.1088/1751-8113/43/48/485204 http://dx.doi.org/10.1088/1751-8113/41/24/244013 Denis Borisov Acta Polytechnica [9] R.R. Gadyl’shin. Local perturbations of the Schrödinger operator on the axis.Theoretical and Mathematical Physics. 2002. V. 132, No. 1. P. 976-982. doi: 10.4213/tmf349 [10] D. Borisov. Discrete spectrum of a pair of non-symmetric waveguides coupled by a window.Sbornik Mathematics. 2006. V. 197. No. 4. P. 475-504. doi: 10.1070/SM2006v197n04ABEH003767 [11] D. I. Borisov. On a model boundary value problem for Laplacian with frequently alternating type of boundary condition.Asymptotic Analysis. 2003. V. 35. No. 1. P. 1-26. [12] C.M. Bender, S. Boettcher. Real spectra in non-hermitian hamiltonians having PT symmetry.Physics Review Letters. 1998. V. 80. No. 24. P. 5243-5246. doi: 10.1103/PhysRevLett.80.5243 [13] A. Mostafazadeh. Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian.Journal of Mathematical Physics. 2002. V. 43. No. 1. P. 205-214. doi: 10.1063/1.1418246 [14] A. Mostafazadeh. Pseudo-Hermiticity versus PT-Symmetry II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum.Journal of Mathematical Physics. 2002. V. 43. No. 5. P. 2814-2816. doi: 10.1063/1.1461427 [15] A. Mostafazadeh. Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries.Journal of Mathematical Physics. 2002. V. 43. No. 8. P. 3944-3951. doi: 10.1063/1.1489072 [16] A. Mostafazadeh. On the Pseudo-Hermiticity of a Class of PT-Symmetric Hamiltonians in One Dimension.Modern Physics Letters A. 2002. V. 17. No. 30. P. 1973-1977. doi: 10.1142/S0217732302008472 [17] M. Znojil. Exact solution for Morse oscillator in PT-symmetric quantum mechanics.Physics Letters A. 1999. V. 264. No. 2. P. 108-111. doi: 10.1016/S0375-9601(99)00805-1 [18] M. Znojil. Non-Hermitian matrix description of the PT-symmetric anharmonic oscillators.Journal of Physics A: Mathematics and General. 1999. V. 32. No. 42. P. 7419-7428. doi: 10.1088/0305-4470/32/42/313 [19] M. Znojil. PT-symmetric harmonic oscillators.Physics Letters A. 1999. V. 259. No. 3-4. P. 220-223. doi: 10.1016/S0375-9601(99)00429-6 [20] G. Levai and M. Znojil. Systematic search for PT -symmetric potentials with real energy spectra.Journal of Physics A: Mathematics and General. 2000. V. 33. No. 40. P. 7165-7180. doi: 10.1088/0305-4470/33/40/313 [21] C.M. Bender. Making sense of non-Hermitian Hamiltonians.Reports on Progress in Physics. 2007. V. 70. No. 6. P. 947-1018. doi: 10.1088/0034-4885/70/6/R03 100 http://dx.doi.org/10.4213/tmf349 http://dx.doi.org/10.1070/SM2006v197n04ABEH003767 http://dx.doi.org/10.1103/PhysRevLett.80.5243 http://dx.doi.org/10.1063/1.1418246 http://dx.doi.org/10.1063/1.1461427 http://dx.doi.org/10.1063/1.1489072 http://dx.doi.org/10.1142/S0217732302008472 http://dx.doi.org/10.1016/S0375-9601(99)00805-1 http://dx.doi.org/10.1088/0305-4470/32/42/313 http://dx.doi.org/10.1016/S0375-9601(99)00429-6 http://dx.doi.org/10.1088/0305-4470/33/40/313 http://dx.doi.org/10.1088/0034-4885/70/6/R03 Acta Polytechnica 54(2):93–100, 2014 1 Introduction and main results 2 Proofs of main results Acknowledgements References