On a Quantum Waveguide with a Small-symmetric Perturbation

We consider a quantum waveguide with a small PT-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.

In this paper we consider an example of a quantum waveguide with a small PT-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 PT-symmetric operators.

Let x x x
= ( , ) 1 2 be Cartesian coordinates in R 2 , 2 be an infinite straight strip.By V V x = ( ) we denote a real-valued function defined on P having bounded support and belonging to L ¥ ( ) P .We assume that it satisfies the following assumption, The main object of our study is the operator on P subject to the Dirichlet boundary condition.We define it rigorously as an unbounded operator in L 2 ( ) P with the domain W 2 0 2 , ( ) P , where the latter is a subspace of W Our aim is to study the spectrum of the operator H e .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 a .

Theorem 1
The residual spectrum of H e is empty and the continuous one coincides 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 H 0 is purely continuous and coincides with [ ) 1, + ¥ .The small perturbation eV 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 ( ) , ( ): Employing these functions, we define The function ũ is well-defined and belongs to W Finally, we introduce a number K by the formula On a Quantum Waveguide with a Small PT -symmetric Perturbation

D. Borisov
We consider a quantum waveguide with a small PT-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, PT-symmetric potential, spectrum 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 H e , converging to the threshold of the continuous spectrum.This eigenvalue is simple, real and satisfies the asymptotic formula l e e e e = -+ ® + 1 4 0 If K<0 , the operator H e has no eigenvalues converging to the threshold of the continuous spectrum as e ® +0 .In particular, if (5) the number K is positive, and if the number K is negative.
Proof.We introduce the numbers : , : the operator H e has no eigenvalues converging to the threshold as e ® +0.Thus, it is sufficient to calculate the numbers The identity (1) implies that v 1 is an odd function, and hence Therefore, it is sufficient to calculate k 2 and check its sign.The mean value of v 1 being zero, the function u 1 is constant as x 1 is large enough.This allows us to write v u x u u x u x Hence, We substitute this formula and (10) into (8), and by ( 7), (9) we conclude that if K > 0 , there exists the unique eigenvalue of H e 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 l e is an eigenvalue of H e 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 Ṽ = 0, ũ = 0 and If the relation (6) holds true, the function u 1 is identically zero, and Employing (3), by analogy with (11) in the same way we check ~~~( It follows from the definition of the function ~( , ) .
In view of (12), ( 13) and this estimate we conclude that In conclusion, we observe that the results of [1,2] allow one also to study also the existence of the eigenvalues emerging from the higher thresholds in the continuous spectrum that are j 2 .It was shown in [2], that if it exists, this eigenvalue is unique.As in Theorem 2, this fact implies that the eigenvalue is real and therefore in this case we are dealing with embedded eigenvalues.
there exists the unique eigenvalue of H e converging to the threshold of the continuous spectrum, and the asymptotics of this eigenvalue reads as follows