Inverse problem for a loop-shaped waveguide
by
Vyacheslav Pivovarchik
Odessa State Academy of Civil Engineering andArchitecture

The following problem

y\pp+(\lambda2-q(x) )y=0,

(1)

y(\lambda, 0)=y(\lambda, a-0)=y(\lambda, a+0),

(2)

y\p(\lambda, 0)+y\p(\lambda, a+0)-y\p(\lambda, a-0)=0.

(3)

describes wave propagation in a loop-shaped waveguide. Let the real-valued potential q(x) be finite, i.e.

q(x)= q1(x) in L2(0, a),   if   x in (0, a), 0,   if   x in [a, \infty) .

(4)

The corresponding operator A acting in L₂(0, \infty) according to the formulae: Ay=-y\pp +q(x)y, D(A)={y in L₂(0, \infty), -y\pp +q(x)y in L₂(0, \infty), y(0)=y(a-0)=y(a+0), y\p (0)+y\p (a+0)-y\p (a-0)=0} is selfadjoint and bounded below. The essential spectrum of A covers the semiaxis [0, \infty). There can occur a finite or an infinite number of simple eigenvalues on the essential spectrum and a finite number of normal (isolated Fredholm) negative eigenvalues of geometric multiplicity <= 2.
The solution of (1), which satisfies conditions (2) and (3), for x >= a and \lambda in \R is of the form

\phi(\lambda, x) = x --> \infty  C(\lambda)(e-i\lambdax-S(\lambda)ei\lambdax),

(5)

where the "S-matrix"

S(\lambda)=e2i\lambdaa\frace(\lambda)e(-\lambda),

with

e(\lambda)=2-c(\lambda, a)-s\p(\lambda, a)-i\lambdas(\lambda, a).

(6)

Here s(\lambda, a) (c(\lambda, a)) is the solution of (1), which satisfies the conditions: s(\lambda, 0)=s\p(\lambda, 0)-1=0 (c(\lambda, a)-1=c\p(\lambda, 0)=0). It should be mentioned that if \lambda in \R and ?(\lambda)=0, then \lambda is an eigenvalue on the essential spectrum.

Theorem 1. Let real-valued q(x) in W₂³(0, a) be such that A >> 0, then:
1. e(-[`(\lambda)])=[`(e(\lambda))],
2. The functions e(\lambda) and 4-e(\lambda) belong to [`HB] (generalized Hermite-Biehler class, see [1]).

3.  e(\lambda)=2-2cos\lambdaa-isin\lambdaa-2K\fracsin\lambdaa\lambda+   iK\fraccos\lambdaa\lambda+K2\fraccos\lambdaa\lambda2+iB\fracsin\lambdaa\lambda2

+F\fracsin\lambdaa\lambda3+iE\fraccos\lambdaa\lambda3+J\fraccos\lambdaa\lambda4+iT\fracsin\lambdaa\lambda4+\frac\psi(\lambda)\lambda4,

where K, B, F, E, J, T are real constants, and \psi(\lambda) is an entire function of exponential type <= a, which belongs to L₂(-\infty, \infty).

Theorem 2. Let e(\lambda) be an entire function of exponential type a which satisfies conditions 1-3 (see Theorem 1). Then there exists a real-valued q(x) in W₂¹(0, a) (in general not unique) such that problem (1)-(4) possesses the "S-matrix" of the form (5).

References

[1] B.Ja. Levin, Distribution of Zeroes of Entire Functions. Transl. Math. Monographs, 5, AMS, Providence, Rhode Island, (1964), 493 pp.

Date received: May 12, 2000