Spectral properties of operators arising in acoustic wave propagation in an ocean of variable depth

Abstract

We prove ann-dimensional version of the following theorem: Letu(x, y) be a solution to

$$c^2 (y)\rho (y)\left( {\frac{1}{{\rho (y)}}u_y } \right)_y + c^2 (y)u_{xx} + k^2 u = 0 (k > 0)$$

in Ω≡{y>0}∖B, continuous in$\bar \Omega$,B being a disc centered at the origin, andρ(y) andc(y) being strictly positive functions constant outside of a bounded set,C ⁽²⁾ except for a finite number of jumps. Then ifu(x,0)→0 exponentially as |x|→∞ andu∈L ²(Ω),u≡0 in Ω.