Property $C$ and applications to inverse problems

Let ${\mathrm{\ell }}_j:=-\frac{d^2}{d{x}^2}+k^2{q}_j(x),$ $k=const>0,j=1,2,$ $0<c_0\leq q_j(x)\leq c_1,$$q\in BV([0,1])$,$q$hasfinitelymanydiscontinuitypoints$x_m\in [0,1],$andisreal-analyticontheintervals$[x_m,x_{m+1}]$ between these points. The set of such functions $q$ is denoted by $M\mathrm{.}$ Only the following property of $M$ is used: if $q_j\in M$, $j=1,2,$ then the function $p(x):=q_2(x)-q_1(x)$ changes sign on the interval $[0,1]$ at most finitely many times. Suppose that $()\quad \int_0^1p(x)u_1(x,k)u_2(x,k)dx=0,\quad \forall k>0,$where$p\in M$isanarbitraryfixedfunction,and$u_j$ solves the problem $\ell_ju_j=0,\quad 0\leq x\leq 1,\quad u’_j(0,k)=0,\quad u_j(0,k)=1.$If$()$implies$h=0$,thenthepair${\ell_1,\ell_2}$ is said to have property $C$ on the set $M$. This property is proved for the pair $\{{\mathrm{\ell }}_1,{\mathrm{\ell }}_2\}$. Applications to some inverse problems for a heat equation are given. the set $M$. This property is proved for the pair