On solvability of three spectra problem (Q2830663)

For \(j=1,2\) let \(q_j\in L_2(0,a')\) be real-valued where \(a'=a/2>0\) and consider the three eigenvalue problems on \([0,a']\): NEWLINE\[NEWLINE-y_j''+q_jy_j=\lambda ^2y_j,\;j=1,2,\;y_1'(0)=y_2(0)=0,\;y_1^{(j)}(a')=(-1)^jy_2^{(j)}(a'),\;j=1,2,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE-y_2''+q_2y_2=\lambda ^2y_2,\;y_2(0)=y_2(a')=0,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE-y_1''+q_1y_1=\lambda ^2y_1,\;y_1'(0)=y_1(a')=0.\tag{3}NEWLINE\]NEWLINE It is shown that the eigenvalues \((\lambda _k)_{k=-\infty ,k\neq 0}^\infty \) of (1), \((\nu _k^{(2)})_{k=-\infty ,k\neq 0}^\infty \) of (2), and \((\mu _k^{(1)})_{k=-\infty ,k\neq 0}^\infty \) of (3) have the asymptotic behavior NEWLINE\[NEWLINE\lambda _k=\frac{\pi(k-\frac12)}a+\frac{A_0}{\pi k}+\frac{\beta _k^{(1)}}k, \tag{4}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\nu _k^{(2)}=\frac{2\pi k}a+\frac{A_2}{\pi k}+\frac{\beta _k^{(2)}}k, \tag{5}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mu _k^{(1)}=\frac{\pi(2k-1)}a+\frac{A_1}{\pi k}+\frac{\beta _k^{(3)}}k, \tag{6}NEWLINE\]NEWLINE where \(A_j=\frac12\int_0^{a'}q_j(x)\,dx\), \(j=1,2\), \(A_0=A_1+A_2\), \((\beta _k^{(j)})_{k=1}^\infty \in l_2\) for \(j=1,2,3\), and with \((\theta _k)_{k=-\infty ,k\neq 0}^\infty =(\mu _k^{(1)})_{k=-\infty ,k\neq 0}^\infty \cup(\nu _k^{(2)})_{k=-\infty ,k\neq 0}^\infty \), the eigenvalues interlace as follows: NEWLINE\[NEWLINE\lambda _1^2<\theta _1^2\leq \lambda _1^2\leq \theta _2^2 \leq \lambda _3^2\leq\dots.\tag{7}NEWLINE\]NEWLINE The main result of the paper deals with the inverse problem: If the three given sequences \((\lambda _k)_{k=-\infty ,k\neq 0}^\infty \), \((\nu _k^{(2)})_{k=-\infty ,k\neq 0}^\infty \), and \((\mu _k^{(1)})_{k=-\infty ,k\neq 0}^\infty\) satisfy the asymptotics (4)--(6) and the interlacing property (7), then there are potentials \(q_1\) and \(q_2\) such that these are sequences of eigenvalues of a problem (1)--(3). If the interlacing in (7) is strict, then the pair \((q_1,q_2)\) is unique.