Properties of the spectral function of a Sturm-Liouville operator with discontinuous coefficients (Q1177814)

The authors extend their work [Mat. Zametki 42, No. 6, 819-830 (1987; Zbl 0652.34027)] and investigate the spectral function of the Sturm- Liouville operator \(L\) generated from the differential expression \[ \ell(u)=1/s(x)\{[p(x)u'(x)]'+q(x)u(x)\}, \leqno (1) \] which is given on the finite intervals \((a,x_ 0)\) and \((x_ 0,b)\); and from self-adjoint boundary conditions at the interval \((a,b)\) where (2) \(p(x)\), \(s(x)\in C^{(1)}[a,x_ 0]\cap C^{(1)}[x_ 0,b]\), (3) \(q(x)\in L_ 2(a,b)\). Consider the functions: \[ y=\int^{x_ 0}_{x_ 0-\rho_ 1(y)}\sqrt{s_ 1(\tau)/p_ 1(\tau)}d\tau,\quad y=\int_{x_ 0}^{x_ 0+\rho_ 2(y)}\sqrt{s_ 2(\tau)/p_ 2(\tau)}d\tau,\quad 0\leq y\leq y_ 0, \] \[ \bar\rho_ 1(x_ 0-x)=\int_ x^{x_ 0}\sqrt{s_ 1(\tau)/p_ 1(\tau)}d\tau,\quad \bar\rho_ 2(x-x_ 0)=\int^ x_{x_ 0}\sqrt{s_ 2(\tau)/p_ 2(\tau)}d\tau. \] Put \(\bar\rho(| x-x_ 0|)-\bar\rho_ 1(x_ 0-x)\) for \(x\leq x_ 0\), \(\bar\rho(| x-x_ 0|)=\bar\rho_ 2(x-x_ 0)\) for \(x\geq x_ 0\), \(g(x)=p(x)s(x)\), \(a(x_ 0)=2/\{\pi[\sqrt{g(x_ 0+0)}+\sqrt{g(x_ 0-0)}]\}\), \(\hat\theta_ \mu(x)=a(x_ 0)\sin\mu\bar\rho(| x-x_ 0|)/\bar\rho(| x-x_ 0|)\) for \(x_ 0-R_ 1\leq x\leq x_ 0+R_ 2\), where \(R_ 1\) and \(R_ 2\) are sufficiently small positive numbers, \(\hat\theta_ \mu(x)=0\) outside the interval \([x_ 0-R_ 1, x_ 0+R_ 2]\), and \(\hat\theta_ \mu(x_ 0)=\hat\theta_ \mu(x_ 0+0)=\hat\theta_ \mu(x_ 0-0)\). The function \(\theta(x,y,\mu)=\sum_{\sqrt\lambda_ n<\mu}u_ n(x)u_ n(y)\) is the spectral function of the operator \(L\), where \(\{u_ n(x)\}\) is the system of eigenfunctions. The basic results of the article is stated in the following theorem: Let the coefficients of the operator (1) satisfy the conditions (2), (3), and some other suitable conditions. Then for any \(a'\) and \(b'\), \(a\leq a'\leq b'\leq b\), and for any \(\sigma_ 1\) and \(\sigma_ 2\), \(a'\leq\sigma_ 1\leq\sigma_ 2\leq b'\) and for any \(\mu\geq 1\), the following estimate for the spectral function \(\theta(x,y,\mu)\), at the point \(x_ 0\), is valid \[ \int^{\sigma_ 2}_{\sigma_ 1}[\theta(x_ 0,y,\mu)-\hat\theta_ \mu(y)]s(y)dy=O(1/\mu ). \]