The eigenvalue and trace of the Sturm-Liouville operator as differentiable functions of a summable potential
The authors consider a Sturm-Liouville equation of the form \[ y''+(\lambda-q(x))y=0 \] on the interval \([0,l]\), where \(q\) is a Lebesgue integrable function, and the boundary conditions of the type \[ (\cos\phi_0) y(0)+(\sin\phi_0) y'(0)=0, \quad (\cos\phi_l) y(l)+(\sin\phi_l) y'(l)=0. \] The boundary value problem admits a strictly increasing infinite sequence of eigenvalues, that will be denoted by \(\lambda_n(q)\), as functions of the map \(q\in L^1([0,l])\); the corresponding normalized eigenfunction in the Sobolev space \(W^{(2)}_1([0,l])\) will be denoted by \(y_n(q,\cdot)\). The authors prove the following results: 1. For all \(n\in \mathbb{N}\) and every \(q\in L^1([0,l])\) there exist \(\delta>0\) and \(C>0\) such that for all \(q_1\in L^1([0,l])\) with \(\|q_1\|_{L^1}\leq \delta\), the following estimate holds: \[ \left|\lambda_n(q+q_1)-\biggl(\lambda_n(q)+\int_0^lq_1(x)y_n(q,x)^2 \text{ d}x\biggr)\right|\leq C\|q_1\|_{L^1}^2. \] 2. If \(q_1,q_2\in L^1([0,l])\) are such that \(q_1(x)\leq q_2(x)\) almost everywhere, then \(\lambda_n(q_1)\leq\lambda_n(q_2)\) for all \(n\). If \(q_1(x)<q_2(x)\) on a set of positive measure, then \(\lambda_n(q_1)<\lambda_n(q_2)\) for all \(n\). 3. For all \(q\in L^1([0,l])\) and all \(q_1\in\;L^\infty([0,l])\), \(|\lambda_n(q+q_1)-\lambda_n(q)|\leq\|q_1\|_{L^\infty}\) for all \(n\). 4. If the angles \(\phi_0\) and \(\phi_l\) are integer multiples of \({\pi\over 2}\), then for all \(q\in L^1([0,l])\) the series \(\sum_{n=0}^\infty(\lambda_n(q)+(\lambda_n(0)+\lambda_n'(0)q))\) converges to zero.
scientific article; zbMATH DE number 1557651