arXiv subjects
Publications and source records attributed to Jiangang Qi.
In this paper, further to the point interaction method for inverse Sturm-Liouville problems on finite intervals firstly proposed in our previous work, we will continue to generalize this method to the inverse eigenvalue problems for singular Sturm-Liouville problems on the half real axis.
In the present paper, motivated by point interaction, we propose a new and explicit approach to inverse Sturm-Liouville eigenvalue problems under Dirichlet boundary. More precisely, when a given Sturm-Liouville eigenvalue problem with the unknown integrable potential interacts with $δ$-function potentials, we obtain a family of perturbation problems, called point interaction models in quantum mechanics. Then, only depending on the first eigenvalues of these perturbed problems, we define and study the first eigenvalue function, by which the desired potential can be expressed explicitly and uniquely. As by-products, using the analytic function theoretic tools, we also generalize several fundamental theorems of classical Sturm-Liouville problems to measure differential equations.
In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied.
Assume the mapping $$A:\left\{ \begin{array}{ll} x_{1}=x+ω+y+f(x,y), y_{1}=y+g(x,y), \end{array} \right. (x, y)\in \mathbb{T}^{d}\times B(r_{0}) $$ is reversible with respect to $G: (x, y)\mapsto (-x, y),$ and $| f | _{C^{\ell}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0}, | g |_{C^{\ell+d}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0},$ where $B(r_{0}):=\{|y|\le r_0:\; y\in\mathbb R^d\},$ $\ell=2d+1+μ$ with $0<μ\ll 1.$ Then when $\varepsilon_{0}=\varepsilon_{0}(d)>0$ is small enough and $ω$ is Diophantine, the map $A$ possesses an invariantS torus with rotational frequency $ω.$ As an application of the obtained theorem, the Lagrange stability is proved for a class of reversible Duffing equation with finite smooth perturbation.
The present paper shows that the eigenvalue sequence $\{λ_n(q)\}_{n\geqslant 1}$ of regular Sturm-Liouville eigenvalue problem with certain monotonic weights is uniformly Lipschitz continuous with respect to the potential $q$ on any bounded subset of $L^1([a,b],\mathbb{R})$.
The present paper gives a priori bounds on the possible non-real eigenvalues of regular indefinite Sturm-Liouville problems and obtains sufficient conditions for such problems to admit non-real eigenvalues.
Regular Sturm-Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this note we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.