We present a simple exact analytical solution, using the Weyl-Titchmarsh-Kodaira spectral theorem, for the spectral function of the one-dimensional diatomic molecule model consisting of two attractive delta function wells in the presence of a static external electric field. For sufficiently deep and far apart wells, this molecule supports both an even and an odd state, and the introduction of a...

The generalized Sturm-Liouville problems in this paper stem from the ideas of Christ Shubin and Stolz, based on introducing singularities at a countable number of regular points on the real line. This idea is generalized to the introduction of a countable number of regular or limitcircle singular points. These results are shown to link with the work of Everitt and Zettl concerned with operator ...

We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh m-functions, mj(z), of two Schrödinger operators Hj = − d2 dx2 + qj , j = 1, 2 in L2((0, R)), 0 < R ≤ ∞, are exponentially close, that is, |m1(z) − m2(z)| = |z|→∞ O(e −2 Im(z)a), 0 < a < R, then q1 = q2 a.e. on [0, a]. The result applies to any boundary conditions at x = 0 and x = R and...

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh m-functions. Also we obtain necessary conditions for regularity of the critical points 0 and∞ of J-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular cri...

In this paper, we investigate the resolvent operator of singular q-Sturm-Liouville problem defined as − ( 1 / q ) D ⁻ ¹ [D y x )] + [r - λ ]y )=0 −(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x) , with boundary condition 0 c o s β i n = y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0 where ∈ C λ∈C $r$ is a real function on $[0,∞)$, continuous at zero and r L l ∞ r∈Lq,loc¹(0,∞) . We give an integral representation for some properti...

We present a simple exact analytical solution, using the Weyl-Titchmarsh-Kodaira spectral theorem, for the spectral function of the one-dimensional diatomic molecule model consisting of two attractive delta function wells in the presence of a static external electric field. For sufficiently deep and far apart wells, this molecule supports both an even and an odd state, and the introduction of a...

We present a simple exact analytical solution, using the Weyl-Titchmarsh-Kodaira spectral theorem, for the spectral function of the one-dimensional diatomic molecule model consisting of two attractive delta function wells in the presence of a static external electric field. For sufficiently deep and far apart wells, this molecule supports both an even and an odd state, and the introduction of a...