Dependence of the Weyl coefficient on singular interface conditions

Abstract

We investigate the influence of interface conditions at a singularity of an indefinite canonical system on its Weyl coefficient. An explicit formula which parameterizes all possible Weyl coefficients of indefinite canonical systems with fixed Hamiltonian function is derived. This result is illustrated with two examples: the Bessel equation, which has a singular endpoint, and a Sturm-Liouville equation whose potential has an inner singularity, which arises from a continuation problem for a positive definite function

Semiconductor Device Problems

“…We close this section with answering the question how the Titchmarsh-Weyl coefficient of a general Hamiltonian h ∈ H α transforms when the data part 'ö, b j , d j ' of h is altered but the Hamiltonian function H 1 is kept fixed. This generalizes the case '(σ 0 , σ 1 ) indivisible' of a previous result in [LW1] to higher negative indices. In [LW1,Theorem 5.4] we have answered the corresponding question for general Hamiltonians with ind − h = 1 (not necessarily satisfying (gH α )).…”

“…This generalizes the case '(σ 0 , σ 1 ) indivisible' of a previous result in [LW1] to higher negative indices. In [LW1,Theorem 5.4] we have answered the corresponding question for general Hamiltonians with ind − h = 1 (not necessarily satisfying (gH α )). However, the case when (σ 0 , σ 1 ) is indivisible already there played a special role, cf.…”

“…However, the case when (σ 0 , σ 1 ) is indivisible already there played a special role, cf. [LW1,Corollary 5.5].…”

“…As usual, the corresponding versions for other values of α ∈ [0, π) can be deduced by applying rotation isomorphisms. Note that in [LW1,Corollary 5.5] the case α = π 2 was considered; in this situation one has to replace q h by − 1 q h and q h0 by − 1 q h 0 in the corollary below. Then…”

Boundary Value Problems, Weyl Functions, and Differential Operators

“…The determination of the Kreȋn-von Neumann extension and other nonnegative extensions can be found in [212,350]. For singular perturbations associated with Sturm-Liouville operators, see for instance [331] and the later papers [9,28,29,124,171,182,282,315,316,479,480,481,482,507,531,532,549,550]; for δ-point interactions we refer to [8,293]. Special properties of the Titchmarsh-Weyl coefficient have been studied in many papers; we just mention [130,292,366,367].…”