Nonsingularity of critical points of some differential and difference operators

Fleige, Andreas; Najman, Branko

doi:10.1007/978-3-0348-8789-2_8

Cited by 16 publications

(21 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In some cases our results lead to a new point of view at some results from [15,16,25,26,31,32,33,34]. We also get some new results for the case of nonsymmetric coefficients and the case when A is a coupling of two differential operators of different order.…”

Section: Introductionmentioning

confidence: 56%

See 1 more Smart Citation

Partially Fundamentally Reducible Operators in Kreĭn Spaces

Ćurgus

Derkach

2014

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 56%

“…Now, Theorem 5.12 implies that A 1 is fundamentally reducible in the Kreȋn space L 2 w (R). In the case α ± = β ± = 0 this result was proved in [16] and for α + = α − and β ± = 0 in [26].…”

Section: Examplesmentioning

confidence: 67%

Partially Fundamentally Reducible Operators in Kreĭn Spaces

Ćurgus

Derkach

2014

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

“…Recently, the question as to whether A is similar to a self-adjoint operator was studied in [1]- [4], [7], [8], and [11]- [13] (see also references therein). The interest in this question is partly due to some problems of mathematical physics and the theory of random processes (see [1], [5]).…”

Section: The Main Object Of the Present Paper Is An Indefinite Sturm-mentioning

confidence: 99%

“…Although several criteria for the regularity of the critical points of an abstract J -nonnegative operator were discovered as early as in the 1970-80s, it has been only recently that the first results on the regularity of the critical point 0 of the operator (1) were obtained in [3] and [4]. In these papers,Ćurgus' regularity criterion was used to prove the regularity of the critical point 0 for the case in which q ≡ 0 and ω(x) = |x| α (α > −1).…”

mentioning

confidence: 99%

On the similarity of a J-nonnegative Sturm-Liouville operator to a self-adjoint operator

Karabash

Kostenko

2009

Funct Anal Its Appl

View full text Add to dashboard Cite

In terms of Weyl-Titchmarsh m-functions, we obtain a new necessary condition for an indefinite Sturm-Liouville operator to be similar to a self-adjoint operator. This condition is used to construct examples of J -nonnegative Sturm-Liouville operators with singular critical point zero.Key words: J -nonnegative operator, critical point, similarity to a self-adjoint operator. 1.The main object of the present paper is an indefinite Sturm-Liouville operator of the formWe assume that both the potential q and the weight function ω are real-valued. Besides, we assume that q, ω ∈ L 1 loc (R) and ω(x) > 0 for almost all x ∈ R. Thus, the differential expression (1) is regular on any finite interval [a, b] but is singular at +∞ and −∞. We will assume that (1) is in the limit point case both at +∞ and −∞; i.e., the operatoris self-adjoint in L 2 (R, ω). We will also assume that L is nonnegative, L = L * 0. In other words, the operator A is J -self-adjoint and J -nonnegative (where J is the operator of multiplication by sgn x, J = J * = J −1 ). Clearly, A = A * . However, the following lemma holds. Lemma 1. If the operator A of the form (1) is J -self-adjoint and J -nonnegative, then σ(A) ⊆ R.Recently, the question as to whether A is similar to a self-adjoint operator was studied in [1]-[4], [7], [8], and [11]-[13] (see also references therein). The interest in this question is partly due to some problems of mathematical physics and the theory of random processes (see [1], [5]).2. One main tool in the study of spectral properties of the operator (1) is the spectral theory of J -nonnegative operators. (The main definitions and results can be found in [9].) It is known [9] that the spectrum of A is real if ρ(A) = ∅. Moreover, in this case the operator A admits a spectral function E A ( · ), which is a family of J -orthogonal projections and whose properties essentially differ from those of the spectral function of a self-adjoint operator only near singular critical points of A. Namely, E A ( · ) is unbounded in the vicinity of such points. The critical points that are not singular are said to be regular. Note that only 0 and ∞ can be critical points of A. Lemma 1 allows one to restate the question concerning the similarity of the operator (1) to a self-adjoint operator in terms of regularity of the critical points of its spectral function.Theorem 1. An operator A of the form (1) is similar to a self-adjoint operator in L 2 (R, ω) if and only if (i) ker A = ker A 2 .(ii) 0 and ∞ are not singular critical points of A.

show abstract

“…The question of nonsingularity of 0 is much harder. It was shown in [6,10,14,17,22,23,27,28,29] that 0 is a regular critical point for several model classes of differential operators. In [23] several necessary similarity conditions in terms of Weyl functions were obtained also.…”

Section: Introductionmentioning

confidence: 99%

Similarity of Indefinite Sturm-Liouville Operators with Singular Potential to a Self-Adjoint Operator

Kostenko

2005

Math Notes

View full text Add to dashboard Cite

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 critical point zero. In particular, it is shown that 0 is a singular critical point of the operator − (sgn x)dx 2 acting in the Hilbert space L 2 (R, (3|x| + 1) −4/3 dx) and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type (sgn x)(−d 2 /dx 2 + q(x)) with the same properties.

show abstract

Nonsingularity of critical points of some differential and difference operators

Cited by 16 publications

References 4 publications

Partially Fundamentally Reducible Operators in Kreĭn Spaces

Partially Fundamentally Reducible Operators in Kreĭn Spaces

On the similarity of a J-nonnegative Sturm-Liouville operator to a self-adjoint operator

Similarity of Indefinite Sturm-Liouville Operators with Singular Potential to a Self-Adjoint Operator

Contact Info

Product

Resources

About