Non-Selfadjoin Difference Operators and Jacobi Matrices with Spectral Singularities

Bairamov, Elgiz; Çakar, Öner; Krall, Allan M.

doi:10.1002/1522-2616(200109)229:1<5::aid-mana5>3.0.co;2-c

Cited by 42 publications

(26 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Lemma Assume that the 2 π periodic function h is analytic in the open upper half plane, all of its derivatives are continuous in the closed upper half plane, and

\sup_{z \in P} |h^{(k)} (z)| \leq A_{k}, k \in N \cup \{0\} .

The set

M \subset ([], - \frac{π}{2}, \frac{3 π}{2})

with linear Lebesgue measure zero is the set of all zeros of the function h with infinite multiplicity in P . If

\int_{0}^{w} \ln T (s) d μ (M_{s}) = - \infty,

where

T (s) = \inf_{k} \frac{A_{k} s^{k}}{k!}, k \in N \cup \{0\},

and μ ( M s ) is the linear Lebesgue measure of s ‐neighborhood of M , w ∈ (0,2 π ) is an arbitrary constant, then h ≡0 …”

Section: Eigenvalues and Spectral Singularitiesmentioning

confidence: 99%

“…() In the case that

({}, a_{n})

and

({}, b_{n})

are real sequences for all

n \in double-struckZ

satisfying a n > 0 and

\sum_{n \in double-struckZ} (||, n) ((), (||, 1 - a_{n}) + (||, b_{n})) < \infty,

Guseinov investigated the inverse problem of scattering theory for . In Bairamov et al and Krall et al, the same equation is considered on the semiaxis under certain conditions on

({}, a_{n})

and

({}, b_{n})

complex sequences, and important spectral properties were obtained. In Adivar and Bairamov,() the difference operator associated with was taken under investigation when

({}, a_{n})

and

({}, b_{n})

are again complex sequences for all

n \in double-struckZ

satisfying .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Difference equations with a point interaction

Bairamov

Cebesoy

Erdal

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

In this paper, we consider an impulsive second‐order difference equation on the whole axis. We determine eigenvalues, spectral singularities, continuous spectrum corresponding to this difference equation with an impulsive condition by using the asymptotic properties of Jost functions, and uniqueness theorems of analytic functions. Finally, we demonstrate that the impulsive difference equation has finite number of eigenvalues and spectral singularities with finite multiplicities under certain conditions.

show abstract

“…Lemma Assume that the 2 π periodic function h is analytic in the open upper half plane, all of its derivatives are continuous in the closed upper half plane, and

\sup_{z \in P} |h^{(k)} (z)| \leq A_{k}, k \in N \cup \{0\} .

The set

M \subset ([], - \frac{π}{2}, \frac{3 π}{2})

with linear Lebesgue measure zero is the set of all zeros of the function h with infinite multiplicity in P . If

\int_{0}^{w} \ln T (s) d μ (M_{s}) = - \infty,

where

T (s) = \inf_{k} \frac{A_{k} s^{k}}{k!}, k \in N \cup \{0\},

and μ ( M s ) is the linear Lebesgue measure of s ‐neighborhood of M , w ∈ (0,2 π ) is an arbitrary constant, then h ≡0 …”

Section: Eigenvalues and Spectral Singularitiesmentioning

confidence: 99%

“…() In the case that

({}, a_{n})

and

({}, b_{n})

are real sequences for all

n \in double-struckZ

satisfying a n > 0 and

\sum_{n \in double-struckZ} (||, n) ((), (||, 1 - a_{n}) + (||, b_{n})) < \infty,

Guseinov investigated the inverse problem of scattering theory for . In Bairamov et al and Krall et al, the same equation is considered on the semiaxis under certain conditions on

({}, a_{n})

and

({}, b_{n})

complex sequences, and important spectral properties were obtained. In Adivar and Bairamov,() the difference operator associated with was taken under investigation when

({}, a_{n})

and

({}, b_{n})

are again complex sequences for all

n \in double-struckZ

satisfying .…”

Section: Introductionmentioning

confidence: 99%

Difference equations with a point interaction

Bairamov

Cebesoy

Erdal

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…Also under the condition (3.11) the function E 0 (λ) has a finite number of zeros with finite multiplicities in [−2, 2] as well as in G ( [2]) .…”

Section: Also (33) and (34) Implies Thatmentioning

confidence: 99%

Spectral Analysis of Non-Selfadjoint Discrete Schrödinger Operators with Spectral Singularities

2001

Self Cite

View full text Add to dashboard Cite

“…It is known that one of the most important properties of non-selfadjoint differential equations is that of having spectral singularities [5][6][7][8][9][10][11]. In [12] it is proved by examples that non-selfadjoint difference equations of second order have spectral singularities. So the theory of these equations becomes interesting.…”

Section: Introductionmentioning

confidence: 99%