Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

Yıl 2019, Cilt: 40 Sayı: 1, 24 - 34, 22.03.2019

Hayati Olğar

https://doi.org/10.17776/csj.451174

Cited By: 2

Öz

Bu makalenin amacı çok-aralığında tanımlı olan, özdeğer parametresini doğrusal
olarak sınır şartlarında bulunduran ve iki tane ek geçiş şartı içeren Sturm-Liouville problemini araştırmaktır. Klasik
Sturm-Liouville teorisi bu tipten çok-aralıklı sınır-değer-geçiş problemlerini
kapsamaktadır. Klasik Sturm-Liouville problemleri için kendine-eşleniklik,
rezolventin kompaktlığı, spektrumun diskretliği ve uygun özfonksiyonların iyi
bilinenHilbert uzayında ortogonal baz oluşturma özelliği
sağlanmaktadır. Genellikle sınır-değer-geçiş problemleri kendine-eşlenik
değildir ve özfonksiyonlar sistemi klasik Hilbert uzayında baz
oluşturmuyor. Bunu dikkate alarak, bu tipten geçiş problemlerinin
kendine-eşlenik biçimde sonuçlanabilmesi için yeni bir yaklaşım önermişiz.
Bunun dışında uygun operatör-demetinin pozitivliğini gösterebilmek için bazı
yeni Hilbert uzayları tanımladık. İlk olarak bu türden spektral problemlerin
genelleştirilmiş özfonksiyonları kavramını tanımladık. Özel olarak gösterdik
ki, eğerpotansiyeli sürekli ise, o halde genelleşmiş özfonksiyonlar
incelediğimiz problemi klasik anlamda da sağlıyor. Daha sonra bazı kompakt
operatörleri öyle tanımladık ki araştırılan sınır-değer-geçiş problemlerini
uygun operatör demetine dönüştürmek mümkün olsun. Son olarak özdeğer
parametresinin mutlak değerce yeteri kadar büyük negativ değerleri için bu
operatör demetinin kendine eşlenik ve pozitiv olduğunu ispat ettik. Elde edilen
sonuçların düzgün Sturm-Liouville problemlerinin sağladığı klasik sonuçları
genelleştirmesi önem arz etmektedir.

Anahtar Kelimeler

Sınır değer problemleri, özfonksiyonlar, sınır ve geçiş şartları, pozitiv operatörler

Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems

Öz

It is purpose of this paper to investigate
Sturm-Liouville equation on many-interval with the eigenvalue parameter appearing linearly in the
boundary conditions and with two supplementary transmission conditions. The
classical Sturmian theory did not cover such type of many-interval boundary
value transmission problems. For the classical Sturm-Liouville problems it is
guaranteed that the problem is self-adjoint with compact resolvent, the
spectrum is disctrete and consist of eigenvalues and the corresponding
eigenfunctions form an orthogonal basis in the well-known Hilbert space . But the boundary-value-transmission problems are not
self-adjoint and the system of eigenfunctions did not form a basis in the
classical Hilbert space in general. Taking in
view this fact we suggest a new approach for self-adjoint realization of such
type transmission problems. Moreover, we define some new Hilbert spaces to
establish positiveness of corresponding operator-pencil. At first we define a
concept of generalized eigenfunctions for this kind of spectral problems. In
particular it is shown that if the potential  is continuous then the
generalized eigenfunctions satisfies the considered problem is the classical
sense. Then we introduce to the consideration some compact operators such a way
that the considered boundary-value-transmission problem can be reduced to the
appropriate operator-pencil equation. Finally, we prove that this
operator-pencil is self-adjoint and positive definite for sufficiently large
negative values of the eigenparameter. It is important to note that the
obtained results extends classical results associated with regular
Sturm-Liouville problems.

Anahtar Kelimeler

Boundary value problems, eigenfunctions, boundary and transmission conditions, positive operators

Kaynakça

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