Negatif Yoğunluk Fonksiyonuna Sahip Kendine Eşlenik Olmayan Schrödinger Operatörü Üzerine Bir Çalışma

Year 2023, Volume: 28 Issue: 1, 220 - 229, 30.04.2023

Nimet Coskun

https://doi.org/10.53433/yyufbed.1139044

Abstract

Bu çalışmada kendine eşlenik olmayan, singüler ve standard dışı bir ağırlık fonksiyonuyla birlikte tanımlanmış operatörün spektral özellikleri ele alınacaktır. Bir boyutlu, zamana bağımlı Schrödinger tipli diferansiyel denklem
-y^''+q(x)y=μ^2 ρ(x)y,x∈[0,∞),
y(0)=0,
başlangıç koşulu ve tamamen negatif olarak tanımlı
ρ(x)=-1,
yoğunluk fonksiyonuyla birlikte göz önüne alınsın. Pozitif değerli sürekli ve süreksiz yoğunluk fonksiyonuna sahip operatörler için literatürde çok sayıda çalışma bulunmaktadır. Yoğunluk fonksiyonunun yapısı operatörün analitik özelliklerini ve çözümlerin gösterimini etkilemektedir. Klasik literatürden farklı olarak, bu çalışmada hiperbolik tipli temel çözümler operatörün spektrumunu belirlemek için kullanılmıştır. Buna ek olarak, özdeğerlerin ve spektral tekilliklerin sonluluğu için gerekli koşullar elde edilmiştir. Böylece, Naimark ve Pavlov koşulları, negatif yoğunluk fonksiyonuna sahip operatör durumunda çözülmüştür.

Keywords

Negatif yoğunluk fonksiyonu, Spektral analiz, Spektral tekillikler

A Study on the Non-selfadjoint Schrödinger Operator with Negative Density Function

Abstract

This study focuses on the spectral features of the non-selfadjoint singular operator with an out-of-the-ordinary type weight function. Take into consideration the one-dimensional time-dependent Schrödinger type differential equation
-y^''+q(x)y=μ^2 ρ(x)y,x∈[0,∞),
holding the initial condition
y(0)=0,
and the density function defined with a completely negative value as
ρ(x)=-1.
There is an enormous number of the papers considering the positive values of ρ(x) for both continuous and discontinuous cases. The structure of the density function affects the analytical properties and representations of the solutions of the equation. Unlike the classical literature, we use the hyperbolic type representations of the equation’s fundamental solutions to obtain the operator’s spectrum. Additionally, the requirements for finiteness of eigenvalues and spectral singularities are addressed. Hence, Naimark’s and Pavlov’s conditions are adopted for the negative density function case.

Keywords

Negative density function, Spectral analysis, Spectral singularities

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