Geodesic
Discrete spectrum and principal functions of non-selfadjoint differential operator
Czechoslovak Mathematical Journal, Tome 49 (1999) no. 4, pp. 689-700
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In this article, we consider the operator $L$ defined by the differential expression \[ \ell (y)=-y^{\prime \prime }+q(x) y ,\quad - \infty x \infty \] in $L_2(-\infty ,\infty )$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition \[ \sup _{-\infty x \infty } \Big \lbrace \exp \bigl (\epsilon \sqrt{|x|}\bigr ) |q(x)|\Big \rbrace \infty , \quad \epsilon > 0 \] holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.
In this article, we consider the operator $L$ defined by the differential expression \[ \ell (y)=-y^{\prime \prime }+q(x) y ,\quad - \infty x \infty \] in $L_2(-\infty ,\infty )$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition \[ \sup _{-\infty x \infty } \Big \lbrace \exp \bigl (\epsilon \sqrt{|x|}\bigr ) |q(x)|\Big \rbrace \infty , \quad \epsilon > 0 \] holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.
Classification : 34B24, 34L05, 34L15, 34L40, 47E05 
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Tunca, Gülen Başcanbaz; Bairamov, Elgiz. Discrete spectrum and principal functions of non-selfadjoint differential operator. Czechoslovak Mathematical Journal, Tome 49 (1999) no. 4, pp. 689-700. http://geodesic.mathdoc.fr/item/CMJ_1999_49_4_a2/

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