| Summary: | This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, which belongs to the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>, is dense within the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.
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