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Akhtyamov A.M. Survey of studies on degenerate boundary conditions and finite spectrum. Multiphase Systems. 14 (2019) 3. 184–201 (in Russian).
2019. Vol. 14. Issue 3, Pp. 184–201
URL: http://mfs.uimech.org/mfs2019.3.025
DOI: 10.21662/mfs2019.3.025
Survey of studies on degenerate boundary conditions and finite spectrum
Akhtyamov A.M.
Mavlutov Institute of Mechanics, UFRC RAS, Ufa, Russia
Bashkir State University, Ufa, Russia

Abstract

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.

Keywords

degenerate boundary conditions,
finite spectrum,
tenth and eleventh John Locker problems

Article outline

The work is devoted to the description of degenerate two-point boundary conditions of a homogeneous spectral problem for the diffusion operator. All degenerate boundary conditions of the spectral problem are described for the diffusion operator, for the third-order differentiation operator, for the fourth-order differentiation operator, and for differential equations of any odd order. One of the problems of John Locker (the tenth problem) related to the question: can a spectral boundary-value problem have a finite spectrum be also solved?

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric operator of diffusion, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false-periodic boundary conditions, and is identically equal to a constant other than zero, if and only if its boundary conditions are generalized Cauchy conditions.

All degenerate boundary conditions for a spectral problem with a third-order differential equation y'''(x) = λy(x) are described. It is proved that the boundary conditions for this spectral problem are degenerate if and only if the coefficient matrix of the boundary conditions of size 2 by 6 consists of two square diagonal matrices of order three, on one diagonal of which there are units, and on the other, the roots of some numbers. consists of two diagonal matrices, on one diagonal of which there are units, and on the other - roots of minus one. It is shown that the third-order differential equation y'''(x) = λy(x) with general boundary conditions (not containing a spectral parameter) cannot have a finite spectrum.

The general form of degenerate boundary conditions for the fourth-order differentiation operator D4 is found. Shown, that the coefficient matrix of degenerate boundary conditions of size 4×8 consists of two fourth-order square diagonal matrices. One of the diagonal matrices is unit, and the diagonal of the second diagonal matrix consists of some numbers. We study the operators D4, whose spectrum fills the entire complex plane. A well-known example is a boundary value eigenvalue problem for an even-order differential operator whose spectrum fills the entire complex plane. These boundary conditions have the form Uj(y) = y(j−1)(0) + (−1)j−1 y(j−1)(1) = 0, j = 1, 2, 3, 4. However, in connection with this, another question arises whether there are other examples of such operators. This section shows that such examples exist. In addition, all 12 classes of boundary value eigenvalue problems are described for the operator D4, whose spectrum fills the entire complex plane. Each of the classes contains an arbitrary constant. Therefore, there is a continuum of degenerate boundary conditions for the fourth-order differentiation operator D4.

It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem: are there similar problems for odd-order differential equations? This section gives a positive answer to this question. It is2 proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved.

John Locker posed a problem (tenth problem): can a spectral boundary-value problem have a finite spectrum? Boundary value problems are considered. This paragraph shows that the corresponding boundary-value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. So John Locker’s tenth problem is solved.

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