ISSN 2658–5782

DOI 10.21662

DOI 10.21662

Electronic Scientific Journal

2019. Vol. 14. Issue 3, Pp. 184–201

URL: http://mfs.uimech.org/mfs2019.3.025,en

DOI: 10.21662/mfs2019.3.025

URL: http://mfs.uimech.org/mfs2019.3.025,en

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

Bashkir State University, Ufa, Russia

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 ^{4} is found. 12 classes of boundary
value eigenvalue problems are described for the operator ^{4}, 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.

degenerate boundary conditions,

finite spectrum,

tenth and eleventh John Locker problems

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

The general form of degenerate boundary conditions for the fourth-order differentiation
operator ^{4} 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 ^{4}, 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
_{j}^{(j−1)}(0) +
(−1)^{j−1} ^{(j−1)}(1) = 0, ^{4},
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 ^{4}.

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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