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Some properties of third order differential operators

Mariella Cecchi, Zuzana Došlá, Mauro Marini (1997)

Czechoslovak Mathematical Journal

Consider the third order differential operator L given by L ( · ) 1 a 3 ( t ) d d t 1 a 2 ( t ) d d t 1 a 1 ( t ) d d t ( · ) and the related linear differential equation L ( x ) ( t ) + x ( t ) = 0 . We study the relations between L , its adjoint operator, the canonical representation of L , the operator obtained by a cyclic permutation of coefficients a i , i = 1 , 2 , 3 , in L and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).

Spectral properties of fourth order differential operators

Ondřej Došlý, Roman Hilscher (1997)

Mathematica Bohemica

Necessary and sufficient conditions for discreteness and boundedness below of the spectrum of the singular differential operator ( y ) 1 w ( t ) ( r ( t ) y ) , t [ a , ) are established. These conditions are based on a recently proved relationship between spectral properties of and oscillation of a certain associated second order differential equation.

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