Poor, Or More Than A Few Classes Before The Sturm-Liouville Problem Two Eigenvalues

In this paper, we discuss the gap and ratio between the first two eigenvalues of Sturm-Liouville operator with different boundary conditions.Firstly, we introduce the new concepts of the approximate single-well and the approxi-mate single-barrier function, then obtain the spectral gap of Schrodinger equation-y"+qy=Ay with Dirichlet boundary conditions and with an approximate single-well potential on [0, π] is λ2-λ1≥3, equality holds if and only if q is constant, and the ratio of string vibrating equation y"+λρy=0with the Dirichlet boundary conditions and with an approximate single-barrier on [0, π] is, equality holds if and only if p is constant; Secondly, for Schrodinger equation-y"+qy=λy with Neumann boundary conditions and with symmetric double-well function on [0,π], we get the spectral gap λ2-λ1≤1with the technique of S. Abramovich [4], equality holds if and only if q is constant; Finally, for string vibrating equa-tions y"+λρy=0with single barrier density function on [0, π] and with Neumann boundary conditions, we get the spectral ratio with the technique of M. Horvath [6], equality holds if and only if p is constant.