A criterion for compactness in L-p(R) of the resolvent of the maximal general form Sturm-Liouville operator

作者：Chernyavskaya N A^*; Shuster L A

来源：Proceedings of the Royal Society of Edinburgh: Section A Mathematics , 2016, 146(3): 483-508.

DOI：10.1017/S0308210515000694

• Sturm-Liouville operator
• compactness of resolvent
• estimates of minimal eigenvalues

We consider the equation
-(r(x)y' (x))' + q(x)y(x) = f(x), x is an element of R,
where f is an element of L-p(R), p is an element of (1, infinity) and
r > 0, q >= 0, 1/r is an element of L-1(loc)(R), q is an element of L-1(loc)(R),
lim(vertical bar d vertical bar ->infinity) integral(x)(x-d) dt/r(t) integral(x)(x-d) q(t)dt = infinity, x is an element of R
We assume that this equation is correctly solvable in L-p(R). Under these assumptions, we study the problem of compactness of the resolvent L-p(-1) : L-p(R) -> L-p(R) of the maximal continuously invertible Sturm-Liouville operator L-p : D-p(R) -> L-p(R). Here
L(p)y = -(ry')' + qy, y is an element of D-p,
D-p = {y is an element of L-p(R) : y, ry' is an element of AC(loc)(R), -(ry')' + qy is an element of L-p(R)}.
In the case p = 2, for the compact operator L-2(-1) : L-2(R) -> L-2(R), we obtain two-sided sharp-by-order estimates of the maximal eigenvalue.

• 出版日期2016-6