I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C\_1 \leq V(x) \leq C\_2$ for all $x \in [0,\infty)$ for some constants $C\_2 > C\_1 >0$.


I am interested to know that a solution (I have some more specific equation of this type, whose behaviour at $\infty$ I roughly know) will be bounded, for example. I imagine it oscillating like a sine wave, with bounded peaks, but don't know how to formalize this.


Grönwall's inequality sounds somewhat related, but it is for first order differential equations. Also, some conservation law of some sum of squares could be hoped to show boundedness, but I could not see if such an equation has a conservation law.


Is there some (hopefully not too advanced and focused on this specific case) reference or advice regarding such problems?



Second order differential equation with oscillating behavior
Under your conditions, not all solutions are bounded. Take $V=V\_0+\lambda$,
where $V\_0$ is a periodic function, and $\lambda$ is going to be large positive.
Then, for generic $V\_0$ we have the following picture: there is a sequence
of real numbers $\lambda\_k\to+\infty$ which divides the positive ray into
intervals. Every other interval is a stability zone (two linearly independent solutions are bounded) and the remaining intervals are gaps (one solution oscillates with exponentially increasing amplitude, another decreases.)
The endpoints $\lambda\_k$ are actually eigenvalues of the periodic and antiperiodic boundary problems. For generic $V\_0$ we have infinite sequences
of both types of intervals. Therefore, for those $\lambda$ in the gaps, $V=V\_0+\lambda$ is bounded from above and below by positive constants, while
one solution of the differential equation is unbounded.


Refs. Levitan, B. M.; Sargsyan, I. S.
Sturm-Liouville and Dirac operators. Dordrecht etc.: Kluwer Academic Publishers, 1990


Marchenko, V., A. Sturm-Liouville operators and applications. Revised ed. of the 1986
Providence, RI: AMS Chelsea (2011).


