Estimating for a string

The great thing about a string is that it is easy to get lower bounds for the lowest eigenvalue by exhibiting an appropriate superharmonic function. Of course the same thing works in higher dimensions, but it is easier to cook up a superharmonic function on the line than in n-space.

Rather than appeal to established theory, we will find it simplest to concoct our own proof of how to get a lower bound for out of a suitable superharmonic function. This proof, which is based on some ideas of Holland [7], may seem a little mysterious. Its advantage is that it works directly with the definition of that we have been using, rather than the definition of as some kind of eigenvalue.

Lemma. If is piecewise , and if there is a function that satisfies

then .

Proof. Setting

we find that

Let be in . Then