I managed to solve it with api getContentPane.
I added the following code in the top of the script:
$(document).ready(function() {
jQuery('.searchresult').jScrollPane({
showArrows: true,
autoReinitialise: false
});
})
And I replaced this:
document.getElementById("searchresult").innerHTML = xmlhttp.responseText;
document.getElementById("searchresult").style.display = "inline";
$('.searchresult').jScrollPane({autoReinitialise: true});
with:
api = jQuery("#searchresult").data('jsp');
api.getContentPane().html(xmlhttp.responseText);
document.getElementById("results").style.display="inline";
api.reinitialise();