Does King Claudius know that Hamlet is a threat?

In Act 2, scene 1 King Claudius is about to be informed by Voltimand on national concerns and issues. The king is too caught up in thinking about why Hamlet is going mad to even care about what information is presented to him. "More than his father's death, that thus hath put him so much from th' understanding of himself." (8). He continues to ponder and invites both Rosencrantz and Guildenstern to join in his discussion. "I entreat you both...you may glean, whether aught to us unknown afflicts him thus." (10-17). It is evident that he is only concerned with domestic family affairs.

In Scene 2:2 (53), King Claudius speaks to Gertrude about his concerns for Hamlet's madness. "He tells me, my dear Gertrude, he hath found the head and source of all your son's distemper."
He is interested in the knowledge that Polonius supposedly has about the reason for his step son's insanity.

When Queen Gertrude responds to this hypothesis, she says "I doubt it is no other but the main. His father's death and our o'erhasty marriage." (57). King Claudius responds to her naive remark by saying "Well we shall sift him." (58). This implies that he is still further interested and concerned with the reason for Hamlet's distraught state. He knows that Hamlet could be a threat and he is becoming worried. After Voltimand's lengthy speech about the state of national affairs, the King only replies with six short lines to this. He was undoubtedly distracted from displaying a reasonably thankful response, suggesting that he still was too concered with Hamlet.