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I still can't seem to grasp his abilities and character.

Do we have more information regarding Schrödinger? He's everywhere and nowhere, but how does that work in case his head is blown off?

Could anyone try and shed some light?

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According to the Schrödinger page at the Hellsing Wiki, his abilities are:

Abilities: Can spontaneously appear and disappear at will. Unable to be killed as long as he wills it. Invulnerability. Immortality.

It's not clearly explained what it means for him to "will" it. I guess his head can't be blown off if he doesn't wish it to be or if it happens, he can still will his head back. But I guess like his name and appearance (cat), he's supposed to be a reference to Schrödinger's cat, a paradox, being simultaneously alive and dead. Other tidbits from the article:

It is confirmed by the Major that Schrödinger only exists for as long as he is aware of himself. He was able to appear in Hellsing's conference room without difficulty, despite the high security of the building at the time. He also appeared inside Zorin Blitz's illusions while Blitz was attempting to attack Seras Victoria's mind, indicating that he can appear in mental realms as well.
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Even if Schrödinger is fatally wounded, if he believes that he is alive and unscathed, then he would be. This can be seen when Alucard drinks the blood of Rip Van Winkle. He is sitting on the ship with the hand television, while in next scene we can see him standing on the right side of the Major.

So that sort of explains the immortality part. Except at the end of the OVA:

.. now that he [Schrödinger] is consumed by Alucard and is mixed with millions of other minds, he can no longer recognize himself, and by extension neither can Alucard, the one who has absorbed him. Whilst neither are technically dead, they do not exist in one place because they cannot recognize themselves, and have thus become "No more than a set of imaginary numbers."

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