How do psychoanalysts interpret the epistemological concept of "proof" in their theoretical work? Not necessarily of psychoanalysis.
I am not sure what kind of "proof" is referred to, so I'll interpret it as epistemological justification of psychoanalysis, or in the context of psychoanalysis. Psychoanalytic epistemology has been discussed at length, here is an encyclopedia article about it, which characterizes it as "epistemology specific to psychoanalytic knowledge as well as the psychoanalysis of mental processes required in the construction of knowledge", and here is Roustang's paper On the Epistemology of Psychoanalysis. Roustang is critical, but he gives a good summary of Freud's and Lacan's positions.
Freud largely subscribed to the usual hypothetico-deductive epistemology from phenomena to theory in science declaring:
"...the unconscious is a hypothesis needed to explain various psychic facts that escape consciousness: principally dreams, symptoms, slips of the tongue, and jokes. Freud's reasoning is very clear: in order to account for these phenomena, which occur as the absences, lacunae, defects, failures and disorders of conscious speech, and to avoid treating such anomalies as purely absurd or purely mysterious, one must resort to the hypothesis of an unconscious psychism".
According to Roustang, this reasoning requires extending physical determinism to psychological events to justify "filling in" the apparent gaps in them, but that is not something many scientists would necessarily resist. His more serious objection is that Freud's approach just names the unknown instead of explaining it. Lacan's epistemology is more elaborate and links psychoanalysis to linguistic structuralism:
"Lacan grafted language onto the unconscious by affirming that language was the condition of the unconscious. This must be an admissible move, since if men did not speak the unconscious would be impossible to imagine... This is the first step. A second step is taken with the posing of the adage: the unconscious is structured like a language... the adage allows for a third step: in order to know the unconscious, it is enough to know language. More specifically, in order to understand unconscious processes, it is enough to study figures of discourse. Hence the famous passage from condensation to metaphor and from displacement to metonymy".
Thus, Lacan can be seen as extending the empirical base of psychoanalysis from clinical psychology to linguistics by analysis of "deep structure" in language. Since language is a more stable object of study than individual psyche one can hope for a firmer ground to stand on. Roustang describes the move as illegitimate however, because it "rests upon a series of confusions. Unconscious processes are confused with those of the dream or of neuroses; hypothetical processes are confused with real ones, real processes are confused with linguistic forms". And he quotes Pirard's extension of the tautology objection against Freud to Lacan, whose treatment of language
"is more apt to imitate the unconscious than to teach us the meaning of speech... If the unconscious takes over language to the point of becoming equivalent to it, we might as well say that there is no longer, nor was there ever, an unconscious".