GLYPH<1>$_{GLYPH<2>}$ GLYPH<1>$_{GLYPH<3>}$ GLYPH<1>$_{GLYPH<4>}$ GLYPH<5>$_{GLYPH<2>}$ GLYPH<5>$_{GLYPH<3>}$ GLYPH<5>$_{GLYPH<4>}$ GLYPH<1>$_{GLYPH<6>}$ GLYPH<5> GLYPH<6> GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7> GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7> GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7> GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7> GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7>GLYPH<8>GLYPH<7> GLYPH<9> GLYPH<10> GLYPH<6> GLYPH<9> GLYPH<10> GLYPH<4>GLYPH<11>GLYPH<6> GLYPH<9> GLYPH<10> GLYPH<4> GLYPH<9> GLYPH<10> GLYPH<6> GLYPH<9> GLYPH<10> GLYPH<6> GLYPH<9> GLYPH<10> GLYPH<2> GLYPH<9>$_{GLYPH<2>}$ GLYPH<9>$_{GLYPH<6>}$ GLYPH<9>$_{GLYPH<3>}$ GLYPH<9>$_{GLYPH<4>}$ FIG. 2: A schematic diagram describing the relation between the root functions ξ$_{n}$ , the stationary states Ψ$_{n}$, and the ladder operators a$_{n}$ , $_{a}$† $_{n}$. We are now ready to find the ladder operators and eigenenergies for our problem. First, we consider Eq. (6) for n = 1 From the form of the Hamiltonian (2) and Eq. (11) we can rewrite Eq. (12) as or equivalently Note that in contrast to Eq. (2), Eq. (14) is a nonlinear first-order differential equation. To solve Eq. (14), we consider the left-hand and right-hand sides of the delta function potential separately. For these regions, Eq. (14) reduces to which has the solution 5 f$_{1}$ ( x ) = √ 2 mE$_{1}$ $_{cot[}$√$_{2}$$_{mE}$$_{1}$ ¯ h ( x − b )] , (16) 1 2 m f 2 1 + ¯ h 2 m df$_{1}$ dx + E$_{1}$ = 0 ( x = pa ) , (15) 1 2 m f 2 1 + ¯ h 2 m df$_{1}$ dx + E$_{1}$ = λδ ( x − pa ) . (14) P 2 2 m + 1 2 m f 2 1 + ¯ h 2 m df$_{1}$ dx + E$_{1}$ = P 2 2 m + λδ ( x − pa ) , (13) a † $_{1}$a$_{1}$ + E$_{1}$ = H. (12)