  We consider type 0A matrix model in the presence of a spacelike D
brane,localized in matter direction at any arbitrary point. It appears that in
order to have an appropriate string/MQM correspondence we must need to impose a
constraint on the matrix model side which is equivalent to an operator
constraint on the matter part of the boundary state that arises from open
string boundary condition. This condition constrains the Hilbert space
generated by the macroscopic loop operator but the bulk matrix model remains
unaffected, thereby describing a situation parallel to string theory. We have
analyzed the constrained theory with uncompactified as well as compactified
time and have shown that it is in good agreement with string theory. Without
this constraint, the grand canonical partition function for MQM on a circle in
the presence of the brane diverges while the constrained partition function is
finite and corresponds to that of a deformed Fermi surface, as expected within
the compactified theory. Matrix model path integral on a circle in the presence
of the brane is expressed as Fredholm determinant. We consider matrix model in
the presence of the brane, to be perturbed by momentum modes. We obtain the
expression of MQM wave function in this background from collective field theory
analysis. We have shown with the help of this constraint that the grand
canonical partition function has an integrable structure and can be expressed
as tau function of Toda hierarchy. Finally we have analyzed the dispersionless
limit.
