Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions

Inverse Sturm-Liouville problems with eigenparameter-dependent boundary con ditions are considered. Theorems analogous to those of both Hochstadt and G elfand and Levitan are proved. In particular, let ly = (1/r)(-(py')' + qy), ly = (1/(r) over tilde)(-((p) over tilde y')' + (q) over tilde y), Delta = [(a)(c) (b)(d)] and Sigma = [(r)(t) (s)(u)] where det Delta = delta > 0, c not equal 0, det Sigma > 0, t not equal 0 an d (cs + dr - au - tb)(2) < 4(cr - ta) (ds - ub). Denote by (l; alpha; Delta ) the eigenvalue problem ly = lambda y with boundary conditions y(0) cos al pha + y'(0) sin alpha = 0 and (a lambda + b) y(1) = (c lambda + d)(py')(1). Define ((l) over tilde; alpha; Delta) as above but with l replaced by (l) over tilde. Let w(n) denote the eigenfunction of (l; alpha; Delta) having e igenvalue lambda(n) and initial conditions w(n)(0) = sin alpha and pw'(n)(0 ) = -cos alpha and let y(n) = -aw(n)(1) + cpw'(n)(1). Define (w) over tilde and (y) over tilde(n) similarly. As sample results, it is proved that if (l; alpha; Delta) and ((l) over til de; alpha; Delta) have the same spectrum, and (l; alpha; Sigma) and ((l) ov er tilde; alpha; Sigma) have the same spectrum or integral(0)(1)/W-n/(2) r dt + (/gamma(n)/(2)/delta) = integral(0)(1)/(w) over tilde(n)/(2) (r) over tilde dt + (/<(gamma)over tilde>(n)/(2)/delta) for all n, then q/r = (q) ov er tilde/(r) over tilde.