On a spectral problem in the theory of the heat operator (Q2654942)

The author derives a new property of the explicitly known eigenvalues and eigerfunctions \(\{X_n(x),\lambda_n\}\), \(n= 1,2,\dots\), of the boundary value problem for the diffusion equation with constant coefficients: (i) \(u_{tt}= c^2 u_{xx}\), \(0< x< 1\), \(0< t< T\), (ii) \(u(1,t)= 0\), \(u_x(0,t)+ du_{tt}(0,t)= 0\), (iii) \(u(x,0)= f(x)\). It is proved that for \(n\geq 3\) a system \(Y_n(x)\) may be found biorthogonal to \(X_n(x)\) and \(X_n(x)\), \(n\geq 3\), satisfy (iv) \(X^{\prime\prime}_1+\lambda x= 0\), (v) \(X(1)=0\), (vi) \(\int^1_0 (\sin\lambda^{{1\over 2}}_2\sin \lambda^{{1\over 2}}_1(1- x)-\sin\lambda^{{1\over 2}}_1\sin \lambda^{{1\over 2}}_2(1- x))X(x)\,dx= d(\lambda_2- \lambda_1)\sin\lambda^{{1\over 2}}_1\sin \lambda^{{1\over 2}}_2 X(0)\).