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Register a new userFractional Sturm-Liouville eigenvalue problems, II
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We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $alpha$, $0<alpha<1$, there is a finite set of real eigenvalues and that, for $alpha$ near $1/2$, there may be none at all. As $alpha to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.
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We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<alpha<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $alpha to 1^-$, and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as $alpha to 1^-$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $alpha$.
On the basis of the theory of Sturm--Liouville problem with distribution coefficients we get the infima and suprema of the first eigenvalue of the problem $-y + (q-lambda) y=0, y(0) -k_0^2 y(0) = y(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set of constant-sign summable functions on $[0,1]$ such that $int_0^1 q dx=pm 1$.
We take advantage of the Sturm-Liouville eigenvalue problem to analytically study the holographic insulator/superconductor phase transition in the probe limit. The interesting point is that this analytical method can not only estimate the most stable mode of the phase transition, but also the second stable mode. We find that this analytical method perfectly matches with other numerical methods, such as the shooting method. Besides, we argue that only Dirichlet boundary condition of the trial function is enough under certain circumstances, which will lead to a more precise estimation. This relaxation for the boundary condition of the trial function is first observed in this paper as far as we know.
We show that there is no classical regular Sturm-Liouville problem on a finite interval whose spectrum consists of infinitely many distinct primes numbers. In particular, this answers in the negative a question raised by Zettl in his book on Sturm-Liouville theory. We also show that there {it may} exist such a problem if the parameter dependence is nonlinear.
We consider Sturm-Liouville problems on the finite interval. We show that spectral data for the case of Dirichlet boundary conditions are equivalent to spectral data for Neumann boundary conditions. In particular, the solution of the inverse problem for the first one is equivalent to the solution of the inverse problem for the second one. Moreover, we discuss similar results for other Sturm-Liouville problems, including a periodic case.
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