Titchmarsh-Weyl \(m\) -functions for second-order Sturm-Liouville problems with two singular endpoints. (English) Zbl 1165.34011
The purpose of the present paper is to associate a Titchmarsh-Weyl function to a class of Sturm-Liouville problems on the half line, where the spectrum is simple and both endpoints \(x=0\) and \(x=\infty\) are singular. More precisely, the author considers the Sturm-Liouville equation
\[ -y^{\prime\prime}+qy=\lambda y\text{ on }(0,\infty) \]
under the assumptions \(q(x)\rightarrow 0\) as \(x\rightarrow\infty\) and either of the following hold.
Case I: \(q(x)=q_0x^{-2}+q_1x^{-1}+\sum_{n=0}^\infty q_{n+2}x^n\) for all \(x\in (0,\infty)\), \(q_n\) real for all \(n\), the series is convergent on \((0,\infty)\) and \(-\tfrac{1}{4}\leq q_0<\infty\), \(q_0\), \(q_1\) not both zero.
Case II: For some \(a>0\) the assumptions in Case I hold on the interval \((0,a)\) and \(q\) is real and locally summable in \([a,\infty)\).
Under these assumptions the spectrum of the usual selfadjoint differential operator associated to \(-\tfrac{d^2}{dx^2}+q\) is simple, the continuous spectrum is contained in \((0,\infty)\) and the spectrum in \((-\infty,0]\) is discrete. With the help of suitably normalized Frobenius solutions at the left endpoint a Titchmarsh-Weyl function for the doubly singular problem on \((0,\infty)\) is introduced by passing to the limit \(b\rightarrow\infty\) of a meromorphic function having its poles on the eigenvalues of an associated Sturm-Liouville problem on \((0,b]\). Thus, the definition of the Titchmarsh-Weyl function is analogous to the usual definition if \(x=0\) is a regular endpoint [see E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I. 2nd ed. Oxford: Clarendon Press (1962; Zbl 0099.05201)] and, in particular, the use of solutions defined by initial conditions at an interior point in \((0,\infty)\) is avoided. The eigenfunction expansion is constructed for both situations: \(x=0\) in the limit circle case and \(x=0\) in the limit point case.
It is worth to mention that in the limit point case the Titchmarsh-Weyl function does not belong to the class of Nevanlinna or Riesz-Herglotz functions, but to some generalized Nevanlinna class. The theory is applied to the Bessel equation and to the radial part of the separated hydrogen atom problem.
\[ -y^{\prime\prime}+qy=\lambda y\text{ on }(0,\infty) \]
under the assumptions \(q(x)\rightarrow 0\) as \(x\rightarrow\infty\) and either of the following hold.
Case I: \(q(x)=q_0x^{-2}+q_1x^{-1}+\sum_{n=0}^\infty q_{n+2}x^n\) for all \(x\in (0,\infty)\), \(q_n\) real for all \(n\), the series is convergent on \((0,\infty)\) and \(-\tfrac{1}{4}\leq q_0<\infty\), \(q_0\), \(q_1\) not both zero.
Case II: For some \(a>0\) the assumptions in Case I hold on the interval \((0,a)\) and \(q\) is real and locally summable in \([a,\infty)\).
Under these assumptions the spectrum of the usual selfadjoint differential operator associated to \(-\tfrac{d^2}{dx^2}+q\) is simple, the continuous spectrum is contained in \((0,\infty)\) and the spectrum in \((-\infty,0]\) is discrete. With the help of suitably normalized Frobenius solutions at the left endpoint a Titchmarsh-Weyl function for the doubly singular problem on \((0,\infty)\) is introduced by passing to the limit \(b\rightarrow\infty\) of a meromorphic function having its poles on the eigenvalues of an associated Sturm-Liouville problem on \((0,b]\). Thus, the definition of the Titchmarsh-Weyl function is analogous to the usual definition if \(x=0\) is a regular endpoint [see E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I. 2nd ed. Oxford: Clarendon Press (1962; Zbl 0099.05201)] and, in particular, the use of solutions defined by initial conditions at an interior point in \((0,\infty)\) is avoided. The eigenfunction expansion is constructed for both situations: \(x=0\) in the limit circle case and \(x=0\) in the limit point case.
It is worth to mention that in the limit point case the Titchmarsh-Weyl function does not belong to the class of Nevanlinna or Riesz-Herglotz functions, but to some generalized Nevanlinna class. The theory is applied to the Bessel equation and to the radial part of the separated hydrogen atom problem.
Reviewer: Jussi Behrndt (Berlin)
MSC:
| 34B20 | Weyl theory and its generalizations for ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 34B27 | Green’s functions for ordinary differential equations |
| 34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 35Q40 | PDEs in connection with quantum mechanics |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
Keywords:
Titchmarsh-Weyl \(m\)-function; Sturm-Liouville theory; Frobenius theory; eigenfunction expansion; Green’s function; continuous spectrum; radial hydrogen equation; Schrödinger wave mechanics; Whittaker functionCitations:
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