Self-Adjoint and Positive Operators

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \] HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(|\ldots |\) in the standard integral is crucial for this.

Integrate \[ \begin {array}{rcl} \text {(a)}&\displaystyle \int _0^{2\pi }\frac {d\phi }{(p+q\cos \phi )^2} &\text {(parameters }p>q>0\text {)},\\[2.2ex] \text {(b)}&\displaystyle \int _{-\infty }^{\infty }\frac {e^{\alpha x}}{1+e^x}\,dx &\text {(parameter }\alpha \in (0,1)\text {)},\\[2.2ex] \text {(c)}&\displaystyle \int _0^{\infty }\sin (a^2x^2)\,dx &\text {(positive parameter }a\text {)}. \end {array} \]

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \]

HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(\abs {\cdots }\) in the standard integral is crucial for this.

Assume a bounded, self-adjoint operator \(T:H\to H\), where \(H\) is a Hilbert space, satisfies \[ \langle Tx,x\rangle \ge \beta \lVert x\rVert ^2, \tag {1} \] for some \(\beta >0\). The goal of this problem is to show that \(T\) is one-to-one and onto.

1. Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal {N}(T)=\{0\}\).
2. Show that the range \(\mathcal {R}(T)\) is closed.
3. Show that \(\mathcal {R}(T)^\perp =\{0\}\).
4. Why do we have \(\mathcal {R}(T)=H\)?

Find the leading behavior of the integral \[ I=\int _0^3\frac {\cos x}{\sqrt {x}}e^{-sx}\,dx \] for large positive \(s\).

Define the operator \(K\) on \(L^2[0,1]\) by \[ Ku(x)=\int _0^1 k(x,y)u(y)\,dy, \] for \(x\in [0,1]\), where \(k(x,y)\) is a real-valued, bounded, continuous function, with \(k(x,y)>0\).

1. Find the \(L^2\) adjoint \(K^*\) for \(K\), and find a condition on \(k\) which guarantees that \(K\) is self-adjoint.
2. Describe the properties of the set \[ \{\lambda \in \mathbb {C}:(K-\lambda I)u=0\text { for some }u\ne 0\} \] (cardinality, bounds, etc..).
3. Consider the problem \(Ku=f\). Let \(X\) be the set of all \(f\in L^2[0,1]\) such that a unique solution \(u=K^{-1}f\) exists. Can we be assured that there is a constant \(C\) such that \(\lVert K^{-1}f\rVert \le C\lVert f\rVert \), for all \(f\in X\)? Why or why not?

Let \(A(x)\) be an \(n\times n\) matrix-valued function defined on the smooth bounded connected domain \(U\subset \mathbb {R}^n\). Assume \(A(x)=(a_{i,j}(x))\) is uniformly symmetric positive definite, and \(a_{i,j}\in C^\infty (\overline {U})\) for all \(1\le i,j\le n\). Consider the problem \[ \begin {aligned} -\operatorname {div}(ADu) &= \lambda u, && \text {in }U,\\ u &= 0, && \text {on }\partial U. \end {aligned} \]

1. Characterize the principal (smallest) eigenvalue \(\lambda _1\) in terms of the Rayleigh quotient for this problem.
2. Prove that \(\lambda _1>0\), and that the associated eigenfunction \(u_1\) is a minimizer of the Rayleigh quotient.

Evaluate the integral \[ I=\int _{-\infty }^{+\infty } e^{iax^2}\,dx \] (\(a\) is a positive parameter).

Consider the differential operator \(Lu=u''+k^2u\), \(k\ne 0\), subject to homogeneous boundary conditions \(u(0)=u(1)=0\). Show by direct calculation that the associated Green’s function is \[ g(x,y)= \begin {cases} \dfrac {\sin kx\,\sin k(y-1)}{k\sin k}, & 0\le x<y\le 1,\\[1.2ex] \dfrac {\sin ky\,\sin k(x-1)}{k\sin k}, & 0\le y<x\le 1, \end {cases} \] provided \(\sin k\ne 0\). Using the fact that \(L\) is self-adjoint, show that the solution to the equation \[ u''(x)+k^2u(x)=f(x) \] with inhomogeneous boundary data \(u(0)=\alpha \), \(u(1)=\beta \) is \[ u(x)=\int _0^1 g(x,y)f(y)\,dy-\frac {\alpha }{\sin k}\sin k(x-1) +\frac {\beta }{\sin k}\sin kx \] provided \(\sin k\ne 0\). What happens when \(k=m\pi \)?

1. Prove Bendixson’s Criterion: Let \(\mathbf {f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(\mathbf {f}\) is not identically zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then \(\dot {x}=\mathbf {f}(x)\) has no closed orbit which is entirely in \(E\).
2. Extend the above proof to cover Dulac’s Criterion: Let \(B(x_1,x_2)\) and \(\mathbf {f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(B\mathbf {f}\) is not identically zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then \(\dot {x}=\mathbf {f}(x)\) has no closed orbit which is entirely in \(E\).
3. Consider the nondimensionalized Lotka-Volterra competition equations \[ \begin {aligned} \dot {u}&=ru(1-u-\alpha _1v),\\ \dot {v}&=v(1-v-\alpha _2u), \end {aligned} \] where \(r\), \(\alpha _1\) and \(\alpha _2\) are positive parameters. Apply Dulac’s criterion to this system for \(u,v>0\). (HINT: try \(B(u,v)=1/uv\).) Does Bendixson’s criteria give a similar result?
4. Draw a biological conclusion.

Consider the problem \[ \begin {aligned} Lu &\equiv -\frac {d}{dx}\left (p\frac {du}{dx}\right )+qu = \lambda u,\\ \alpha u(0)+\beta \frac {du}{dx}(0) &= 0,\\ \gamma u(1)+\delta \frac {du}{dx}(1) &= 0 \end {aligned} \tag {5.1} \] where \(x\in [0,1]\), \(p\in C^1[0,1]\), \(q\in C[0,1]\), and \(\alpha ,\beta ,\gamma ,\delta \) are constants.

1. Show that (5.1) is self-adjoint if and only if \(p\) and \(q\) are real, \(\gamma \bar {\delta }=\bar {\gamma }\delta \), and \(\alpha \bar {\beta }=\bar {\alpha }\beta \).
2. Suppose hereafter that \(p=q=1\), \(\alpha =1\), \(\beta =0\), \(\gamma =0\), \(\delta =1\). Compute the eigenvalues and eigenfunctions for \(L\) with the given boundary conditions.
3. Compute the Green’s function for \(L\) (with coefficients and boundary conditions as in (b)).
4. Show how to solve the nonhomogeneous problem \[ Lu=f(x) \] with \(L\) and the boundary conditions as in (c). What conditions on \(f\) suffice to make \(u\) a classical solution?

Prove:

Let \(f\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(f\) is not identically zero and either nonnegative in \(E\) or non-positive in \(E\). Then \(x'=f(x)\) has no closed orbit which is entirely in \(E\).