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Let \( a, b \in \mathbb{R}, a < b,\alpha ,\gamma \in \mathbb{R},\beta \in \mathbb{C} \). We define the form \( \mathfrak{t} \) on \( \mathcal{D}\left( \mathfrak{t}\right) = {H}^{1}\left( \mathbb{R}\right) \) by \[ \mathrm{t}\left\lbrack {f, g}\right\rbrack = \left\langle {-\mathrm{i}{f}^{\prime }, - \mathrm{i}{g}^{\pri... | Proof Define \( {T}_{0} = - \frac{{d}^{2}}{d{x}^{2}} \) on \( \mathcal{D}\left( {T}_{0}\right) = \left\{ {f \in {H}^{2}\left( \mathbb{R}\right) : f\left( a\right) = f\left( b\right) = {f}^{\prime }\left( a\right) = }\right. \) \( \left. {{f}^{\prime }\left( b\right) = 0}\right\} \) . As in Example 1.6, it follows that ... | Yes |
Let \( \mathcal{J} \) be an open interval of \( \mathbb{R} \), and \( \mathcal{H} = {L}^{2}\left( \mathcal{J}\right) \) . Let \( p \) and \( q \) be real functions from \( {L}^{\infty }\left( \mathcal{J}\right) \) . Suppose that there is a constant \( c > 0 \) such that \( p\left( x\right) \geq c \) a.e. on \( \mathcal... | If \( M \) is a constant such that \( \left| {p\left( x\right) }\right| \leq M \) and \( \left| {q\left( x\right) }\right| \leq M \) a.e. on \( \mathcal{J} \), we estimate \[ M\parallel f{\parallel }_{{H}^{1}\left( \mathcal{J}\right) }^{2} \geq \mathfrak{t}\left\lbrack f\right\rbrack \geq c{\begin{Vmatrix}{f}^{\prime }... | Yes |
Lemma 10.18 Suppose that \( B \in \mathbf{B}\left( \mathcal{H}\right) \) has a bounded inverse \( {B}^{-1} \in \mathbf{B}\left( \mathcal{H}\right) \) . If \( T \) is a densely defined operator on \( \mathcal{H} \), then \( {\left( {B}^{ * }TB\right) }^{ * } = {B}^{ * }{T}^{ * }B \) . If \( T \) is closed or self-adjoin... | Proof Put \( S \mathrel{\text{:=}} {B}^{ * }{TB} \) . Clearly, \( {S}^{ * } \supseteq {B}^{ * }{T}^{ * }B \) . Let \( x \in \mathcal{D}\left( {S}^{ * }\right) \) . It suffices to prove that \( x \in \mathcal{D}\left( {{B}^{ * }{T}^{ * }B}\right) \) . If \( y \in \mathcal{D}\left( T\right) \), then \( {y}^{\prime } \mat... | Yes |
Theorem 10.19 Suppose that \( \Omega \subseteq {\mathbb{R}}^{d} \) is an open bounded set of class \( {C}^{2} \) . Then the Dirichlet Laplacian \( - {\Delta }_{D} \) is the positive self-adjoint operator on \( {L}^{2}\left( \Omega \right) \) defined by\n\n\[ \left( {-{\Delta }_{D}}\right) f = - {\Delta f}\;\text{ for }... | The form of \( - {\Delta }_{D} \) is given by (10.28), and the form Hilbert space \( \mathcal{D}\left\lbrack {-{\Delta }_{D}}\right\rbrack \) is the Sobolev space \( {H}_{0}^{1}\left( \Omega \right) \) . The operator \( - {\Delta }_{D} \) has a purely discrete spectrum contained in \( \left\lbrack {{c}_{\Omega }, + \in... | Yes |
Theorem 10.20 Let \( \Omega \subseteq {\mathbb{R}}^{d} \) be an open bounded set of class \( {C}^{2} \) . Then the Neumann Laplacian \( - {\Delta }_{N} \) is the positive self-adjoint operator on \( {L}^{2}\left( \Omega \right) \) which acts by \( \left( {-{\Delta }_{N}}\right) f = - {\Delta f} \) for \( f \) in the do... | The Dirichlet Laplacian was defined as the Friedrichs extension of the minimal operator \( {L}_{\min } \) . It can be shown that the Neumann Laplacian can be obtained in a similar manner as the Friedrichs extension of the operator \( {L}^{N} = - \Delta \) with domain\n\n\[ \mathcal{D}\left( {L}^{N}\right) = \left\{ {f ... | Yes |
Theorem 10.21 (KLMN theorem) Let \( A \) be a positive self-adjoint operator on a Hilbert space \( \mathcal{H} \) . Suppose that \( \mathfrak{s} \) is a relatively \( {\mathfrak{t}}_{A} \) -bounded symmetric form on \( \mathcal{H} \) with \( {\mathfrak{t}}_{A} \) -bound \( {\beta }_{{\mathfrak{t}}_{A}}\left( \mathfrak{... | Proof We define the form \( \widetilde{\mathfrak{t}} \) by \( \widetilde{\mathfrak{t}}\left\lbrack {x, y}\right\rbrack = A\left\lbrack {x, y}\right\rbrack + \mathfrak{s}\left\lbrack {x, y}\right\rbrack \) for \( x, y \in \mathcal{D}\left( \widetilde{\mathfrak{t}}\right) \mathrel{\text{:=}} \mathcal{D}\left\lbrack A\rig... | Yes |
Proposition 10.22 Let \( A \) and \( B \) be lower semibounded self-adjoint operators such that \( \mathcal{D}\left\lbrack A\right\rbrack \cap \mathcal{D}\left\lbrack B\right\rbrack \) is dense in the underlying Hilbert space \( \mathcal{H} \). (i) There is a unique self-adjoint operator \( C \), called the form sum of... | Proof (i): Since the lower semibounded form \( {\mathfrak{t}}_{A} + {\mathfrak{t}}_{B} \) is densely defined by assumption and closed by Corollary 10.2, the assertion follows at once from Theorem 10.7. (ii): For \( x \in \mathcal{D}\left( A\right) \cap \mathcal{D}\left( B\right) \) and \( y \in \mathcal{D}\left\lbrack ... | Yes |
Proposition 10.23 Let \( \\mathcal{F} \) be a finite subset of \( {\\mathbb{R}}^{d} \), and \( V \) a nonnegative Borel function on \( {\\mathbb{R}}^{d} \) such that \( V \\in {L}^{1}\\left( K\\right) \) for each compact set \( K \\subseteq {\\mathbb{R}}^{d} \) satisfying \( K \\cap \\mathcal{F} = \\varnothing \) . The... | Proof Obviously, \( \\mathcal{D} \\mathrel{\\text{:=}} {C}_{0}^{\\infty }\\left( {{\\mathbb{R}}^{d} \\smallsetminus \\mathcal{F}}\\right) \) is dense in \( {L}^{2}\\left( {\\mathbb{R}}^{d}\\right) \) . Let \( f \\in \\mathcal{D} \) . Then supp \( f \) is compact, and (supp \( f) \\cap \\mathcal{F} = \\varnothing \), so... | Yes |
Example 10.10 (Form sum versus operator sum) Let \( \omega \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \) be a nonnegative function such that \( \operatorname{supp}\omega \subseteq \left\lbrack {-2,2}\right\rbrack \) and \( \omega \left( x\right) = 1 \) for \( x \in \left\lbrack {-1,1}\right\rbrack \) . Let \( \left\... | \[ V\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{2}^{-n}\frac{\omega \left( {x - {r}_{n}}\right) }{{\left| x - {r}_{n}\right| }^{\alpha }} \] if \( x \) is irrational. Since the integral of \( \left| {\omega \left( {x - {r}_{n}}\right) }\right| {\left| x - {r}_{n}\right| }^{-\alpha } \) over \( \mathbb{R... | Yes |
Example 10.11 (Form sum \( - \frac{{d}^{2}}{d{x}^{2}}\dot{ + }{\alpha \delta } \) ) Let \( T = - \mathrm{i}\frac{d}{dx} \) and \( A \mathrel{\text{:=}} {T}^{2} = - \frac{{d}^{2}}{d{x}^{2}} \) be the self-adjoint operators on \( {L}^{2}\left( \mathbb{R}\right) \) from Example 1.7. Then \( \mathfrak{t}\left\lbrack {f, g}... | \[ {\mathfrak{s}}_{\alpha }\left\lbrack {f, g}\right\rbrack = {\alpha f}\left( 0\right) \overline{g\left( 0\right) }\;\text{ for }f, g \in \mathcal{D}\left( {\mathfrak{s}}_{\alpha }\right) \mathrel{\text{:=}} \mathcal{D}\left( T\right) = {H}^{1}\left( \mathbb{R}\right) . \] From Lemma 1.12 it follows that the form \( {... | Yes |
Statement \( 1{\mathfrak{s}}_{F} \) is relatively \( {\mathfrak{t}}_{A} \) -bounded if and only if there is a vector \( u \in \mathcal{H} \) such that \( F\left( x\right) = \left\langle {{A}^{1/2}x, u}\right\rangle, x \in \mathcal{D}\left\lbrack A\right\rbrack \) . If this is true, the \( {\mathfrak{t}}_{A} \) -bound o... | Proof The if assertion is easily verified. We carry out the proof of the only if part.\n\nAssume that \( {\mathfrak{s}}_{F} \) is relatively \( {\mathfrak{t}}_{A} \) -bounded. Then there are positive constants \( a, b \) such that (10.40) is satisfied. Since \( A \geq I \) ,(10.40) implies that\n\n\[ \left| {{\mathfrak... | Yes |
Lemma 11.1 A form \( \mathrm{t} \) is bounded on \( V \) if and only if there is constant \( M > 0 \) such that \( \left| {\mathfrak{t}\left\lbrack u\right\rbrack }\right| \leq M\parallel u{\parallel }_{V}^{2} \) for \( u \in \mathcal{D}\left( \mathfrak{t}\right) \) . | Proof The only if direction is trivial by setting \( u = v \) in (11.2). Hence, it suffices to verify the if part. If \( \parallel u{\parallel }_{V} \leq 1 \) and \( \parallel v{\parallel }_{V} \leq 1 \), then we have \( \mathfrak{t}\left\lbrack {u + {\tau v}}\right\rbrack \leq {4M} \) for \( \tau = 1, - 1,\mathrm{i}, ... | Yes |
Lemma 11.2 (Lax-Milgram lemma) Let \( \mathfrak{t} \) be a bounded coercive form on a Hilbert space \( \left( {V,\langle \cdot , \cdot {\rangle }_{V}}\right) \) such that \( \mathcal{D}\left( \mathrm{t}\right) = V \) . Then there exists an operator \( B \in \mathbf{B}\left( V\right) \) with inverse \( {B}^{-1} \in \mat... | Proof Let \( u \in V \) . By (11.2), \( {F}_{u}\left( \cdot \right) \mathrel{\text{:=}} \overline{\mathrm{t}\left\lbrack {u, \cdot }\right\rbrack } \) is a bounded linear functional on \( V \) with \( \begin{Vmatrix}{F}_{u}\end{Vmatrix} \leq C\parallel u{\parallel }_{V} \) . From the Riesz representation theorem it fol... | Yes |
Theorem 11.3 Let \( \mathfrak{t} \) be a bounded coercive form on the Hilbert space \( \left( {V,\langle \cdot , \cdot {\rangle }_{V}}\right) \). Then the operator \( {A}_{\mathfrak{t}} \) associated with \( \mathfrak{t} \), considered as a form on \( \mathcal{H} \) with domain \( \mathcal{D}\left( \mathrm{t}\right) = ... | Proof The scalar product \( \mathfrak{s} \mathrel{\text{:=}} \langle \cdot , \cdot {\rangle }_{V} \) of \( V \) is a positive form on \( \mathcal{H} \). Since the form norm of \( \mathfrak{s} \) and the norm of \( V \) are equivalent by (11.4) and \( V \) is a Hilbert space, \( \mathfrak{s} \) is closed. Let \( {A}_{\m... | Yes |
Corollary 11.4 Let \( {\mathrm{t}}_{1} \) and \( {\mathrm{t}}_{2} \) be bounded coercive forms on Hilbert spaces \( \left( {{V}_{1},\langle \cdot , \cdot {\rangle }_{{V}_{1}}}\right) \) and \( \left( {{V}_{2},\langle \cdot , \cdot {\rangle }_{{V}_{2}}}\right) \) which are densely and continuously embedded into the Hilb... | Proof Since \( {\mathrm{t}}_{1}\left\lbrack {x, y}\right\rbrack = \left\langle {{A}_{{\mathrm{t}}_{1}}x, y}\right\rangle = \left\langle {{A}_{{\mathrm{t}}_{2}}x, y}\right\rangle = {\mathrm{t}}_{2}\left\lbrack {x, y}\right\rbrack \) for all \( x, y \in \mathcal{D} \mathrel{\text{:=}} \) \( \mathcal{D}\left( {A}_{{\mathr... | Yes |
Corollary 11.5 Let \( \mathfrak{t} \) be a bounded coercive form on the Hilbert space \( \left( {V,\langle \cdot , \cdot {\rangle }_{V}}\right) \) . Then \( {A}_{\mathrm{t}} \) is self-adjoint if and only if \( \mathrm{t} \) is symmetric. | Proof By Theorem 11.3(ii), \( {A}_{\mathrm{t}} \) is self-adjoint if and only if \( {A}_{\mathrm{t}} = {A}_{{\mathrm{t}}^{ * }} \) . By Corollary 11.4, \( {A}_{\mathrm{t}} = {A}_{{\mathrm{t}}^{ * }} \) is equivalent to \( \mathrm{t} = {\mathrm{t}}^{ * } \) . | Yes |
Theorem 11.8 (Representation theorem for sectorial forms) Suppose that \( \mathfrak{t} \) is a densely defined closed sectorial form on \( \mathcal{H} \). Then the operator \( {A}_{\mathfrak{t}} \) is m-sectorial, and we have:\n\n(i) \( \mathcal{D}\left( {A}_{\mathfrak{t}}\right) \) is a dense linear subspace of the Hi... | Proof Because \( \mathrm{t} \) is closed, \( V \mathrel{\text{:=}} \left( {\mathcal{D}\left( \mathrm{t}\right) ,\parallel \cdot {\parallel }_{\operatorname{Re}\mathrm{t}}}\right) \) is a Hilbert space. Since \( \mathcal{D}\left( \mathrm{t}\right) \) is dense in \( \mathcal{H} \) and \( \parallel \cdot \parallel \leq \p... | Yes |
Corollary 11.9 The map \( \mathrm{t} \rightarrow {A}_{\mathrm{t}} \) gives a one-to-one correspondence between densely defined closed sectorial forms and m-sectorial operators on a Hilbert space. | Proof The existence and injectivity of this mapping follow from Theorem 11.8 and Corollary 11.4. It remains to prove the surjectivity.\n\nLet \( T \) be an \( m \) -sectorial operator, and let \( \mathfrak{t} \) be the closure of the form \( {\mathfrak{s}}_{T} \) from Proposition 11.7. By Proposition 3.19(iii), \( T \)... | Yes |
Proposition 11.10 The form \( \mathrm{t} \) on \( V \) is an elliptic form according to Definition 11.2. More precisely, for any \( \gamma < \alpha \), there exists a real constant \( {c}_{\gamma } \) such that\n\n\[ \operatorname{Re}t\left\lbrack u\right\rbrack - {c}_{\gamma }\parallel u{\parallel }^{2} \geq \gamma \p... | Proof Since \( {a}_{kl},{b}_{k},{c}_{k}, q \in {L}^{\infty }\left( \Omega \right) \), it follows at once from the Cauchy-Schwarz inequality that \( \mathfrak{t} \) is bounded on \( V \), that is, there is a constant \( C > 0 \) such that\n\n\[ \left| {\mathfrak{t}\left\lbrack {u, v}\right\rbrack }\right| \leq C\paralle... | Yes |
Lemma 11.11 \( {L}_{\min } \subseteq {A}_{\mathrm{t}} \subseteq {L}_{\max } \) . | Proof Let \( u \in {C}_{0}^{\infty }\left( \Omega \right) \) . We argue in a similar manner as in the proof of Eq. (10.29). We choose a bounded open set \( \widetilde{\Omega } \subseteq \Omega \) of class \( {C}^{2} \) such that \( \operatorname{supp}u \subseteq \widetilde{\Omega } \) and apply the Green formula (D.5) ... | Yes |
Proposition 11.12 Let \( V = {H}_{0}^{1}\left( \Omega \right) \) and \( {a}_{kl},{b}_{k},{c}_{k}, q \in {C}^{\infty }\left( \Omega \right) \cap {L}^{\infty }\left( \Omega \right) \) for all \( k \) , \( l \) . Assume that the ellipticity assumption (11.12) holds. Then \( {A}_{\mathfrak{t}} \) is the restriction of the ... | Proof Since \( {A}_{\mathrm{t}} \subseteq {L}_{\max } \) by Lemma 11.11 and \( \mathcal{D}\left( {A}_{\mathrm{t}}\right) \subseteq \mathcal{D}\left\lbrack {A}_{\mathrm{t}}\right\rbrack = {H}_{0}^{1}\left( \Omega \right) \), we have \( \mathcal{D}\left( {A}_{\mathrm{t}}\right) \subseteq {H}_{0}^{1}\left( \Omega \right) ... | Yes |
Corollary 11.13 Retain the assumptions of Proposition 11.12 and suppose that the conditions (11.15) hold. Then the Friedrichs extension \( {\left( {L}_{0}\right) }_{F} \) of the lower semi-bounded symmetric operator \( {L}_{0} \) and of its closure \( {L}_{\min } \) are given by | \[ \mathcal{D}\left( {\left( {L}_{0}\right) }_{F}\right) = {H}_{0}^{1}\left( \Omega \right) \cap \mathcal{D}\left( {L}_{\max }\right) \;\text{ and }\;{\left( {L}_{0}\right) }_{F}f = {L}_{\max }f\;\text{ for }f \in \mathcal{D}\left( {\left( {L}_{0}\right) }_{F}\right) . \] | Yes |
Proposition 11.14 Suppose that \( \Omega \) is of class \( {C}^{2} \) . For the operator \( {L}_{N} \), we have \( {L}_{\min } \subseteq {L}_{N} \subseteq {L}_{\max } \) . A function \( f \in {H}^{2}\left( \Omega \right) \) belongs to \( \mathcal{D}\left( {L}_{N}\right) \) if and only if it satisfies the boundary condi... | Proof The relations \( {L}_{\min } \subseteq {L}_{N} \subseteq {L}_{\max } \) are already contained in Lemma 11.11.\n\nLet \( f \in {H}^{2}\left( \Omega \right) \) . Clearly, \( f \in \mathcal{D}\left( {L}_{\max }\right) \) . For \( v \in V = {H}^{1}\left( \Omega \right) \), we compute\n\n\[ \mathfrak{t}\left\lbrack {f... | Yes |
Lemma 12.2 Set \( {\mathcal{E}}_{\lambda } \mathrel{\text{:=}} {E}_{A}\left( \left( {-\infty ,\lambda }\right) \right) \mathcal{H} \) and \( d\left( \lambda \right) \mathrel{\text{:=}} \dim {\mathcal{E}}_{\lambda } \) for \( \lambda \in \mathbb{R} \) . Then\n\n\[ d\left( \lambda \right) < n\;\text{ if }\lambda < {\mu }... | Proof Assume to the contrary that (12.5) is false, that is, \( d\left( \lambda \right) \geq n \) and \( \lambda < {\mu }_{n}\left( A\right) \) . Let \( \mathcal{D} \in {\mathfrak{F}}_{n - 1} \) be given. Since \( d\left( \lambda \right) = \dim {\mathcal{E}}_{\lambda } \geq n \), there exists a unit vector \( x \in {\ma... | Yes |
Corollary 12.3 Let \( A \) and \( B \) be lower semibounded self-adjoint operators on (infinite-dimensional) Hilbert spaces \( \mathcal{G} \) and \( \mathcal{H} \), respectively, such that \( \mathcal{G} \) is a subspace of \( \mathcal{H} \). Suppose that \( A \geq B \) or \( A \succcurlyeq B \). Then we have \( {\lamb... | Proof From the relation \( A \geq B \) resp. \( A \succcurlyeq B \) it follows immediately that \( {\widetilde{\mu }}_{n}\left( A\right) \geq \) \( {\widetilde{\mu }}_{n}\left( B\right) \) resp. \( {\mu }_{n}\left( A\right) \geq {\mu }_{n}\left( B\right) \) for \( n \in \mathbb{N} \), so \( {\lambda }_{n}\left( A\right... | Yes |
Corollary 12.4 Let \( A \) be a positive self-adjoint operator, and \( B \) a self-adjoint operator on \( \mathcal{H} \). Suppose that \( B \) is relatively \( A \) -bounded with \( A \) -bound zero and \( {\sigma }_{\mathrm{{ess}}}\left( {A + {\beta B}}\right) = \lbrack 0, + \infty ) \) for \( \beta \geq 0 \). Then th... | Proof First, we note that by the Kato-Rellich Theorem 8.5 the operator \( A + {\beta B} \) is self-adjoint on \( \mathcal{D}\left( {A + {\beta B}}\right) = \mathcal{D}\left( A\right) \). From the assumption \( {\sigma }_{\mathrm{{ess}}}\left( {A + {\beta B}}\right) = \lbrack 0, + \infty ) \) and the above definition of... | Yes |
Proposition 12.5 Let \( A \) be a lower semibounded self-adjoint operator on \( \mathcal{H} \) . Let \( V \) be a d-dimensional subspace of \( \mathcal{D}\left( A\right) \), and \( {A}_{V} \mathrel{\text{:=}} {PA} \mid V \) the compression of \( A \) to \( V \), where \( P \) is the orthogonal projection onto \( V \) .... | Proof Let \( k \in \{ 1,\ldots, d\} \) . Since \( {\lambda }_{k}\left( A\right) = {\mu }_{k}\left( A\right) \) and \( {\lambda }_{k}\left( {A}_{V}\right) = {\mu }_{k}\left( {A}_{V}\right) \) by (12.4 (see, e.g., Remark 2.), it suffices to prove that \( {\mu }_{k}\left( {A}_{V}\right) \geq {\mu }_{k}\left( A\right) \) .... | Yes |
Proposition 12.6 (Temple's inequality) Let \( A \) be a lower semibounded self-adjoint operator such that \( {\lambda }_{1}\left( A\right) < {\lambda }_{2}\left( A\right) \), and let \( \alpha \in \mathbb{R}, x \in \mathcal{D}\left( {A}^{2}\right) ,\parallel x\parallel = 1 \) . Suppose that \( \langle {Ax}, x\rangle < ... | Proof Set \( {\lambda }_{1} \mathrel{\text{:=}} {\lambda }_{1}\left( A\right) \) . Since \( {\lambda }_{1}\left( A\right) < \alpha < {\lambda }_{2}\left( A\right) \), we have \( \sigma \left( A\right) \cap \left( {{\lambda }_{1},\alpha }\right) = \varnothing \) . Hence, \[ \left\langle {\left( {A - {\lambda }_{1}I}\rig... | Yes |
Proposition 12.7 Let \( V \) be a nonnegative function of \( {L}_{\mathrm{{loc}}}^{\infty }\left( {\mathbb{R}}^{d}\right) \) such that \( \mathop{\lim }\limits_{{\parallel x\parallel \rightarrow \infty }}V\left( x\right) = + \infty \) . Then the form sum \( A = - \Delta + V \) is a positive self-adjoint operator with p... | Proof First, we note that the form sum \( A = - \Delta \dot{ + }V \) is a well-defined positive selfadjoint operator by Proposition 10.23, since \( V \in {L}_{\mathrm{{loc}}}^{\infty }\left( {\mathbb{R}}^{d}\right) \) . Let \( M > 0 \) be given. Since \( \mathop{\lim }\limits_{{\parallel x\parallel \rightarrow + \infty... | Yes |
Proposition 12.8 Let \( A \) be a self-adjoint operator on a Hilbert space \( \mathcal{H} \) such that \( {\sigma }_{\text{ess }}\left( A\right) \subseteq \lbrack 0, + \infty ) \). (i) A has a negative eigenvalue if and only if \( \langle {Ax}, x\rangle < 0 \) for some \( x \in \mathcal{D}\left( A\right) \). (ii) A has... | Proof Let \( {E}_{A} \) denote the spectral measure of \( A \) . We freely use the description of the spectrum given in Proposition 5.10. Since \( {\sigma }_{\mathrm{{ess}}}\left( A\right) \subseteq \lbrack 0, + \infty ) \), all points of \( \sigma \left( A\right) \cap \) \( \left( {-\infty ,0}\right) \) are eigenvalue... | Yes |
Proposition 12.9 Let \( V \) be a real-valued function of \( {L}^{2}\left( {\mathbb{R}}^{3}\right) + {L}^{\infty }{\left( {\mathbb{R}}^{3}\right) }_{\varepsilon } \) . Suppose that there exist positive constants \( c,{R}_{0} \), and \( \delta \) such that \( \delta < 2 \) and\n\n\[ V\left( x\right) \leq - c\parallel x{... | Proof By Theorems 8.8 and \( {8.19}, A \mathrel{\text{:=}} - \Delta + V \) is a self-adjoint operator on \( {L}^{2}\left( {\mathbb{R}}^{3}\right) \) and \( {\sigma }_{\text{ess }}\left( A\right) = \lbrack 0, + \infty ) \) . Hence, the assumptions of Proposition 12.8 are satisfied.\n\nLet us fix a function \( \varphi \i... | Yes |
Proposition 12.10 Let \( V \) be a real-valued Borel function on \( {\mathbb{R}}^{d} \) which is relatively \( \left( {-\Delta }\right) \) -bounded with \( \left( {-\Delta }\right) \) -bound less than one. Suppose that there exists a number \( \beta \in \left( {0,2}\right) \) such that\n\n\[ V\left( {ax}\right) = {a}^{... | Proof By the Kato-Rellich Theorem 8.5, the operator \( - \Delta + V \) is self-adjoint on \( \mathcal{D}\left( {-\Delta }\right) \) . Let \( \lambda \) be an eigenvalue of \( - \Delta + V \) with eigenvector \( f,\parallel f\parallel = 1 \) . Put \( {f}_{a}\left( x\right) = f\left( {ax}\right) \) for \( a \in \left( {0... | Yes |
Example 12.1 (Coulomb potential \( V\left( x\right) = - \gamma \parallel x{\parallel }^{-1} \) on \( {L}^{2}\left( {\mathbb{R}}^{3}\right) \) ) First, suppose that \( \gamma \in \mathbb{R} \) . Recall from Example 8.5 that \( V \in {L}^{2}\left( {\mathbb{R}}^{3}\right) + {L}^{\infty }{\left( {\mathbb{R}}^{3}\right) }_{... | The potential \( V \) satisfies the homogeneity condition (12.12) with \( \beta = 1 \) . Hence, by Proposition 12.10, the self-adjoint operator \( - \Delta + V \) has no eigenvalue in \( \lbrack 0, + \infty ) \) . Further, let \( \lambda \) be an eigenvalue of \( - \Delta + V \) with normalized eigenfunction \( f \) . ... | Yes |
Lemma 12.11 Let \( \Omega \) and \( \widetilde{\Omega } \) be open subsets of \( {\mathbb{R}}^{d} \). (i) If \( \Omega \subseteq \widetilde{\Omega } \), then \( - {\Delta }_{D,\widetilde{\Omega }} \leq - {\Delta }_{D,\Omega } \). (ii) \( - {\Delta }_{N,\Omega } \leq - {\Delta }_{D,\Omega } \). | Proof Both inequalities follow at once from the definitions of the order relation \ | No |
Lemma 12.12 Let \( {\Omega }_{1},\ldots ,{\Omega }_{q} \) be pairwise disjoint open subsets of \( {\mathbb{R}}^{d} \). (i) If \( \Omega = \mathop{\bigcup }\limits_{{j = 1}}^{q}{\Omega }_{j} \), then \( - {\Delta }_{D,\Omega } = {\bigoplus }_{j = 1}^{q} - {\Delta }_{D,{\Omega }_{j}} \) on \( {L}^{2}\left( \Omega \right)... | Proof For a function \( f \) on \( \Omega \) we denote its restriction to \( {\Omega }_{j} \) by \( {f}_{j} \). (i): Since \( \Omega \) is the union of the disjoint sets \( {\Omega }_{j} \), for \( f, g \in {C}_{0}^{\infty }\left( \Omega \right) \), we have \[ {\mathfrak{t}}_{D,\Omega }\left\lbrack {f, g}\right\rbrack ... | Yes |
Example 12.2 \( \\left( {\\Omega = \\left( {a, a + l}\\right), d = 1, l > 0}\\right) \) In this case, \( - {\\Delta }_{D,\\Omega } \) is the differential operator \( - \\frac{{d}^{2}}{d{x}^{2}} \) with boundary conditions \( f\\left( a\\right) = f\\left( {a + l}\\right) = 0 \) . Easy computations show that \( - {\\Delt... | \[ \\sigma \\left( {-{\\Delta }_{D,\\Omega }}\\right) = \\left\\{ {{n}^{2}{\\pi }^{2}{l}^{-2} : n \\in \\mathbb{N}}\\right\\} \] consisting of simple eigenvalues \( {n}^{2}{\\pi }^{2}{l}^{-2} \) and the corresponding eigenfunctions \( {\\varphi }_{n} \), \( n \\in \\mathbb{N} \), are \[ {\\varphi }_{2k}\\left( x\\right... | Yes |
Example 12.3 \( \left( {\Omega = \left( {{a}_{1},{a}_{1} + l}\right) \times \cdots \times \left( {{a}_{d},{a}_{d} + l}\right) \subseteq {\mathbb{R}}^{d}, l > 0}\right) \) By separation of variables we easily determine the spectra and the eigenfunctions of the operators \( - {\Delta }_{D,\Omega } \) and \( - {\Delta }_{... | \[ \sigma \left( {-{\Delta }_{D,\Omega }}\right) = \left\{ {\left( {{n}_{1}^{2} + \ldots + {n}_{d}^{2}}\right) {\pi }^{2}{l}^{-2} : {n}_{1},\ldots ,{n}_{d} \in \mathbb{N}}\right\} , \] \[ {\varphi }_{n}\left( x\right) = {\varphi }_{{n}_{1}}\left( {x}_{1}\right) \cdots {\varphi }_{{n}_{d}}\left( {x}_{d}\right) ,\;n = \l... | Yes |
Lemma 12.13 Let \( \Omega = \left( {{a}_{1},{a}_{1} + l}\right) \times \cdots \times \left( {{a}_{d},{a}_{d} + l}\right) \subseteq {\mathbb{R}}^{d}, l > 0 \) . For \( \lambda > 0 \) , \[ \left| {{N}_{D,\Omega }\left( \lambda \right) - {\lambda }^{d/2}{\omega }_{d}{l}^{d}{\left( 2\pi \right) }^{-d}}\right| \leq \mathop{... | Proof From Example 12.3 and formula (12.20) it follows that \( {N}_{D,\Omega }\left( \lambda \right) \) is the number of points \( n = \left( {{n}_{1},\ldots ,{n}_{d}}\right) \in {\mathbb{N}}^{d} \) such that \( \left( {{n}_{1}^{2} + \ldots + {n}_{d}^{2}}\right) {\pi }^{2}{l}^{-2} < \lambda \), or equivalently, the num... | Yes |
Lemma 13.1 \( {d}^{i}\left( V\right) = \dim \mathcal{R}{\left( V\right) }^{ \bot } \) and \( {d}^{e}\left( V\right) = \dim \mathcal{D}{\left( V\right) }^{ \bot } \) . | Proof By (13.1), we have \( {d}^{i}\left( V\right) = {d}_{0}\left( V\right) = \dim \mathcal{R}{\left( V\right) }^{ \bot } \). Fix \( \mu \in \mathbb{C},0 < \left| \mu \right| < 1 \). Since \( \left( {{V}^{-1} - \mu }\right) {Vx} = \left( {I - {\mu V}}\right) x = - \mu \left( {V - {\mu }^{-1}}\right) x \) for \( x \in \... | Yes |
Lemma 13.2 If \( V \) is an isometric operator on \( \mathcal{H} \) and \( \mathcal{R}\left( {I - V}\right) \) is dense in \( \mathcal{H} \), then \( \mathcal{N}\left( {I - V}\right) = \{ 0\} \) | Proof Let \( x \in \mathcal{N}\left( {I - V}\right) \) . For \( v \in \mathcal{D}\left( V\right) \), we have\n\n\[ \langle \left( {I - V}\right) v, x\rangle = \langle v, x\rangle - \langle {Vv}, x\rangle = \langle v, x\rangle - \langle {Vv},{Vx}\rangle = \langle v, x\rangle - \langle v, x\rangle = 0. \]\n\nHence, \( x ... | Yes |
Proposition 13.4 \( {T}_{V} \) is a densely defined symmetric operator which has the Cayley transform \( V \) . | Proof Let \( x \in \mathcal{D}\left( {T}_{V}\right) \) . Then \( x = \left( {I - V}\right) y \) for some \( y \in \mathcal{D}\left( V\right) \) . Using the assumption that \( V \) is isometric, we compute\n\n\[ \left\langle {{T}_{V}x, x}\right\rangle = \left\langle {{T}_{V}\left( {I - V}\right) y,\left( {I - V}\right) ... | Yes |
Corollary 13.6 A densely defined symmetric operator \( T \) is self-adjoint if and only if its Cayley transform \( {V}_{T} \) is unitary. | Proof By Proposition 3.12, the symmetric operator \( T \) is self-adjoint if and only if \( \mathcal{R}\left( {T - \bar{\lambda }I}\right) = \mathcal{H} \) and \( \mathcal{R}\left( {I - {\lambda I}}\right) = \mathcal{H} \) . Since \( \mathcal{D}\left( {V}_{T}\right) = \mathcal{R}\left( {T - \bar{\lambda }I}\right) \) a... | Yes |
Corollary 13.7 A unitary operator \( V \) is the Cayley transform of a self-adjoint operator if and only if \( \mathcal{N}\left( {I - V}\right) = \{ 0\} \) . | Proof By Theorem 13.5 and Corollary 13.6, \( V \) is the Cayley transform of a selfadjoint operator if and only if \( \mathcal{R}\left( {I - V}\right) \) is dense. Since \( V \) is unitary, it is easily checked that the latter is equivalent to the relation \( \mathcal{N}\left( {I - V}\right) = \{ 0\} \) . | Yes |
Corollary 13.8 If one of the deficiency indices \( {d}_{ + }\left( T\right) \) or \( {d}_{ - }\left( T\right) \) of a densely defined symmetric operator \( T \) on \( \mathcal{H} \) is finite, then each symmetric operator \( S \) on \( \mathcal{H} \) satisfying \( S \supseteq \bar{T} \) is closed. | Proof We assume without loss of generality that \( {d}_{ + }\left( T\right) < \infty \) (otherwise, we replace \( T \) by \( - T \) and use the relation \( {d}_{ \pm }\left( T\right) = {d}_{ \mp }\left( {-T}\right) \) ). By Proposition 13.3(iv), the Cayley transform \( {V}_{S} \) of a symmetric extension \( S \) of \( ... | Yes |
Theorem 13.9 Let \( T \) be a densely defined symmetric operator on \( \mathcal{H} \). Suppose that \( {\mathcal{G}}_{ + } \subseteq \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) and \( {\mathcal{G}}_{ - } \subseteq \mathcal{N}\left( {{T}^{ * } - \bar{\lambda }I}\right) \) are closed linear subspaces of \( \mat... | Proof Since any closed extension of \( T \) is an extension of \( \bar{T} \) and \( {\left( \bar{T}\right) }^{ * } = {T}^{ * } \), we can assume that \( T \) is closed. Then, by Proposition 13.3, \( \mathcal{D}\left( {V}_{T}\right) \) is closed and closed symmetric extensions of \( T \) are in one-to-one correspondence... | Yes |
Theorem 13.10 A densely defined symmetric operator \( T \) on \( \mathcal{H} \) possesses a selfadjoint extension on \( \mathcal{H} \) if and only if \( {d}_{ + }\left( T\right) = {d}_{ - }\left( T\right) \) . | Proof By Corollary 13.6, an operator \( {T}_{U} \) from Theorem 13.9 is self-adjoint if and only if its Cayley transform is unitary, or equivalently, if \( {\mathcal{G}}_{ + } = \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) and \( {\mathcal{G}}_{ - } = \mathcal{N}\left( {{T}^{ * } - \bar{\lambda }I}\right) \) i... | Yes |
Corollary 13.11 Each densely defined symmetric operator \( T \) on \( \mathcal{H} \) has a selfadjoint extension acting on a possibly larger Hilbert space. | Proof Define a symmetric operator \( S \mathrel{\text{:=}} T \oplus \left( {-T}\right) \) on the Hilbert space \( \mathcal{K} \mathrel{\text{:=}} \) \( \mathcal{H} \oplus \mathcal{H} \) . Since \( {S}^{ * } = {T}^{ * } \oplus \left( {-{T}^{ * }}\right) \), we have \( \mathcal{N}\left( {{S}^{ * } \pm \mathrm{i}I}\right)... | Yes |
Example 13.1 (Examples 1.4 and 1.5 continued) Let us abbreviate \( {\mathcal{N}}_{\pm \mathrm{i}} \mathrel{\text{:=}} \) \( \mathcal{N}\left( {{T}^{ * } \mp \mathrm{i}I}\right) \) and recall from Example 3.2 that \( {\mathcal{N}}_{\mathrm{i}} = \mathbb{C} \cdot {e}^{-x} \) and \( {\mathcal{N}}_{-\mathrm{i}} = \mathbb{C... | Since \( {e}^{a + b - x} \in {\mathcal{N}}_{\mathrm{i}} \) and \( {e}^{x} \in {\mathcal{N}}_{-\mathrm{i}} \) have equal norms in \( {L}^{2}\left( {a, b}\right) \), the isometric mappings of \( {\mathcal{N}}_{\mathrm{i}} \) onto \( {\mathcal{N}}_{-\mathrm{i}} \) are parameterized by \( w \in \mathbb{T} \) and determined... | Yes |
Theorem 13.12 (Ando-Nishio theorem) A positive symmetric operator \( T \) on \( \mathcal{H} \) admits a positive self-adjoint extension on \( \mathcal{H} \) if and only if \( \mathcal{E}\left( T\right) \) is dense in \( \mathcal{H} \) . If this is true, there exists a unique smallest (according to Definition 10.5) amon... | \[ \mathcal{D}\left\lbrack {T}_{N}\right\rbrack = \mathcal{D}\left( {T}_{N}^{1/2}\right) = \mathcal{E}\left( T\right) \] \[ {T}_{N}\left\lbrack y\right\rbrack = {\begin{Vmatrix}{T}_{N}^{1/2}y\end{Vmatrix}}^{2} = {v}_{T}\left( y\right) ,\;y \in \mathcal{E}\left( T\right) . \] | Yes |
Lemma 13.13 \( \mathcal{D}\left( A\right) \subseteq \mathcal{E}\left( T\right) \) for any positive symmetric extension \( A \) of \( T \) . | Proof By the Cauchy-Schwarz inequality, for \( x \in \mathcal{D}\left( T\right) \) and \( y \in \mathcal{D}\left( A\right) \) ,\n\n\[ \n{\left| \langle Tx, y\rangle \right| }^{2} = {\left| \langle Ax, y\rangle \right| }^{2} \leq \langle {Ax}, x\rangle \langle {Ay}, y\rangle = \langle {Ay}, y\rangle \langle {Tx}, x\rang... | Yes |
Proposition 13.14 The Friedrichs extension \( {T}_{F} \) of the densely defined positive symmetric operator \( T \) and its form are given by\n\n\[ \n{T}_{F} = {Q}^{ * }\bar{Q} \equiv {Q}^{ * }{Q}^{* * }\;\text{ and }\;{T}_{F}\left\lbrack {x, y}\right\rbrack = \langle \bar{Q}x,\bar{Q}y{\rangle }^{\prime }\;\text{ for }... | Proof Since \( \mathcal{D}\left( T\right) = \mathcal{D}\left( Q\right) \) is dense, \( \bar{Q} \) is a densely defined closed operator of \( \mathcal{H} \) into \( {\mathcal{K}}_{T} \) . Let \( \mathfrak{t} \) be the corresponding closed form defined in Example 10.5, that is,\n\n\[ \n\mathrm{t}\left\lbrack {x, y}\right... | Yes |
Let \( T \) be a densely defined positive symmetric operator on \( \mathcal{H} \). Then the Friedrichs extension \( {T}_{F} \) is the largest, and the Krein-von Neumann extension \( {T}_{N} \) is the smallest among all positive self-adjoint extensions of \( T \) on \( \mathcal{H} \). That is, \( {T}_{N} \leq A \leq {T}... | Proof Since \( \mathcal{E}\left( T\right) \supseteq \mathcal{D}\left( T\right) \) by Lemma 13.13, \( \mathcal{E}\left( T\right) \) is dense in \( \mathcal{H} \), so by Theorem 13.12 the Krein-von Neumann extension \( {T}_{N} \) exists and is the smallest positive self-adjoint extension. By Theorem 10.17(ii), the Friedr... | Yes |
Let \( S \) be a given positive self-adjoint operator on \( \mathcal{H} \). Then \( T \) has a positive self-adjoint extension \( A \) on \( \mathcal{H} \) satisfying \( A \leq S \) if and only if\n\n\[{\left| \langle Tx, y\rangle \right| }^{2} \leq \langle {Tx}, x\rangle \langle {Sy}, y\rangle \;\text{ for all }x \in ... | Proof First, we assume that (13.18) holds. Then \( \mathcal{D}\left( S\right) \subseteq \mathcal{E}\left( T\right) \), so \( \mathcal{E}\left( T\right) \) is dense and the Krein-von Neumann extension \( {T}_{N} \) exists by Theorem 13.12. Using (13.14), (13.12), and (13.18), we conclude that \( {T}_{N}\left\lbrack y\ri... | Yes |
Proposition 13.18 Suppose that \( \mathcal{D}\left( T\right) \) is dense in \( \mathcal{H} \) and \( \mathcal{N}\left( T\right) = \{ 0\} \) . Then \( {T}^{-1} \) has a positive self-adjoint extension on \( \mathcal{H} \) if and only if \( \mathcal{N}\left( {T}_{F}\right) = \{ 0\} \) . If this is fulfilled, then \( {\le... | Proof Obviously, \( {T}^{-1} \) is also positive and symmetric. Suppose that \( S \) is a positive self-adjoint extension of \( {T}^{-1} \) . Since \( \mathcal{R}\left( S\right) \) contains the dense set \( \mathcal{D}\left( T\right) \), we have \( \mathcal{N}\left( S\right) = \mathcal{R}{\left( S\right) }^{ \bot } = \... | Yes |
Proposition 13.19 Suppose that \( \mathcal{E}\left( T\right) \) is dense in \( \mathcal{H} \) . Then we have \( \mathcal{N}\left( {T}_{N}\right) = \{ 0\} \) if and only if \( \mathcal{R}\left( T\right) \) is dense in \( \mathcal{H} \) . If this is true, then \( {T}^{-1} \) is a densely defined positive symmetric operat... | Proof Since \( {T}_{N} = {J}^{* * }{J}^{ * } \), we have \( \mathcal{N}\left( {T}_{N}\right) = \mathcal{N}\left( {J}^{ * }\right) = \mathcal{R}{\left( J\right) }^{ \bot } = \mathcal{R}{\left( T\right) }^{ \bot } \) . Therefore, \( \mathcal{N}\left( {T}_{N}\right) \) is trivial if and only if \( \mathcal{R}\left( T\righ... | Yes |
Lemma 13.20 Let \( A \) be a positive self-adjoint operator on a Hilbert space \( \mathcal{H} \) such that \( \mathcal{N}\left( A\right) = \{ 0\} \) . A vector \( y \in \mathcal{H} \) belongs to \( \mathcal{D}\left( {A}^{-1/2}\right) \) if and only if \[ {\varphi }_{A}\left( y\right) \mathrel{\text{:=}} \mathop{\sup }\... | Proof Throughout this proof all suprema are over \( x \in \mathcal{D}\left( {A}^{1/2}\right), x \neq 0 \) . First, suppose that \( y \in \mathcal{D}\left( {A}^{-1/2}\right) \) . Since \( \mathcal{N}\left( A\right) = \{ 0\} \), we have \( \mathcal{N}\left( {A}^{1/2}\right) = \{ 0\} \) . Therefore, \( \mathcal{R}\left( {... | Yes |
The self-adjoint operator \( S \) is positive if and only if\n\n\[ \n{B}^{ * }{x}_{2} \in \mathcal{D}\left( {A}^{-1/2}\right) \;\text{ and }\;{\begin{Vmatrix}{A}^{-1/2}{B}^{ * }{x}_{2}\end{Vmatrix}}^{2} \leq \left\langle {C{x}_{2},{x}_{2}}\right\rangle \;\text{ for }{x}_{2} \in \mathcal{D}\left( C\right) .\n\] | Proof Clearly, \( S \) is positive if and only if\n\n\[ \n\left\langle {S\left( {{x}_{1} + \lambda {x}_{2}}\right) ,{x}_{1} + \lambda {x}_{2}}\right\rangle = \left\langle {A{x}_{1},{x}_{1}}\right\rangle + 2\operatorname{Re}\lambda \left\langle {{B}^{ * }{x}_{2},{x}_{1}}\right\rangle + {\left| \lambda \right| }^{2}\left... | Yes |
Proposition 13.21 \( {C}_{m} \mathrel{\text{:=}} {C}_{ + } - \parallel B\parallel \cdot I \) is the smallest, and \( {C}_{M} \mathrel{\text{:=}} \parallel B\parallel \cdot I - {C}_{ - } \) is the largest among all bounded self-adjoint extensions of \( B \) on \( \mathcal{H} \) which have the same norm as \( B \) . | Proof By construction, \( {C}_{m} \supseteq {B}_{ + } - \parallel B\parallel \cdot I = B \) and \( {C}_{M} \supseteq \parallel B\parallel \cdot I - {B}_{ - } = B \) . Clearly, the relations \( 0 \leq {C}_{ \pm } \leq 2\parallel B\parallel \cdot I \) imply that \( - \parallel B\parallel \cdot I \leq {C}_{m} \leq \parall... | Yes |
Proposition 13.23 For \( S,{S}_{1},{S}_{2} \in \mathcal{P}\left( \mathcal{H}\right) \) we have:\n\n(i) \( S \) is self-adjoint if and only if \( \mathcal{D}\left( {B}_{S}\right) = \mathcal{H} \), that is, if \( {B}_{S} \) is self-adjoint. | Proof (i): Since \( S \geq 0 \), Proposition 3.2(i) implies that -1 is in \( \pi \left( S\right) \) . Hence, \( S \) is self-adjoint if and only if \( \mathcal{R}\left( {S + I}\right) \equiv \mathcal{D}\left( {B}_{S}\right) \) is \( \mathcal{H} \) by Proposition 3.11. | No |
Theorem 13.24 Let \( S \) be a densely defined positive symmetric operator on \( \mathcal{H} \). If \( C \) is a bounded self-adjoint extension of the Krein transform \( {B}_{S} \) such that \( \parallel C\parallel \leq 1 \), then \( C \in {\mathcal{S}}_{1}\left( \mathcal{H}\right) \), and the inverse Krein transform \... | Proof Since \( C \) is symmetric, \( \mathcal{N}\left( {I - C}\right) \bot \mathcal{R}\left( {I - C}\right) \). Therefore, since \( \mathcal{D}\left( S\right) \) is dense and \( \mathcal{D}\left( S\right) = \mathcal{R}\left( {I - {B}_{S}}\right) \subseteq \mathcal{R}\left( {I - C}\right) \), it follows that \( \mathcal... | Yes |
The standard example of a conjugation \( J \) is the complex conjugation of functions on a Hilbert space \( {L}^{2}\left( {X,\mu }\right) \), that is, \( \left( {Jf}\right) \left( t\right) = \overline{f\left( t\right) } \) . | Obviously, a multiplication operator \( {M}_{\varphi } \) is \( J \) -real if and only if \( \varphi \left( t\right) \) is real \( \mu \) -a.e. on \( X \) . | Yes |
Proposition 13.25 Let \( J \) be a conjugation, and let \( T \) be a densely defined symmetric operator on a Hilbert space \( \mathcal{H} \) . Suppose that \( T \) is \( J \) -real. Then we have:\n\n(i) \( {T}^{ * } \) is \( J \) -real. | Proof In this proof we abbreviate \( {\mathcal{N}}_{\lambda } \mathrel{\text{:=}} \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) and \( {\mathcal{N}}_{\bar{\lambda }} \mathrel{\text{:=}} \mathcal{N}\left( {{T}^{ * } - \bar{\lambda }I}\right) \) .\n\n(i): For \( u, v \in \mathcal{H} \), it follows from (13.20) th... | Yes |
Let \( J \) be the conjugation on \( \mathcal{H} = {L}^{2}\left( {-a, a}\right), a > 0 \), given by \( \left( {Jf}\right) \left( x\right) = \overline{f\left( {-x}\right) } \). Then the symmetric operator \( T = - \mathrm{i}\frac{d}{dx} \) with domain \( \mathcal{D}\left( T\right) = {H}_{0}^{1}\left( {-a, a}\right) \) i... | Indeed, if \( f \in \mathcal{D}\left( T\right) \), then \( {Jf} \in \mathcal{D}\left( T\right) \), and\n\n\[ \left( {JTf}\right) \left( x\right) = \overline{\left( {Tf}\right) \left( {-x}\right) } = \mathrm{i}\overline{\frac{d}{dx}f\left( {-x}\right) } = - \mathrm{i}\overline{{f}^{\prime }\left( {-x}\right) } = \left( ... | Yes |
Proposition 14.1 Let \( T \) be a closed relation on \( \mathcal{H} \). A complex number \( \lambda \) is in \( \rho \left( T\right) \) if and only if \( \mathcal{N}\left( {T - {\lambda I}}\right) = \{ 0\} \) and \( \mathcal{R}\left( {T - {\lambda I}}\right) = \mathcal{H} \). | Proof This proof is based on two simple identities which are easily verified:\n\n\[ \mathcal{N}\left( {T - {\lambda I}}\right) = \mathcal{M}\left( {\left( T - \lambda I\right) }^{-1}\right) ,\;\mathcal{R}\left( {T - {\lambda I}}\right) = \mathcal{D}\left( {\left( T - \lambda I\right) }^{-1}\right) . \]\n\n(14.4)\n\nLet... | Yes |
Proposition 14.2 There is a one-to-one correspondence between the sets of operators \( B \in \mathcal{S}\left( \mathcal{H}\right) \) and of self-adjoint relations \( \mathcal{B} \) on \( \mathcal{H} \) given by \[ \mathcal{B} = \mathcal{G}\left( B\right) \oplus \left( {\{ 0\} \oplus {\left( {\mathcal{H}}_{B}\right) }^{... | Proof It is easily verified that (14.5) defines a self-adjoint relation \( \mathcal{B} \) for \( B \in \mathcal{S}\left( \mathcal{H}\right) \) . Conversely, suppose that \( \mathcal{B} \) is a self-adjoint relation on \( \mathcal{H} \) . Let \( B \mathrel{\text{:=}} {\mathcal{B}}_{s} \) and \( {\mathcal{H}}_{B} \mathre... | Yes |
Lemma 14.3 Let \( \mathcal{B} \) be a linear relation on \( \mathcal{H} \) such that \( {\mathcal{B}}^{-1} \) is the graph of a positive self-adjoint operator \( A \) on \( \mathcal{H} \). Then \( \mathcal{B} \) is a positive self-adjoint relation. | Proof Since \( \mathcal{N}\left( A\right) \) is a reducing subspace for \( A \), we can write \( A = C \oplus 0 \) on \( \mathcal{H} = \) \( {\mathcal{H}}_{B} \oplus \mathcal{N}\left( A\right) \), where \( C \) is a positive self-adjoint operator with trivial kernel on \( {\mathcal{H}}_{B} \mathrel{\text{:=}} \) \( \ma... | Yes |
Proposition 14.4 A linear relation \( \mathcal{B} \) on \( \mathcal{H} \) is self-adjoint if and only if there is a unitary operator \( V \) on \( \mathcal{H} \) such that\n\n\[ \mathcal{B} = \{ \left( {x, y}\right) \in \mathcal{H} \oplus \mathcal{H} : \left( {I - V}\right) y = \mathrm{i}\left( {I + V}\right) x\} . \] | Proof Suppose that \( \mathcal{B} \) is self-adjoint. Let \( B \in \mathcal{S}\left( \mathcal{H}\right) \) be the corresponding selfadjoint operator from Lemma 14.2. By Corollary 13.6, the Cayley transform \( {V}_{B} = \) \( \left( {B - \mathrm{i}I}\right) {\left( B + \mathrm{i}I\right) }^{-1} \) is a unitary operator ... | Yes |
Let \( a, b \in \mathbb{R}, a < b \), and let \( T = - \mathrm{i}\frac{d}{dx} \) be the symmetric operator on \( \mathcal{D}\left( T\right) = {H}_{0}^{1}\left( {a, b}\right) \) in \( {L}^{2}\left( {a, b}\right) \). By integration by parts (see (1.13)) we have\n\n\[ \mathrm{i}{\left\lbrack f, g\right\rbrack }_{{T}^{ * }... | Therefore, the triplet\n\n\[ \mathcal{K} = \mathbb{C},\;{\Gamma }_{ + }\left( f\right) = \sqrt{2}f\left( a\right) ,\;{\Gamma }_{ - }\left( f\right) = \sqrt{2}f\left( b\right) \]\n\nis a boundary triplet for \( {T}^{ * } \) . Indeed,(14.10) implies (14.8). The surjectivity condition (ii) in Definition 14.2 is obviously ... | Yes |
Let \( a, b \in \mathbb{R}, a < b \) . For the symmetric operator \( T = - \frac{{d}^{2}}{d{x}^{2}} \) on \( \mathcal{D}\left( T\right) = {H}_{0}^{2}\left( {a, b}\right) \) in \( {L}^{2}\left( {a, b}\right) \), integration by parts yields (see (1.14)) | \[ {\left\lbrack f, g\right\rbrack }_{{T}^{ * }} = f\left( b\right) \overline{{g}^{\prime }\left( b\right) } - {f}^{\prime }\left( b\right) \overline{g\left( b\right) } - f\left( a\right) \overline{{g}^{\prime }\left( a\right) } + {f}^{\prime }\left( a\right) \overline{g\left( a\right) }. \] Hence, there is a boundary ... | Yes |
Proposition 14.5 There exists a boundary triplet \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) for \( {T}^{ * } \) if and only if the symmetric operator \( T \) has equal deficiency indices. We then have \( {d}_{ + }\left( T\right) = \) \( {d}_{ - }\left( T\right) = \dim \mathcal{K} \) . | Proof If \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) is a boundary triplet for \( {T}^{ * } \), by Lemma 14.13(ii) below \( {\Gamma }_{0} \) is a topological isomorphism of \( {\mathcal{N}}_{\pm \mathrm{i}} \) onto \( \mathcal{K} \) . Thus, \( {d}_{ + }\left( T\right) = {d}_{ - }\left( T\right) = \dim... | Yes |
Lemma 14.6 Let \( \mathcal{B} \) be a linear relation on \( \mathcal{K} \), and let \( S \) be a linear operator on \( \mathcal{H} \) such that \( T \subseteq S \subseteq {T}^{ * } \) and \( \mathcal{B}\left( S\right) = \mathcal{B} \) . Then:\n\n(i) \( {S}^{ * } = {T}_{{\mathcal{B}}^{ * }} \) . | Proof (i): First note that \( T \subseteq S \subseteq {T}^{ * } \) implies that \( \bar{T} = {T}^{* * } \subseteq {S}^{ * } \subseteq {T}^{ * } \) . Hence, a vector \( y \in \mathcal{D}\left( {T}^{ * }\right) \) belongs to \( \mathcal{D}\left( {S}^{ * }\right) \) if and only if for all \( x \in \mathcal{D}\left( S\righ... | Yes |
Proposition 14.7 There is a one-to-one correspondence between all closed linear relations \( \mathcal{B} \) on \( \mathcal{K} \) and all proper extensions \( S \) of \( T \) given by \( \mathcal{B} \leftrightarrow {T}_{\mathcal{B}} \) . Furthermore, if \( \mathcal{B},{\mathcal{B}}_{0} \), and \( {\mathcal{B}}_{1} \) ar... | Proof All assertions are easily derived from Lemma 14.6. If \( \mathcal{B} \) is a closed relation, then \( {T}_{\mathcal{B}} \) is closed by Lemma 14.6(ii) applied to \( S = {T}_{\mathcal{B}} \) . If \( S \) is a closed operator and \( T \subseteq S \subseteq {T}^{ * } \), then \( \mathcal{B}\left( S\right) \) is clos... | Yes |
Corollary 14.8 If \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) is a boundary triplet for \( {T}^{ * } \), then there exist self-adjoint extensions \( {T}_{0} \) and \( {T}_{1} \) of the symmetric operator \( T \) on \( \mathcal{H} \) defined by \( \mathcal{D}\left( {T}_{0}\right) = \mathcal{N}\left( {\... | Proof Clearly, \( {\mathcal{B}}_{0} = \{ 0\} \oplus \mathcal{K} \) and \( {\mathcal{B}}_{1} = \mathcal{K} \oplus \{ 0\} \) are self-adjoint relations. Hence, \( {T}_{0} = {T}_{{\mathcal{B}}_{0}} \) and \( {T}_{1} = {T}_{{\mathcal{B}}_{1}} \) are self-adjoint operators by Proposition 14.7(v). | Yes |
Corollary 14.9 Let \( \mathcal{B} \) be a closed relation on \( \mathcal{K} \). The operators \( {T}_{\mathcal{B}} \) and \( {T}_{0} \) are disjoint if and only if \( \mathcal{B} \) is the graph of an operator on the Hilbert space \( \mathcal{K} \). | Proof By Proposition 14.7(ii), \( {T}_{\mathcal{B}} \) and \( {T}_{0} \equiv {T}_{{\mathcal{B}}_{0}} \) are disjoint if and only if \( \mathcal{B} \cap {\mathcal{B}}_{0} = \) \( \{ \left( {0,0}\right) \} \), where \( {\mathcal{B}}_{0} = \{ 0\} \oplus \mathcal{K} \), or equivalently, if \( \mathcal{B} \) is the graph of... | Yes |
Theorem 14.10 Suppose that \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) is a boundary triplet for \( {T}^{ * } \) . For any operator \( S \) on \( \mathcal{H} \), the following are equivalent:\n\n(i) \( S \) is a self-adjoint extension of \( T \) on \( \mathcal{H} \) .\n\n(ii) There is a self-adjoint l... | Proof The equivalence of (i) and (ii) is already contained in Proposition 14.7. Recall that \( \mathcal{B} \) is uniquely determined by the operator \( {T}_{\mathcal{B}} \), since \( \mathcal{B}\left( {T}_{\mathcal{B}}\right) = \mathcal{B} \) .\n\n(ii) \( \rightarrow \) (iii): Let \( \mathcal{B} \) be a self-adjoint re... | Yes |
We apply Theorem 14.10 to the boundary triplet from Example 14.1. It follows that all self-adjoint extensions of the operator \( T = - \mathrm{i}\frac{d}{dx} \) on \( \mathcal{D}\left( T\right) = {H}_{0}^{1}\left( {a, b}\right) \) are the operators \( {T}^{z} = - \mathrm{i}\frac{d}{dx} \) with domains \( \mathcal{D}\le... | This fact has been derived in Example 13.1 by using the Cayley transform. Note that \( {T}^{z} \) is just the operator \( {S}_{z} \) from Example 1.5. | Yes |
Statement \( \left( {\mathcal{K},{\Gamma }_{ + },{\Gamma }_{ - }}\right) \) is a boundary triplet for \( {T}^{ * } \) . | Proof Let \( x = {x}_{0} + {x}_{ + } + {x}_{ - } \) and \( y = {y}_{0} + {y}_{ + } + {y}_{ - } \) be vectors of \( \mathcal{D}\left( {T}^{ * }\right) \), where \( {x}_{0},{y}_{0} \in \mathcal{D}\left( \bar{T}\right) \) and \( {x}_{ \pm },{y}_{ \pm } \in {\mathcal{N}}_{\pm \mathrm{i}} \) . A straightforward simple compu... | Yes |
Proposition 14.11 If \( A \) is a self-adjoint extension of \( T \) on \( \mathcal{H} \) and \( \mu \in \rho \left( A\right) \), then\n\n\[ \mathcal{D}\left( {T}^{ * }\right) = \mathcal{D}\left( \bar{T}\right) \dot{ + }{\left( A - \mu I\right) }^{-1}{\mathcal{N}}_{\bar{\mu }}\dot{ + }{\mathcal{N}}_{\mu }, \] | Proof The proof of (14.17) is similar to the proof of formula (3.10) in Proposition 3.7. Since \( \mu \in \rho \left( A\right) \), we have \( \mu \in \pi \left( T\right) \) . Hence, Corollary 2.2 applies and yields\n\n\[ \mathcal{H} = \mathcal{R}\left( {\bar{T} - {\mu I}}\right) \oplus {\mathcal{N}}_{\bar{\mu }},\;\tex... | Yes |
Statement \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) is a boundary triplet for \( {T}^{ * } \) . | In this proof let \( \left( {\mathcal{K},{\Gamma }_{0}^{\prime },{\Gamma }_{1}^{\prime }}\right) \) denote the triplet from Example 14.5. Since\n\n\[ A{\left( A - \mu I\right) }^{-1}{x}_{0} + {\left( A - \mu I\right) }^{-1}{x}_{1} = {x}_{0} + {\left( A - \mu I\right) }^{-1}\left( {\mu {x}_{0} + {x}_{1}}\right) ,\]\n\nw... | Yes |
Example 14.7 \( \left( {d = 1}\right) \) First, let \( n = 1 \) . A Hermitian \( 1 \times 1 \) matrix is just a real number \( B \), and the operator \( {T}_{B} \) is determined by the boundary condition \( B{\Gamma }_{0}\left( x\right) = \) \( {\Gamma }_{1}\left( x\right) \) . If \( n = 0 \), then \( B \) acts on the ... | \[ B{\Gamma }_{0}\left( x\right) = {\Gamma }_{1}\left( x\right) ,\;B \in \mathbb{R} \cup \{ \infty \} . \] Writing \( B \) as \( B = \cot \alpha \), we obtain another convenient parameterization of selfadjoint extensions of \( T \) which will be used in Chap. 15. It is given by \[ {\Gamma }_{0}\left( x\right) \cos \alp... | Yes |
Example 14.8 \( \left( {d = 2}\right) \) Let \( B \in \mathcal{S}\left( \mathcal{K}\right) \) . The operator \( B \) can act on a subspace of dimension 2,1, or 0 of the Hilbert space \( \mathcal{K} = {\mathbb{C}}^{2} \) . That is, we have the following three possible cases. | Case \( 1\left( {{\mathcal{K}}_{B} = \mathcal{K}}\right) \) Then \( B \) corresponds to a hermitian \( 2 \times 2 \) matrix, so the relation \( {P}_{B}{\Gamma }_{1}\left( x\right) = B{\Gamma }_{0}\left( x\right) \) says\n\n\[ \n{\psi }_{1}\left( x\right) = {b}_{1}{\varphi }_{1}\left( x\right) + c{\varphi }_{2}\left( x\... | Yes |
Let us consider the symmetric operator \( T = - \frac{{d}^{2}}{d{x}^{2}} \) with domain \( \mathcal{D}\left( T\right) = {H}_{0}^{2}\left( {0,\infty }\right) \) on \( {L}^{2}\left( {0,\infty }\right) \) . Integration by parts yields\n\n\[ \n{\left\lbrack f, g\right\rbrack }_{{T}^{ * }} = - f\left( 0\right) \overline{{g}... | Then the above assumptions are satisfied with \( d = 1,\varphi \left( f\right) = f\left( 0\right) ,\psi \left( f\right) = {f}^{\prime }\left( 0\right) \) , so there is a boundary triplet for the operator \( {T}^{ * } \) given by\n\n\[ \n\mathcal{K} = \mathbb{C},\;{\Gamma }_{0}\left( f\right) = f\left( 0\right) ,\;{\Gam... | Yes |
Proposition 14.14 For \( z, w \in \rho \left( {T}_{0}\right) \), we have:\n\n(i) \( \gamma {\left( \bar{z}\right) }^{ * } = {\Gamma }_{1}{\left( {T}_{0} - zI\right) }^{-1} \) . | Proof (i): Let \( x \in \mathcal{H} \) . Set \( y = {\left( {T}_{0} - zI\right) }^{-1}x \) . Let \( v \in \mathcal{K} \) . Using the facts that \( {T}^{ * }\gamma \left( \bar{z}\right) v = \bar{z}\gamma \left( \bar{z}\right) v,{\Gamma }_{0}y = 0 \), and \( {\Gamma }_{0}\gamma \left( \bar{z}\right) v = v \) and Definiti... | Yes |
Proposition 14.15 For arbitrary \( z, w \in \rho \left( {T}_{0}\right) \), we have:\n\n(i) \( M\left( z\right) {\Gamma }_{0}u = {\Gamma }_{1}u \) for \( u \in {\mathcal{N}}_{z} \).\n\n(ii) \( M{\left( z\right) }^{ * } = M\left( \bar{z}\right) \).\n\n(iii) \( M\left( w\right) - M\left( z\right) = \left( {w - z}\right) \... | Proof (i): Since \( \gamma \left( z\right) = {\left( {\Gamma }_{0} \upharpoonright {\mathcal{N}}_{z}\right) }^{-1} \), we get \( M\left( z\right) {\Gamma }_{0}u = {\Gamma }_{1}\gamma \left( z\right) {\Gamma }_{0}u = {\Gamma }_{1}u \).\n\n(ii): Let \( u \in {\mathcal{N}}_{z} \) and \( {u}^{\prime } \in {\mathcal{N}}_{\b... | Yes |
Corollary 14.16 The Weyl function \( M\left( z\right) \) is a Nevanlinna function on \( \mathcal{K} \) . | Proof By Proposition 14.15(iv), \( M\left( z\right) \) is a \( \mathbf{B}\left( \mathcal{K}\right) \) -valued holomorphic function on \( {\mathbb{C}}_{ + } \) . Let \( z \in {\mathbb{C}}_{ + } \) and \( y = \operatorname{Im}z \) . From Proposition 14.15,(ii) and (iii), we obtain\n\n\[ M\left( z\right) - M{\left( z\righ... | Yes |
Theorem 14.18 Let \( T \) be a densely defined symmetric operator on \( \mathcal{H} \), and \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) a boundary triplet for \( {T}^{ * } \) . Suppose that \( \mathcal{B} \) is a closed relation on \( \mathcal{K} \) and \( z \in \rho \left( {T}_{0}\right) \) . Then th... | Proof Let \( f \in \mathcal{D}\left( {T}_{\mathcal{B}}\right) \) . Since \( f \in \mathcal{D}\left( {T}^{ * }\right) \), by (14.19) there are vectors \( {x}_{z} \in \mathcal{D}\left( {T}_{0}\right) \) and \( {u}_{z} \in {\mathcal{N}}_{z} \) such that \( f = {x}_{z} + {u}_{z} \) . Writing \( {y}_{z} \mathrel{\text{:=}} ... | Yes |
Recall from Example 14.5 that \( A = {T}_{0} \) and \( \mu \in \rho \left( A\right) \) . Let \( v \in \mathcal{N}\bar{\mu } = \mathcal{K} \) . By the definition of the boundary maps the equality \( v = A{\left( A - \mu I\right) }^{-1}v + {\left( A - \mu I\right) }^{-1}\left( {-{\mu v}}\right) \) yields \( {\Gamma }_{0}... | \[ \gamma \left( z\right) = \left( {A - \bar{\mu }I}\right) {\left( A - zI\right) }^{-1} \upharpoonright \mathcal{K},\;z \in \rho \left( A\right) . \] | Yes |
Example 14.12 (Second standard Example 14.6 continued) Let us recall from Example 14.6 that \( A = {T}_{0},\mu \in \rho \left( A\right) \) is real, and \( \mathcal{K} = {\mathcal{N}}_{\mu } \) . | From the definitions of the boundary maps we get \( {\Gamma }_{0}v = v \) and \( {\Gamma }_{1}v = 0 \) for \( v \in \mathcal{K} \) . Therefore, \( \gamma \left( \mu \right) v = v \) and \( M\left( \mu \right) v = {\Gamma }_{1}\gamma \left( \mu \right) v = 0 \), that is, \( \gamma \left( \mu \right) = {I}_{\mathcal{H}} ... | Yes |
Lemma 14.19 \( {M}_{A,\mathcal{K}}\left( z\right) \) is an operator-valued Nevanlinna function. | Proof Clearly, \( {M}_{A,\mathcal{K}}\left( z\right) \in \mathbf{B}\left( \mathcal{K}\right) \) is holomorphic on \( {\mathbb{C}}_{ + } \) . Let \( z \in {\mathbb{C}}_{ + } \) . Setting \( x = \operatorname{Re}z \) and \( y = \operatorname{Im}z \), a straightforward computation shows that\n\n\[\n\operatorname{Im}{M}_{A... | Yes |
Theorem 14.20 Let \( S \) be a densely defined symmetric operator on \( \mathcal{H} \), and let \( \widetilde{A} \) and \( A \) be self-adjoint extensions of \( S \) on \( \mathcal{H} \). Then there exists a closed linear subspace \( \mathcal{K} \) of \( \mathcal{N}\left( {{S}^{ * } - \mathrm{i}I}\right) \) and a self-... | Proof Clearly, the operator \( T \mathrel{\text{:=}} A \upharpoonright \left( {\mathcal{D}\left( A\right) \cap \mathcal{D}\left( \widetilde{A}\right) }\right) \) is a symmetric extension of \( S \). Hence, \( {T}^{ * } \subseteq {S}^{ * } \) and \( \mathcal{K} \mathrel{\text{:=}} \mathcal{N}\left( {{T}^{ * } - \mathrm{... | Yes |
Proposition 14.21 Suppose that \( {T}_{0} \) is the Friedrichs extension \( {T}_{F} \) of \( T \) . Let \( \mathcal{B} \) be a self-adjoint relation on \( \mathcal{K} \) . If the self-adjoint operator \( {T}_{\mathcal{B}} \) is lower semibounded, so is the relation \( \mathcal{B} \) . More precisely, if \( \lambda < {m... | Proof Fix a number \( {\lambda }^{\prime } < \lambda \) . Since \( {\lambda }^{\prime } < {m}_{T} = {m}_{{T}_{F}} = {m}_{{T}_{0}} \) and \( {\lambda }^{\prime } < {m}_{{T}_{\mathcal{B}}} \), we have \( {\lambda }^{\prime } \in \rho \left( {T}_{\mathcal{B}}\right) \cap \rho \left( {T}_{0}\right) \) . Hence, the resolven... | Yes |
Theorem 14.22 Let \( T \) be a densely defined lower semibounded symmetric operator, and let \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) be a boundary triplet for \( {T}^{ * } \) such that the operator \( {T}_{0} \) is the Friedrichs extension \( {T}_{F} \) of \( T \) . Let \( \lambda \in \mathbb{R},\... | Proof We begin by proving a simple preliminary fact. That is, we show that \[ \left( {T\mathcal{B} - {\lambda I}}\right) \left\lbrack {x,\gamma \left( \lambda \right) u}\right\rbrack = 0 \] (14.58) for \( x \in \mathcal{D}\left\lbrack {T}_{F}\right\rbrack \) and \( \gamma \left( \lambda \right) u \in \mathcal{D}\left\l... | Yes |
Corollary 14.23 Suppose that \( S \) is a lower semibounded self-adjoint extension of the densely defined lower semibounded symmetric operator \( T \) on \( \mathcal{H} \) . Let \( \lambda \in \mathbb{R} \) , \( \lambda < {m}_{T} \), and \( \lambda \leq {m}_{S} \) . Then \( S \) is equal to the Friedrichs extension \( ... | Proof Since then \( \lambda \in \rho \left( {T}_{F}\right) \), Example 14.6, with \( A = {T}_{F},\mu = \lambda \), yields a boundary triplet \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) for \( {T}^{ * } \) such that \( {T}_{0} = {T}_{F} \) . By Propositions 14.7(v) and 14.21, there is a self-adjoint re... | Yes |
The operator \( A = {T}_{0} \) is equal to the Friedrichs extension \( {T}_{F} \) of \( T \) if and only if for all \( u \in \mathcal{K}, u \neq 0 \), we have\n\n\[ \mathop{\lim }\limits_{{t \rightarrow - \infty }}\langle M\left( t\right) u, u\rangle = - \infty \] | Assume without loss of generality that \( \mu = 0 \) . Then \( A \geq 0 \) . By Corollary 14.23, applied with \( S = A \) and \( \lambda = 0 \), it suffices to show that any nonzero vector \( u \in \mathcal{K} = \mathcal{N}\left( {T}^{ * }\right) \) is not in the form domain \( \mathcal{D}\left\lbrack A\right\rbrack \)... | Yes |
Let \( T = - \frac{{d}^{2}}{d{x}^{2}} \) with domain \( \mathcal{D}\left( T\right) = {H}_{0}^{2}\left( {a, b}\right) \) on \( {L}^{2}\left( {a, b}\right), a, b \in \mathbb{R} \). By the Poincaré inequality (10.24) we have \[ \langle {Tf}, f\rangle = {\begin{Vmatrix}{f}^{\prime }\end{Vmatrix}}^{2} \geq {\pi }^{2}{\left(... | Clearly, \( {T}^{ * } = - \frac{{d}^{2}}{d{x}^{2}} \) on \( \mathcal{D}\left( {T}^{ * }\right) = {H}^{2}\left( {a, b}\right) \). Hence, \( \mathcal{N}\left( {T}^{ * }\right) = \mathbb{C} \cdot 1 + \mathbb{C} \cdot x \). Therefore, by (14.67) we have \( \mathcal{D}\left( {T}_{N}\right) = \mathcal{D}\left( T\right) \dotp... | Yes |
Example 14.15 \( \left( {T = - \frac{{d}^{2}}{d{x}^{2}} + {c}^{2}}\right. \) on \( \left. {{H}_{0}^{2}\left( {0,\infty }\right), c > 0}\right) \) Then we have \( \mathcal{N}\left( {T}^{ * }\right) = \) \( \mathbb{C} \cdot {e}^{-{cx}} \) and \( \mathcal{D}\left( {T}_{N}\right) = {H}_{0}^{2}\left( {0,\infty }\right) + \m... | The domain of the Friedrichs extension is \( \mathcal{D}\left( {T}_{F}\right) = \left\{ {f \in {H}^{2}\left( {0,\infty }\right) : f\left( 0\right) = 0}\right\} \) . Since \( h\left( x\right) \mathrel{\text{:=}} {\left( 2c\right) }^{-1}x{e}^{-{cx}} \in \mathcal{D}\left( {T}_{F}\right) \) and \( \left( {{T}_{F}h}\right) ... | Yes |
Let \( f \in {H}^{2}\left( \Omega \right) \). Then \( f \in \mathcal{D}\left( {T}_{N}\right) \) if and only if \( \frac{\partial f}{\partial \nu } \upharpoonright \partial \Omega = \frac{\partial \mathrm{H}\left( f\right) }{\partial \nu } \upharpoonright \partial \Omega \). | Proof Let \( f \in \mathcal{D}\left( {T}_{N}\right) \). By (14.67), \( f \) can be written as \( f = g + h \), where \( g \in \mathcal{D}\left( T\right) \) and \( h \in \mathcal{N}\left( {T}^{ * }\right) \). | No |
Lemma 15.2 If \( f \in \mathcal{D}\left( \bar{T}\right) \) and \( g \in \mathcal{D}\left( {T}^{ * }\right) \), then \( {\left\lbrack f, g\right\rbrack }_{a} = {\left\lbrack f, g\right\rbrack }_{b} = 0 \) . | Proof We carry out the proof of the equality \( {\left\lbrack f, g\right\rbrack }_{a} = 0 \) . Let us choose \( {g}_{0} \in \mathcal{D}\left( {T}^{ * }\right) \) such that \( g = {g}_{0} \) in some neighborhood of \( a \) and \( {g}_{0} = 0 \) in some neighborhood of \( b \) . Then we have \( {\left\lbrack f, g\right\r... | Yes |
Proposition 15.4 The operator \( T \) has deficiency indices \( \left( {0,0}\right) \) or \( \left( {1,1}\right) \) or \( \left( {2,2}\right) \) . | Proof Since \( q \) is real-valued, a function \( f \) belongs to \( \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) if and only if its complex conjugate \( \bar{f} \) is in \( \mathcal{N}\left( {{T}^{ * } - \bar{\lambda }I}\right) \) . Hence, \( T \) has equal deficiency indices. As just noted, the dimension of ... | Yes |
Proposition 15.5 Suppose that the end point \( a \) is regular for \( \mathcal{L} \) .\n\n(i) If \( f \in \mathcal{D}\left( {T}^{ * }\right) \), then \( f \) and \( {f}^{\prime } \) can be extended to continuous functions on \( \lbrack a, b) \) . | Proof Since \( {T}^{ * }f \in {L}^{2}\left( {a, b}\right) \subseteq {L}_{\text{loc }}^{1}\left( {a, b}\right) \), Proposition 15.3(i) applies to \( g \mathrel{\text{:=}} {T}^{ * }f \) and \( \lambda = 0 \) and yields the assertion of (i). | No |
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